The Effects of Number Theory Study on High School Students' Metacognition and Mathematics Attitudes Anthony M. Miele Submitte

The Effects of Number Theory Study on High School Students’ Metacognition and Mathematics Attitudes

Anthony M. Miele

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy under the Executive Committee of the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

2014

ABSTRACT

The Effects of Number Theory Study on High School Students’ Metacognition and Mathematics Attitudes

Anthony M. Miele

The purpose of this study was to determine how the study of number theory might affect high school students’ metacognitive functioning, mathematical curiosity, and/or attitudes towards mathematics.

The study utilized questionnaire and/or interview responses of seven high school students from New York City and 33 high school students from Dalian, China. The questionnaire components served to measure and compare the students’ metacognitive functioning, mathematical curiosity, and mathematics attitudes before and after they worked on a number theory problem set included with the questionnaire. Interviews with 13 of these students also helped to reveal any changes in their metacognitive tendencies and/or mathematics attitudes or curiosity levels after the students had worked on said number theory problems.

The investigator sought to involve very motivated as well as less motivated mathematics students in the study. The participation of a large group of Chinese students enabled the investigator to obtain a diverse set of data elements, and also added an international flavor to the research.

All but one of the 40 participating students described or presented some evidence of metacognitive enhancement, greater mathematical curiosity, and/or improved attitudes towards mathematics after the students had worked on the assigned number theory problems. The results of the study thus have important implications for the value of number theory coursework by high school students, with respect to the students’ metacognitive processes as well as their feelings about mathematics as an academic discipline.

TABLE OF CONTENTS

Page

CHAPTER I – INTRODUCTION ...... 1

Need for the Study ...... 1

Purpose of the Study ...... 2

Procedures of the Study ...... 3

CHAPTER II – LITERATURE REVIEW ...... 7

Attitudes Towards Mathematics ...... 7

Number Theory in High School ...... 16

Metacognition ...... 25

CHAPTER III – METHODOLOGY ...... 37

CHAPTER IV – FINDINGS AND RESULTS ...... 46

Preliminary Trial Summary ...... 46

Preliminary Trial Interviews—Common Theme ...... 48

Preliminary Trial Details...... 48

Student Q1 ...... 48

Student Q2 ...... 51

Student Q3 ...... 53

Main Study Summary ...... 56

Main Study Interviews—Common Themes ...... 58

Main Study Details ...... 59

Student Q4 ...... 59

Student Q5 ...... 61

Page

CHAPTER IV (continued)

Student Q6 ...... 64

Student Q7 ...... 66

Student Q8 ...... 69

Student Q9 ...... 71

Student Q10 ...... 73

Student Q11 ...... 75

Student Q12 ...... 78

Student Q13 ...... 80

Student Q14 ...... 82

Student Q15 ...... 83

Student Q16 ...... 85

Student Q17 ...... 87

Student Q18 ...... 89

Student Q19 ...... 90

Student Q20 ...... 92

Student Q21 ...... 94

Student Q22 ...... 96

Student Q23 ...... 98

Student Q24 ...... 100

Student Q25 ...... 101

Student Q26 ...... 103

Page

CHAPTER IV (continued)

Student Q27 ...... 105

Student Q28 ...... 106

Student Q29 ...... 108

Student Q30 ...... 110

Student Q31 ...... 111

Student Q32 ...... 113

Student Q33 ...... 115

Student Q34 ...... 116

Student Q35 ...... 118

Student Q36 ...... 120

Student Q37 ...... 122

Student Q38 ...... 124

Student Q39 ...... 125

Student Q40 ...... 127

CHAPTER V – SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS ...... 130

Summary ...... 130

Conclusions ...... 131

Recommendations ...... 135

REFERENCES ...... 138

Internet Resources ...... 145

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Page

APPENDICES

Appendix A – Questionnaire Parts A, B, and C ...... 146

Appendix B – Interview Script ...... 153

Appendix C – Certificate of Participation Templates and Consent for Reference Form ...... 156

Appendix D – Institutional Review Board Required Forms ...... 159

Appendix E – Mandarin Translations of the Cover Memo to the High School in China

(with English version), Parts A, B, and C of the Questionnaire, the

Certificate of Participation Templates, and the Institutional Review Board

Required Forms ...... 165

Appendix F – Student Q1 Responses...... 167

Appendix G - Student Q2 Responses...... 176

Appendix H - Student Q3 Responses...... 185

Appendix I - Student Q4 Responses ...... 194

Appendix J - Student Q5 Responses ...... 203

Appendix K - Student Q6 Responses...... 212

Appendix L - Student Q7 Responses ...... 221

Appendix M - Student Q8 Responses ...... 230

Appendix N - Student Q9 Responses...... 239

Appendix O - Student Q10 Responses...... 248

Appendix P - Student Q11 Responses ...... 257

Appendix Q - Student Q12 Responses...... 266

Appendix R - Student Q13 Responses ...... 275

Page

Appendix S - Student Q14 Responses ...... 283

Appendix T - Student Q15 Responses ...... 289

Appendix U - Student Q16 Responses...... 295

Appendix V - Student Q17 Responses...... 301

Appendix W - Student Q18 Responses ...... 307

Appendix X - Student Q19 Responses...... 313

Appendix Y - Student Q20 Responses...... 319

Appendix Z - Student Q21 Responses ...... 326

Appendix AA - Student Q22 Responses ...... 332

Appendix BB - Student Q23 Responses ...... 339

Appendix CC - Student Q24 Responses ...... 345

Appendix DD - Student Q25 Responses ...... 351

Appendix EE - Student Q26 Responses ...... 357

Appendix FF - Student Q27 Responses ...... 363

Appendix GG - Student Q28 Responses ...... 369

Appendix HH - Student Q29 Responses ...... 375

Appendix II - Student Q30 Responses ...... 381

Appendix JJ - Student Q31 Responses ...... 387

Appendix KK - Student Q32 Responses ...... 393

Appendix LL - Student Q33 Responses ...... 399

Appendix MM - Student Q34 Responses ...... 405

Appendix NN - Student Q35 Responses ...... 411

Page

Appendix OO - Student Q36 Responses ...... 417

Appendix PP - Student Q37 Responses ...... 423

Appendix QQ - Student Q38 Responses ...... 429

Appendix RR - Student Q39 Responses ...... 435

Appendix SS - Student Q40 Responses ...... 441

Appendix TT – Preliminary Trial Summary ...... 447

Appendix UU – Main Study Summary ...... 449

Appendix VV – Questionnaire Part A and Part C Scores ...... 451

Appendix WW – Questionnaire Part B Credits ...... 453

ACKNOWLEDGMENTS

As I reflect over my experiences in writing this dissertation, one of the most important lessons I have learned is that doctoral research very frequently reflects the work of many individuals. This is indeed the case with my dissertation. I can only hope that the words which follow will give due credit to the many individuals whose advice and contributions made my work a reality.

I first wish to thank my professors at Teachers College, Columbia University. From my very first class in September 2009, through the final stages of my dissertation writing in 2014,

J. Philip Smith, Ph.D., was my advisor. Dr. Smith provided the guidance and encouragement that enabled me to reach my academic goals. Whether my questions dealt with course content, course selection, study methods or research methodology, Dr. Smith always had the answers which led to the best solutions. My academic achievements at Teachers College, including but by no means limited to this dissertation, were due in large part to Dr. Smith’s help over the past five years. I offer my thanks to Dr. Smith for these benefits.

Since early 2010, I have also benefited greatly by my association with Bruce R. Vogeli,

Ph.D. On innumerable instances, Dr. Vogeli advised me on the techniques and methods I needed to become a successful student, researcher, and scholar. I have always appreciated Dr. Vogeli’s involvement in my dissertation writing and other studies at Teachers College. With Dr. Vogeli’s help, I learned how to progress through the doctoral program and to do so as a true professional.

I express my gratitude to Dr. Vogeli for this valuable assistance.

Stuart A. Weinberg, Ed.D., also provided me with very helpful suggestions as I was working through my dissertation research. I particularly appreciate Dr. Weinberg’s willingness

vii to respond to my questions on short notice, while modifying his own schedule to review and comment on my research and writing. This is much appreciated.

I also wish to thank Erica Walker, Ph.D., who assisted me in the difficult first step of choosing a research topic and beginning the actual writing of the dissertation. Dr. Walker helped me to focus on the crucial questions, to perform the proper investigations, and to draw the appropriate conclusions. Dr. Walker’s ongoing encouragement and support provided me with the confidence necessary to complete my task, and for that I am very grateful.

Alexander P. Karp, Ph.D., was also very helpful to me as I started writing my dissertation in early 2012. Dr. Karp’s useful and practical suggestions enabled me to begin on the right track and to understand the requirements applicable to dissertation writing. I am grateful for that assistance as well.

Completion of my dissertation was also made possible by the input of Felicia Moore

Mensah, Ph.D., and from the Columbia University Mathematics Department, Patrick Gallagher,

Ph.D. Dr. Mensah and Dr. Gallagher, along with Drs. Smith, Vogeli, and Weinberg, served on my faculty committee and agreed to review and comment on my research. I wish to thank Dr.

Mensah and Dr. Gallagher for this, and also for their willingness to rearrange their own schedules to participate in my final defense hearing.

Over the five years I spent at Teachers College, I have greatly appreciated the encouragement, enthusiasm, and positive energy of Dr. Joseph Garrity. Dr. Garrity assisted me with research methods, scholarly writing, and presentation techniques, and helped develop my confidence in my abilities as a doctoral student. I thank Dr. Garrity for his very valuable input.

viii

I also wish to thank Dr. William E. Hixson for his very helpful advice concerning my research methodology. Dr. Hixson assisted me in refining my research methods and in motivating students to excel. This is also very much appreciated.

I was able to complete my dissertation with the help of other members of the Teachers

College community. Sarah Hersch, M.A., Compliance Manager of the Institutional Review

Board, worked with me to ensure that my research and related documentation were all in compliance with applicable regulations. Gary Ardan of the Office of Doctoral Studies assisted me with questions relating to course registration, Ph.D. program requirements, and all relevant deadlines for the successful completion of the degree. Krystle Hecker, Academic Secretary for the Program in Mathematics, was always available to help me with the scheduling of meetings, registration for courses, and resolution of computer issues. Gabriella Oldham, Ed.D., afforded me the benefit of her expertise in typing, proofreading, and formatting. I offer my thanks to them all for being so helpful to me.

One of the challenges I encountered during this process was finding student participants for my research. I met this challenge successfully because of the generous help of several individuals. Some of my contacts expressed the preference to remain anonymous, but I thank them just as well for their involvement and input. Lois Lee and David Lee generously spent much time helping me locate student participants from the New York City area, and Jennifer

Dowie and Sean Dowie together provided me with very useful information about students at pre- college programs throughout the U.S. My wife, Gang Li-Miele, and my sister-in-law, Yang Li, worked very diligently to locate student participants from China, and also spent many hours preparing the Mandarin translations of my questionnaires and related forms. I particularly

ix appreciate Yang Li’s work in typing the Mandarin translations needed for the Chinese-speaking student participants.

I am also grateful for the encouragement and support provided by my parents, Michael J.

Miele and Connie E. Miele, who always had faith in me. My uncle, John A. Miele, Ph.D., helped me build my confidence and my enthusiasm for scholarly pursuits, and this is appreciated as well.

I also thank my wife, Gang Li-Miele, who has been with me from the start of my journey as a doctoral student, and throughout many earlier projects. In 2014, she and I are not only celebrating the completion of my dissertation, but also our 20th anniversary together. This combination of events seems very fitting.

A. M. M.

DEDICATION

This dissertation is dedicated to my wife, Gang Li-Miele, and to my parents, Michael J. Miele and Connie E. Miele.

They have all made everything possible.

Chapter I

INTRODUCTION

Need for the Study

Carl Friedrich Gauss reportedly considered the field of number theory to be the “queen of mathematics.” If Gauss were alive today, he would certainly be startled to discover that the study of number theory does not fill a central or prominent role in the K-12 curriculum (Campbell &

Zazkis, 2002a). He would also be perplexed to find that a limited amount of research in mathematics education has so far been undertaken regarding students’ learning, understanding, and appreciation of number theory and its applications in the information age.

After reviewing the National Council of Teachers of Mathematics (NCTM) 1981 commission report entitled Guidelines for the Preparation of Teachers of Mathematics, Ball

(1988) posed the following pertinent question: “For primary grade teachers, why no course in number theory?” In the Curriculum and Evaluation Standards for School Mathematics (NCTM,

1989), the topic of number theory is included only in Standard 6 for the fifth through eighth grades. If we peruse the NCTM Handbook of Research in Mathematics Teaching and Learning

(Grouws, 1992) or the NCTM Yearbook, Developing Mathematical Reasoning in Grades K-12

(Stiff, 1999), we find close to nothing on the teaching and learning of number theory topics.

While the Common Core Standards for Mathematics (NCTM, 2010) includes some material on greatest common divisors, least common multiples, and prime factorizations, this material comprises a very small portion of the Standards content. Similarly, Principles and Standards for

School Mathematics (NCTM, 2000) addresses number theory topics such as primes, divisibility, and perfect squares, but such topics are not presented therein as major constituents of recommended course content. In the context of this dissertation, “number theory” relates to those structures and properties of the integers which are not fundamentally computational (such as the frequency and distribution of prime numbers, and the relevance of factorizations and congruences).

In view of the issues referenced above, one is prompted to ask if number theory should be more widely taught as a subject in and of itself (at least in high school). One example of the need for a stronger exposure to abstract number theory in the pre-college years is the range of difficulties students experience in understanding the differences between whole numbers and rational numbers (Durkin & Shire, 1991; Greer, 1987; Mack, 1995; Silver, 1992). Actually, both teachers and students need more advanced structural understandings of fundamental mathematical ideas such as this (Kieran, 1992; Ma, 1999). Given such a situation, we need to determine the extent to which an increase in number theory study might have pedagogical as well as psychological benefits for high school students.

Purpose of the Study

The purpose of this dissertation was to explore the potential effects of number theory study on students’ metacognition and general attitudes towards mathematics. The focus of the study was high school mathematics students in the 11th or 12th grade. To achieve its purpose, this study answered the following research questions:

1. What are the levels of the participant students’ metacognitive skills at the start of this

study?

2. What are the participant students’ attitudes towards mathematics at the start of this

study, and to what extent do the students then exhibit curiosity, collaboration, and/or

inventiveness in mathematics?

3. How do the participant students describe their metacognitive functioning and general

attitudes towards mathematics after having worked on a number theory problem set?

4. Can number theory study improve high school students’ metacognitive functioning,

mathematical curiosity, and/or attitudes towards mathematics?

Procedures of the Study

The Main Study utilized a three-part written questionnaire which the investigator distributed to 37 high school students. These students attended high schools in New York City and Dalian, China. In addition, the investigator interviewed 10 of these students using an interview protocol, which was designed to gather more information on the thought processes of the students. The participant students were in either the 11th or 12th grade.

Part A of the questionnaire measures students’ current metacognitive abilities as well as their attitudes towards mathematics and their propensities for inventing, discussing, and exploring mathematical problems and concepts. The first 30 questions (designed to measure metacognition) are primarily based on a questionnaire by Panaoura, Philippou, and Christou

(2001), but also have roots in related inventories of Fortunato, Hecht, Tittle, and Alvarez (1991),

Schraw and Sperling-Denisson (1994), and Sperling, Howard, and Murphy (2002). The remaining few questions in Part A of the questionnaire are geared towards the measurement of students’ attitudes towards mathematics and their levels of curiosity in the mathematical realm.

Part B of the questionnaire consists of eight problems in the field of elementary number theory, which the students worked on independently, collaboratively, and/or using any research 3

materials they chose. Part B presents a much wider exposure to number theory than, say, word problems on the properties of consecutive odd integers that one may find in New York State

Regents examinations in mathematics (http://www.nysedregents.org/IntegratedAlgebra/). Some of the problems in Part B of the questionnaire were purposefully written to encourage the students to monitor their thought processes, engage in critical thinking, and rewrite the problems in a way that was more meaningful to them.

Part C of the questionnaire is a revised version of the questions presented in Part A. Part

C questions measure the students’ metacognitive potential, mathematical attitudes, and levels of inquisitiveness in the context of the elementary number theory problems which they attempted in

Part B.

The students first worked on Parts A and B of the questionnaire. They were given Part C of the questionnaire after they completed their responses to Parts A and B.

After receiving the completed questionnaire components from the students, the investigator reviewed the Part B responses, and compared the responses in Parts A and C, to determine if the Part B number theory exercises appeared to have caused some change in metacognitive functioning, mathematical curiosity or attitudes towards mathematics.

The Interview Script consists of questions which asked the students to provide, in their own words, details about their mathematics attitudes, beliefs, and thought processes. The questions were asked in the context of the students’ work on the number theory problem set in

Part B, as well as with reference to their mathematics activity before and after their work on the problem set. The Interview Script was also used to obtain more information from selected students on their experiences with the number theory problem set and any resultant changes in their mathematics attitudes, beliefs, and metacognitive processes. Interviews with 10 selected

students began at least one week after they completed and submitted all parts of the questionnaire. The 10 students interviewed were students who had performed very well on the number theory problem set and/or students who did not perform well on the problem set.

Roughly four weeks before the start of the above-described study, a Preliminary Trial was also undertaken, involving three New York City high school students (in the 11th or 12th grade) who were asked to complete all parts of the questionnaire. These students were interviewed (using the Interview Script) at least one week after their completion of the questionnaire. The participants in the Preliminary Trial were recruited from groups of high school students living in the vicinity of the investigator’s neighborhood, through discussions with the students and their parents outside of the applicable school settings. All three of these students attended different high schools in the New York City area, and none of said students attended the same high school as any of the students in the Main Study. The Preliminary Trial is referenced as a Pilot Study in the Informed Consent and Research Description Documentation for Parents, appearing in Appendix D and originally approved by the Institutional Review Board.

Each student participant was given a Certificate of Participation. The investigator completed each such Certificate to evidence the student’s involvement in the study (including the student’s name if he or she was interviewed or if the student and his/her parent or legal guardian otherwise agreed to provide the student’s name). The Certificate of Participation showed the student’s questionnaire code number if the student’s parent or legal guardian chose not to provide the student’s name. The three students in the Preliminary Trial and the first four students in the Main Study were given the name of the investigator to use as a reference for their college applications or for any work-related application processes (pursuant to Institutional Review

Board approval).

The investigator obtained the parents’ phone numbers, student names or other information personally identifying the participant students (and linked to the participant students’ individual responses) for purposes of this study. Records of such names and other personal identifiers (if applicable) were destroyed at the conclusion of the investigator’s final dissertation defense, except to the extent that the investigator needed to disclose the names and other personal identifiers for purposes of serving as a reference for the three students in the

Preliminary Trial and/or the first four students in the Main Study. The investigator obtained prior written approvals from the three students in the Preliminary Trial, the first four students in the

Main Study, and their parents/legal guardians for purposes of this disclosure (using a Consent for

Reference Form prepared with Institutional Review Board approval). The Institutional Review

Board subsequently disapproved said Consent for Reference Form for future use with participating students.

Parts A, B, and C of the questionnaire, as well as the Interview Script, and the Certificate of Participation Templates and Consent for Reference Form are included in Appendices A, B, and C, respectively. Institutional Review Board Required Forms are included in Appendix D.

Mandarin translations of the investigator’s cover memo to the high school in China, Parts

A, B, and C of the questionnaire, the Certificate of Participation templates, and Institutional

Review Board Required Forms are included in Appendix E.

The investigator analyzed the data using appropriate statistical techniques.

Chapter II

LITERATURE REVIEW

The current mathematics education literature does not include many investigations of the effects of number theory study on high school students’ metacognitive abilities and mathematics attitudes; however, many scholarly resources address possible ways of improving students’ attitudes towards mathematics coursework and studies have looked at metacognition in mathematics education. In addition, numerous documented efforts have introduced number theory courses into the high school mathematics curricula. Among the justifications for these efforts are the numerous difficulties students experience in distinguishing between whole numbers and rational numbers (Durkin & Shire, 1991; Greer, 1987; Mack, 1995; Silver, 1992), and the need for both teachers and students to develop more advanced structural comprehension of such basic mathematical ideas (Kieran, 1992; Ma, 1999). The purpose of this literature review was to present an analysis of notable research materials pertinent to the topics of mathematics attitudes and high school number theory study and metacognition, and to describe how the present study fits within the broad framework of such research.

Attitudes Towards Mathematics

Before attempting to find a link between high school number theory study and improved mathematics attitudes, one needs to understand the factors which (according to existing research)

do effect a student’s attitudes towards mathematics. A summary of such factors thus follows, based primarily on results reported by Kelly (2011).

A student’s past performance in mathematics courses predicts the extent to which the student will freely choose to undertake more advanced mathematics studies (Ercikan, McCreith,

& Lapointe, 2005; House, 2000). Also, parents’ levels of education and enthusiasm for their children’s academic achievement, as well as the socio-economic status of families, affect students’ willingness and desire to study advanced mathematics (Ercikan, McCreith, & Lapointe,

2005; Flores, 2007). On a more psychological note, human beings usually prefer to do things that they feel they can do, or that they believe will eventually bring them some type of reward

(Schunk, 1987). Wilkins and Ma (2003) noted that as students proceed through middle school and high school, their attitudes towards mathematics decline. They further discovered that the parents, teachers, and peer groups have a major influence on the way students feel about mathematics as a field of study (also noted by Kelly, 2011). The specific effects of each of these factors will be discussed next.

Parental beliefs about the importance of academic activity in general (and mathematics study in particular) strongly influence students’ attitudes towards mathematics study (Eccles &

Jacobs, 1986). Parents who place a great value on academic achievement develop in their children the belief that academics are important (Hwang, 1995; McNair & Johnson, 2009).

Besides the beliefs of parents, the extent to which parents are involved in their students’ education is also relevant. Steinberg, Lamborn, Dornbusch and Darling (1992) defined parental involvement as assisting their children with their course selections and homework assignments, monitoring their children’s progress in school, and accompanying their children to both academic and extracurricular school activities and meetings. This type of consistent parental

involvement increases students’ levels of engagement in school (Steinberg et al., 1992); however, other research has revealed that peer influence can reduce the correlation between parental involvement and students’ achievement in and attitudes towards mathematics (Steinberg et al., 1992).

Other investigators have generalized the concept of parental involvement with high school students to include parents spending time with the students while discussing current events, part-time jobs, and other topics relevant to adolescence (Trusty & Lampe, 1997). The students who were aware of this form of parental involvement in their lives had stronger beliefs that they were in control of the outside events around them (Rotter, 1990).

High school algebra students and their parents constituted the subjects of a study by

Shirvani (2007). In said study, the investigator distributed surveys to treatment and control groups of students and parents, before and after parents in the treatment group received enhanced communications from the teacher about their children’s academic performance in algebra. The results indicated that the students in the treatment group (i.e., those whose parents received the teachers’ feedback) had considerably higher levels of self-confidence in their mathematical abilities and were more engaged in classroom mathematics activities.

The existing research demonstrates that parental involvement has positive effects on students’ attitudes towards mathematics study; however, these positive effects ultimately depend on the amount of time parents are able to devote to their children’s education. It is widely known that in many U.S. families, high divorce rates, financial problems, job stress, and problems with addiction have caused a fragmentation of home life. Challenges of this nature make it very difficult for even the most well-intentioned parents to encourage and maintain a supportive learning environment for their children.

The teaching styles and content knowledge of mathematics teachers also have an impact on students’ mathematics attitudes (Cornell, 1999; Trujillo & Hadfield, 1999). As compared to teachers in other countries, U.S. teachers employ pedagogical methods which are less effective in honing critical thinking skills or tendencies towards mathematical exploration (Trends in

International Mathematics and Science Study [TIMSS], 2003). When students lack a deep understanding of fundamental mathematical ideas, they have difficulty seeing the interconnections among areas of mathematics and do not appreciate the value of mathematics in the world around them (Crespo, 2003). The students then see mathematics as a long series of irrelevant rules and formulas which they need to memorize, and they quickly become bored and distracted. As expected, the end result is that the students have very negative attitudes towards mathematics.

Research has revealed that the level of a teacher’s content knowledge has a major effect on the teacher’s confidence in his or her abilities to teach (Cady & Rearden, 2007; Ross &

Bruce, 2007). When teachers believe they can be successful as teachers, and when they expect they will find a way to help their students learn and understand mathematics, they inevitably set higher academic goals for themselves and for their students. They also impart to students the persistence to work harder to achieve academic goals and to overcome obstacles in the educational process (Ross & Bruce, 2007). A student who does not become discouraged but approaches mathematics study with persistence and ambition is more likely to have an empowering attitude towards mathematics.

The teacher’s confidence level also determines the extent to which students will be required to engage in higher-level thinking tasks, interact with their fellow students on mathematics problems, and use mathematical inventiveness. Such activities involving students

help them develop their own self-confidence in mathematics, which in turn leads to greater levels of mathematical achievement by the student body (Ross & Bruce, 2007).

The successful learning of mathematics depends in large part on the way mathematics is presented to the learner, and the manner in which the learner interacts with the learning environment (Yara, 2009). Students’ positive attitudes towards mathematics can be intensified by the enthusiasm, dependability, helpfulness, and strong content knowledge of teachers. A student’s disposition towards mathematics can be made (or unraveled) by the teacher’s attitudes towards the subject, the students, and the classroom environment (Yara, 2009).

Bandura (1971) showed that human behaviors are acquired by watching a role model

(e.g., a teacher, parent, mentor or friend) performing such behaviors. The role model displays certain behaviors and the learner watches and then attempts to imitate them. Teachers are indeed role models whose behaviors and reactions can be readily copied by their students (Yara, 2009).

In a similar vein, Chacko (1981) demonstrated that the attitudes of teachers towards teaching can significantly predict the students’ attitudes and achievement levels in mathematics.

The collective research available thus supports the contention that mathematics learners absorb the dispositions of teachers to form their own “student attitudes” towards mathematics, which will likely affect student learning outcomes (Yara, 2009).

A student’s attitude towards mathematics can also be affected by the student’s peer group. Some peers encourage their fellow students to strive for higher academic achievement, and foster positive and empowering beliefs about schoolwork. Other members of a student’s peer group can very easily have the opposite effect, resulting in the student viewing school as a boring or hostile environment. The school experiences and achievement levels of a particular student

also depend on the personality traits and behavior patterns of the peers with whom the student spends a considerable amount of time (Bernt & Keefe, 1992).

Negative peer influence can take the form of pressuring a student to hide academic talents and scholarly inclinations in order to avoid critical and disapproving peer responses (Sullivan,

Tobias, & McDonough, 2006). One study conducted by Hufton, Elliott, and Illushin (2002) involved interviews with slightly over 150 students who were 15 years old, and examined the effect that peers have on mathematics classroom performance and basic work ethics in school.

Some of the interviewees reported that students who were categorized by their peers as being too hard-working were called names such as “nerd” (Hufton et al., 2002). The name-calling spread beyond the classroom and became prevalent in both in-school and out-of-school activities

(Hufton et al., 2002). Many students were unwilling to learn and militated against the teachers’ attempts to structure and maintain a directed learning experience (Hufton et al., 2002). Attitudes towards mathematics invariably suffer in such an environment.

Students who choose to associate with peer groups that scoff at academic achievement may decide to relax or even abandon their learning efforts in order to stay on good terms with their friends (Urdan & Maehr, 1995). If talented and motivated students are treated like social misfits because they have high grades in mathematics, other students may conclude that being apathetic towards mathematics is the preferred direction to take.

Peer group influences can also positively affect mathematics attitudes within the student body. In working with members of their peer group in mathematics class, high school students have indicated that asking peers for assistance on mathematics problems is not as intimidating as going to the teacher and asking for help (Nardi & Steward, 2003). In addition, working with their peer group can help students increase their own confidence level. Students tend to have more

optimistic attitudes towards their abilities to solve problems when they see one or more of their peers finding solutions to the same or similar problems (Bandura, 1997; Schunk, 1987).

Furthermore, some students categorize their peer relationships in school, and their overall school setting, as positive aspects of their lives. Such students have more positive attitudes towards schoolwork (including mathematics) and reach greater levels of academic success (McNair &

Johnson, 2009). Even if a student is met with social acceptance from some—but not many— fellow students, more positive attitudes towards school can still be the result (Bandura et al.,

1996).

If students feel comfortable expressing themselves in class, levels of mathematics anxiety are likely to decrease and attitudes towards mathematics are likely to improve. In addition to the influences of teachers, parents and peer groups, a technique that can be used to help mathematics students express themselves freely and comfortably, and thereby enhance mathematics attitudes, is called bibliotherapy. This term refers to the use of reading to create meaningful change, to motivate, and to enable meaningful personality growth (Doll & Doll, 1997; Forgan, 2002, 2003;

Lenkowsky, 1987; Sridhar & Vaughn, 2000). Bibliotherapy can help students understand themselves and cope with anxiety and other negative emotions by providing literature pertaining to their personal situations and developmental needs. Reading influences thinking and behavior, and the specific needs of students can be addressed through selected readings and guided discussions.

Bibliotherapy includes several stages (Forgan, 2002; Halsted, 1994), and each is summarized below:

1. Identification: While reading a book involving a character who is experiencing

negative feelings about mathematics, a student can easily experience a therapeutic

experience when he or she says, “This character is a lot like me. I can relate to this

person because we both share the same types of problems.” This is a form of

identification, which becomes stronger as more similarities between the character and

the reader become apparent.

2. Catharsis: With the identification process described above, there also comes a relief

of tension or a catharsis, an emotional reaction that lets the readers know they are not

alone in their problems with mathematics. (This emotional reaction is also referred to

as universalization [Slavson, 1950]).

3. Insight: As students read books about other children struggling with mathematics,

they have the chance to learn vicariously from these characters, and learn new ways

of looking at their own troublesome emotions and feelings towards mathematics. This

new insight helps students change their attitudes, behaviors, and mindsets as they

encounter real-life situations that are similar to the situations faced by the fictional

characters in the books.

The value of bibliotherapy lies in the fact that books can help guide the emotional development of students much more than intellectual discussions can, because stories can directly affect human emotions (Forgan, 2002, 2003). A talented writer can help students establish an emotional connection with a fictional character who is experiencing similar problems in school. Through the use of bibliotherapy, students who are reluctant to speak about their fears, anxieties, and other negative emotions can identify with fictional characters, experience the catharsis mentioned above, and acquire some useful insights and distinctions

(Halsted, 1994). For bibliotherapy to be successful in easing negative feelings about mathematics, meaningful follow-up discussions are required (Forgan, 2002). Simply reading a

book is not enough. Students also need to become involved in discussions, counseling, role- playing activities, problem solving, relaxation with music and art, and journal preparation

(Hebert, 1991, 1995; Hebert & Furner, 1997; Forgan, 2002, 2003). When structured in this way, the technique of bibliotherapy can be a pleasant experience while also giving students a time for introspection and self-awareness. Said technique provides mathematics-weary students with the opportunity to learn that they are not alone in their distress, they were never alone in that sense, and they can find ways of conquering their negative emotions and reaching their full potential as students.

The research summarized above is useful for identifying the factors and teaching strategies which influence students’ attitudes towards mathematics study; however, said research does have some limitations. Most of the studies presented did not involve interviews with students currently enrolled in high school. The research materials dealing with parental influence relied on a review and analysis of extensive data and written surveys, rather than interviews and group discussions. As for bibliotherapy, one limitation is that some students generally do not like to read. For such students, a possible alternative is to allow them the option of writing their own short stories about fictional students with negative attitudes towards mathematics. These short stories can then be shared with the class, in the hope that they will serve as a lens through which the students can perceive and eventually change their own feelings about and attitudes towards mathematics. Such an approach relies on the assumption that students like to write (or that they can tolerate it).

Another limitation in the literature on high school students’ mathematics attitudes is that it does not generally gauge the potential for improving student attitudes by exposing students to

new mathematics topics (such as number theory). The present study was deigned to remedy that limitation.

Number Theory in High School

Researchers have pointed to the limited emphasis on number theory for teachers and/or students in the pre-college years (Ball, 1988; Campbell & Zazkis, 2002a, 2002b, 2006; Grouws,

1992; Stiff, 1999). A review of various NCTM sources similarly reveals little or no emphasis on number theory topics (e.g., www.corestandards.org/the-standards; www.nctm.org; Guidelines for the preparation of teachers of mathematics [NCTM, 1981]; Curriculum and evaluation standards for school mathematics [NCTM, 1989]; Yearbook, developing mathematical reasoning in grades K-12 [NCTM, 1999]; Principles and standards for school mathematics

[NCTM, 2000]; Common core standards for mathematics [NCTM, 2010]). Perhaps in light of this situation, numerous educators have attempted to incorporate number theory into the high school curriculum. The literature references and examples discussed below were chosen to illustrate major efforts previously undertaken to help students see the usefulness of number theory in high school and to expose them to the wonders of number theory before they enter college.

The study of number theory has been categorized as a useful tool for students in high school, in areas such as Fibonacci numbers, Diophantine equations, continued fractions, and algorithms for computing the value of π (Dence, 1999). Some elementary concepts of number theory, such as lattice points, divisibility, Pythagorean triplets, and Fermat numbers, need to be integrated into high school algebra and geometry courses (Barnett, 1965). Rather than viewing elementary number theory as an advanced course for gifted high school students, teachers should mold number theoretic concepts into their algebra and geometry lesson plans. A variety of

number theory texts would be useful to teachers for this purpose (e.g., Dudley, 1969; Koshy,

2007). Some texts also provide a very wide selection of elementary number theory problems that could be suitable for high school students (e.g., Musser, Burger, & Peterson, 2008). Other examples of such elementary number theory exercises are word problems on the properties of consecutive odd integers on New York State Regents examinations (http://www.nysedregents. org/IntegratedAlgebra/).

Students will reap greater benefits from courses which bring together various topics of mathematics and illustrate how these topics relate and support each other, as compared to courses which present mathematics in a compartmentalized fashion (Barnett, 1965). To this effect, an interesting recommendation is to teach number theory to high school students through the use of a heuristic approach. The idea is to treat theorems as problems and then approach the proofs of those theorems as solutions to problems (Libeskind, 1971). One important caveat in this approach is to ensure that students understand the difference between theorems and problems. A problem is ordinarily classified as a theorem if the problem is unusually interesting in itself or repeatedly valuable to the mathematical community in the solution of other problems.

The teaching of number theory in high school via a heuristic approach can be illustrated by an example involving the Euclidean Algorithm.

The Euclidean Algorithm for computing the greatest common divisor (GCD) of two integers is commonly introduced in the United States during abstract algebra or number theory courses offered at the university level. Given two positive integers a and b (with a > b), the basic process for determining the GCD of a and b involves repeated applications of the Euclidean

Algorithm, as shown below:

First, divide a by b to obtain

a = q0b + r1, 0 ≤ r1< b.

Next, divide b by r1 to obtain

b = q1 r1 + r2, 0 ≤ r2< r1.

Similarly, divide r1 by r2 to obtain

r1 = q2 r2 + r3, 0 ≤ r3< r2.

Continuing the above process, one obtains

r2 = q3 r3 + r4, 0 ≤ r4< r3,

………………………..

rk-2 = qk-1 rk-1 + rk, 0 ≤ rk< rk-1,

rk-1 = rk-1 rk + 0.

The above process concludes when the remainder of 0 is obtained. The theorem is then stated as follows: The last positive remainder which is obtained by the repeated applications of the

Euclidean Algorithm is the GCD of the integers a and b.

Many instructors have found that high school students’ retention spans (and levels of understanding) are considerably limited when they attempt to comprehend the Euclidean

Algorithm process described above. A problem-solving approach to teaching the GCD computation has produced better results, as evidenced by the following experiment involving the teaching of number theory to a selected group of high school students between the ages of 14 and

15 (Libeskind, 1971).

The experiment began with the instructor introducing the notion of GCD in an informal manner, by asking the question “What is the largest number that divides 6 and 15?” Then, the

instructor informed the students that the largest number that divides both integers a and b is the

GCD of a and b and is written (a,b). The instruction continued as follows:

We can write (6, 15) = 3, (6, 18) = 6, etc.

If our given numbers are very small, it is relatively easy to guess what their GCD is.

It is almost impossible to guess (455, 221) or (584, 1606).

It is therefore necessary to find a system enabling us to determine the GCD of a and

As an example, we will compute (455, 221).

If we can find two small numbers (e.g., each less than 10), whose GCD is the same as

the GCD of 455 and 221, then this method would give us the desired result.

Note that (15, 10) = 5, (15 - 10, 10) = 5, (15, 6) = 3, (15 - 6, 6) = 3.

The students then conjectured that (a, b) = (a-b, b). Upon hearing this conjecture, the instructor asked the students to check their conjecture with a few additional numerical examples, and then asked them to prove their conjecture.

One student proposed that the set of all the common divisors of 24 and 18 was the same as the set of all common divisors of ‘24 – 18’ and 18 (namely, the set {1, 2, 3, 6}). The student then suggested that the class should try to prove this in general.

In response to the student’s suggestion, the students and the instructor jointly produced the proof outlined below:

Let S be the set of all common divisors of a and b. Let T be the set of all common divisors of a – b and b. We will first show that S ⊂ T. If d ∈ S, then d │a and d │ b. (Recall that d │ a if a = kd for some integer k.) By a theorem previously proved as part of the students’

mathematics lessons, for all integers a and b, d │a and d │ b => d │ a – b. Since d │ b and d │ a – b, it is clear that d ∈ T. So, S ⊂ T.

Next, we show that T ⊂ S. If d ∈ T, then d │ a – b and d │ b. Since for all integers x and y, d │ x and d │ y => d │ x + y, we have d │ (a – b) + b => d │ a => d ∈ S. So T ⊂ S.

Since S ⊂ T and T ⊂ S, we conclude that S = T. This means that the greatest number in the set S (i.e., (a, b)) is equal to the greatest number in the set T (i.e., (a-b, b)).

The instructor then asked the students to use the equality (a, b) = (a-b, b) as repeatedly as necessary to compute (288, 51). After the students wrote (288, 51) = (237, 51) = (186, 51) =

(135, 51) = (84, 51) = (33, 51), most of them noticed (either independently or after being questioned about a possible shortcut) that they did not have to subtract 51 from 288 five times. It would have been much easier for them to divide 288 by 51 and find the remainder.

On the basis of the above method of instruction, the students eventually could determine the GCD of any two given integers (e.g., 975 and 105) as follows:

(975, 105); 975 ÷ 105 = 9 remainder 30

(975, 105) = (30, 105); 105 ÷ 30 = 3 remainder 15

(975, 105) = (30, 105) = (30, 15); 30 ÷ 15 = 2 remainder 0

(975, 105) = (30, 105) = (30, 15) = (15, 0) = 15

In addition to learning how to use the above-described algorithm, the students displayed the ability to reproduce the proof of (a, b) = (a-b, b) and were also able to generalize this equality to (a, b) = (a-qb, b) for any integers a, b and q. Even more notably, the students could prove this generalization in a manner very similar to that used in the proof of (a, b) = (a-b, b) (Libeskind,

1971).

Most of the students in the above experiment went even further and were able to solve the following type of problem:

Consider (14a + 3, 21a + 4). Find the value of this GCD for several different values of a.

Based on your results, make a conjecture and prove it (Libeskind, 1971).

The experiment involving GCDs demonstrated that if a student tries to learn a “theorem- proof” approach to number theory, he or she will rarely understand how the result was obtained.

Once a theorem is forgotten, it will be difficult for the student to recall it. On the other hand, if a student starts to think of theorems as problems to be solved, he or she can more easily recall a theorem by “solving it” as if it were a problem (Libeskind, 1971).

Another pertinent goal relates to determining the possible benefits that can be gained by introducing high school students to number theory topics. Two benefits arguably attainable by introducing high school students to the idea of prime factorization are the reinforcement of computational abilities and a comprehension of commutativity and associativity in multiplication. Furthermore, this understanding can then be used to introduce to the students two applications: determining the number of divisors of an integer n and computing the sum of the divisors of n (Pagni, 1979).

Hendricks and Millman (2011) conducted a study to determine the ways in which high school students’ participation in an advanced course (covering various proofs and problems in number theory and algebra) affected their mathematical self-efficacy (i.e., their beliefs about their capability to attain a certain level of performance on given tasks or goals (Bandura, 1994)).

This study is included here because its goal of determining how number theory instruction affects student beliefs is rather similar to the objectives of the present study.

The Hendricks and Millman study involved 15 high school students (14 seniors and one junior) who were enrolled in an 18-week course entitled Proofs and Problems in Number Theory and Algebra (PPNTA). Major topics covered in the course included basic properties of the integers, divisibility and prime numbers, the Fundamental Theorem of Arithmetic, polynomials and Diophantine equations, equivalence relations, and applications to combinatorics. To determine the effects of the PPNTA course on the students’ mathematical self-efficacy, the investigators used a pre- and post-course self-efficacy instrument. These instruments asked the students to rate their self-confidence in areas such as knowing how to work in teams to solve number theory and algebra problems, knowing how to explain ideas that motivate proofs, knowing how to define a “mathematical state of mind,” and knowing how to identify fallacious reasoning in incorrect proofs. The students also completed a post-course survey questionnaire about the PPNTA course and their experiences learning the course content.

After reviewing the results of the above study, the investigators reported a 29.16-point difference between the students’ pre- and post-course measures of self-efficacy. While not all the details of the results appear to have been provided, the study apparently suggested that the

PPNTA course improved the students’ mathematical self-efficacy levels. These reported results are encouraging and serve to support the theory that studying number theory can help high school students develop more self-confidence in mathematics. For a future study, it would also be interesting to determine if completion of the PPNTA course helped high school students become more mathematically self-confident and more metacognitive.

Other investigators have suggested high school number theory study as a tool for accomplishing additional pedagogical objectives. The use of number theory to solve the

“birthday problem” can be used as a first-day motivational discussion for the initial class session

of a number theory lecture course geared towards gifted high school students (Srinivasan, 2010).

In this context, the “birthday problem” involves determining a person’s age, month of birth, and the day of the week on which the person was born. The solution of the “birthday problem” requires applications of the division algorithm, binary representations, Diophantine equations, and the integral part of an integer x. It is expected that working through the “birthday problem” will convince high school students that it will be a worthwhile endeavor to continue studying number theory (Srinivasan, 2010).

Another study revealed that high school students’ appreciation of the nature of proofs, as well as the sophistication of their reasoning abilities, increased after having studied number theory concepts (Quinn, 2009). The investigator determined that while only 8% of the participating students utilized inductive reasoning on a pre-study test to investigate if and why a generalization held, 88% of the participating students did so on the post-study test. Another recommendation coming from this study was for teachers to spend time on problems involving patterns that do not continue on and on as students generally anticipate, in order to challenge the students’ belief that inductive reasoning is equivalent to proof.

Our discussion of the efforts to incorporate number theory into the high school mathematics curriculum will conclude with references to specific number theory courses created specifically for high school students. The first is “A Course in Number Theory for High School

Students” (Garlock, 2011), consisting of 13 chapters and available (in part) on the web. Rather than laying out everything in the text (much like a conventional high school mathematics course), said number theory course aims to require the student to assume the responsibility for discovering and proving. The topics covered include finding integer solutions to ax + by = 1, continued fractions, perfect numbers, modular arithmetic, primitive roots, congruences, and

Gaussian integers. Also, the Department of Mathematics of the University of Utah College of

Science offers a four-part Summer Mathematics Program for High School Students, of which the first part is a three-week course entitled “Explorations in Number Theory.” Among the topics covered in that course are the theory of cryptography and the actual encoding and decoding of messages that students send to each other (http://www.math.utah.edu/hsp/desc.html). Similarly, the website http://www.math.cornell.edu/~mec/2008-2009/Anema/numbertheory/ numbers.html contains a lesson plan for high school students entitled “Number Theory and Cryptography,” covering topics such as check digits, Zn, the Euler φ-function, and the RSA algorithm. Other examples of such pre-college course offerings include the Ross Mathematics Program at Ohio

State University (http://www.math.osu.edu/ross/index.html), which includes summer courses on number theory, and the Program in Mathematics for Young Scientists (PROMYS) at Boston

University. PROMYS is a six-week summer academic program designed to encourage highly motivated and industrious high school students to probe the creative and investigative realm of mathematics in a diverse community of fellow students, counselors, research mathematicians, and visiting scholars (www.promys.org). Said program includes a one-hour number theory lecture given to the high school students every weekday morning. Fun-filled presentations of number theory concepts are offered to high school students at the Center for Talented Youth site at Franklin & Marshall College. Among the memorable teaching strategies used in that program are the Seven Skits of Number Theory, and the naming of particular number theoretic results after the students who suggested them to the class (http://www.realcty.org/mw/index.php? title=Number_Theory).

The aforementioned studies, research efforts, and course offerings have improved our understanding of the usefulness of number theory study in the high school classroom (and have

rendered number theory more accessible to high school students). However, they have not generally addressed the possible effects of number theory instruction on high school students’ metacognitive abilities.

Metacognition

Because the present study aimed to determine the extent to which metacognitive abilities can be affected by number theory study in high school, it is first necessary to examine the definition of metacognition, the ways by which metacognitive skills can be developed and assessed, and the nature and results of prior studies on metacognition and mathematics education. It is also important to note that metacognition has great relevance in areas such as problem solving, mathematics study, curriculum, and cognitive psychology (Desoete &

Veenman, 2006; Fortunato, Hecht, Tittle & Alvarez, 1991; Panaoura, Philippou, & Christou,

2001; Schraw & Sperling-Dennison, 1994; Sperling, Howard, & Murphy, 2002; also see edina.k12.mn.us/teach/curriconline/guide/la/secondary/STRP/metacog/ index.htm).

John Flavell originally used the word metacognition in the late 1970s to refer to

“cognition about cognitive phenomena,” or “thinking about thinking” (Flavell, 1979, p. 906).

While cognition can be viewed as the manner in which learners’ minds act on the real world, metacognition is the manner by which learners’ minds act on their cognitive processes. For further clarity, metacognition is defined as any act of thinking or reflecting which operates on a cognitive thought in such a way as to assist the learner in learning a new concept or solving a problem (Holton & Clarke, 2006). Several other alternative formulations of metacognition are widely accepted, such as knowing, evaluating, and regulating one’s own thinking (Wilson &

Clarke, 2002), and self-regulation, beliefs and intuitions, and knowledge of one’s own thought

patterns (Schoenfeld, 1992). Similar conceptions of metacognition have been offered by investigators in cognitive psychology, as exemplified by the following definitions:

“The monitoring and control of thought” (Martinez, 2006, p. 696);

“Awareness and management of one’s own thought” (Kuhn & Dean, 2004, p. 270);

“The knowledge and control children have over their own thinking and learning

activities” (Cross & Paris, 1988, p. 131); and

“Awareness of one’s own thinking, awareness of the content of one’s conceptions, an

active monitoring of one’s cognitive processes, an attempt to regulate one’s cognitive

processes in relationship to further learning, and an application of a set of heuristics

as an affective device for helping people organize their methods of attack on

problems in general” (Hennessey, 1999, p. 3).

As Kuhn and Dean (2004) explain, metacognition is the process that enables a student

(who has already been taught a specific strategy to be used in a specific problem context) to retrieve and utilize that same strategy in a similar problem context. They further define metacognition as an executive control that comprises both monitoring and self-regulation.

Metacognition has also been viewed as a multidimensional set of general skills that are distinct from general intelligence and might even serve to compensate for certain intelligence deficiencies (Schraw, 1998).

Metacognition consists of two constituent parts: knowledge about cognition (cognitive knowledge) and monitoring of cognition (cognitive regulation) (Cross & Paris, 1988; Flavell,

1979). An effective means of understanding these two constituent parts is to provide examples of each.

Types of cognitive knowledge include the following:

Knowledge about oneself as a learner and knowledge about the factors that affect

cognition (Cross & Paris, 1988; Flavell, 1979; Kuhn & Dean, 2004);

Awareness of cognition, including knowledge about strategies (Cross & Paris, 1988;

Flavell, 1979; Kuhn & Dean, 2004; Schraw et al., 2006); and

Knowledge about why and when to use a given strategy (Schraw et al., 2006).

Some examples of cognitive regulation are:

Identification and selection of pertinent strategies and appropriate allocation of

resources (Cross & Paris, 1988; Paris & Winograd, 1990; Schraw et al., 2006);

Awareness of, and attending to, one’s comprehension and performance of tasks

(Cross & Paris, 1988; Paris & Winograd, 1990; Schraw et al., 2006); and

Assessing the processes and results of one’s learning efforts, and reevaluating and

revising one’s learning goals and objectives (Cross & Paris, 1988; Paris & Winograd,

1990; Schraw et al., 2006; Whitebread et al., 2009).

Although the ability to articulate cognitive knowledge tends to improve as a person ages, many adults still have difficulty explaining what they know about their own thinking processes

(Schraw et al., 2006). By contrast, those who are good at critical thinking can be said to be more metacognitive. In fact, some investigators have included critical thinking within the concept of metacognition (Flavell, 1979; Martinez, 2006). In a similar fashion, Hennessey (1999) offered a list of metacognitive skills that are very similar to skills frequently mentioned as examples of critical thinking. Such skills are listed below:

Considering and evaluating the basis of one’s belief systems;

Assessing one’s conceptions in comparison with competing conceptions;

Evaluating one’s conceptions in terms of evidence that may or may not support those

conceptions;

Considering and evaluating the status of one’s conceptions; and

Evaluating the levels of consistency and generalizability of one’s conceptions.

A somewhat different view is that self-regulated learning includes metacognition, motivation, and cognition, which in turn includes critical thinking (Schraw et al., 2006).

Metacognition can also be seen as a condition that supports critical thinking because monitoring the quality of one’s thought processes increases the chances that the person will exhibit high- quality thinking (or critical thinking) (Lai, 2011).

Martinez (2006) further argued that metacognitive strategies can improve a person’s persistence and motivation levels when faced with challenging tasks and obstacles. Paris and

Winograd (1990) presented a similar argument, maintaining that metacognition inevitably includes the element of affect (because as students monitor and evaluate their own cognition, they gradually become more aware of their own strengths and weaknesses).

Considering that the present study attempted to uncover the effects of number theory study on students’ metacognitive abilities, a pertinent question is whether or not metacognition can actually be taught. A number of investigators have offered evidence that metacognition is teachable and can be fostered through the use of certain instructional techniques (Cross & Paris,

1988; Hennessey, 1999; Kramarski & Mevarech, 2003).

In an experiment reported by Cross and Paris (1988), an effort was made to enhance the metacognitive abilities and reading comprehension skills of 171 elementary school students in the third and fifth grades. The students were exposed to a curriculum plan (entitled Informed

Strategies for Learning) that was structured in such a way to increase the students’ awareness

and use of effective reading techniques. During the instructional periods, the students were given strategy training that included precise attention to declarative, conditional, and procedural knowledge about reading techniques. As measured against a comparison group of students, the students receiving the strategy training showed significant improvements in evaluating the difficulties of the reading tasks before them, making plans to reach a specific goal, and monitoring and evaluating their progress towards reaching that goal.

Hennessey (1999) reported a three-year instructional strategy involving 170 elementary school students in the first through sixth grades. The students completed science classes that were designed to measure students’ conceptions of science and the nature of science in general.

The instructional activities were specifically geared towards the development of metacognition.

Such activities included expressing the students’ science conceptions in a visible format, giving the students the opportunity to clarify their science conceptions in small teams, encouraging conceptual debate, and helping the students practice in a variety of contexts. The results of this instruction showed that students’ metacognitive skills increased as the years passed. In fact, the first-grade students displayed the highest levels of metacognition among all the students completing the three-year program.

In a study exploring the effects of metacognitive training on the mathematical reasoning and metacognitive abilities of 384 eighth-grade students, Kramarski and Mevarech (2003) found that the training had positive effects. The students who were exposed to metacognitive instruction in cooperative or individualized learning settings outperformed comparison students in a number of areas, such as the ability to interpret the meaning and significance of graphs, the fluency of accurate mathematical explanations, and the use of logical arguments to bolster mathematical thinking. Such trained students were also more flexible in their mathematical

reasoning, performed better on transfer tasks, and were better able to represent mathematical concepts in a variety of ways.

The cited research demonstrated that metacognitive skills can be taught, at least to some students. In fact, many investigators have recommended various pedagogical techniques that are useful in the teaching of metacognition, such as the delivery of explicit instruction in cognitive knowledge and cognitive regulation (Cross & Paris, 1988; Schraw, 1998; Schraw et al., 2006). In particular, Schraw emphasized the importance of strategy training to help students understand how and when to use strategies and why such strategies are useful to them. Other investigators also highlighted the importance of stressing the value of strategies to motivate students to use them in productive and independent ways (Cross & Paris, 1988; Kramarski & Mevarech, 2003;

Schneider & Lockl, 2002).

In addition to delivering instruction on cognitive knowledge, educators also need to help students develop their skills in monitoring and regulating their cognition. Kuhn (2000) indicated that metacognitive instruction should aim to increase a student’s awareness and control of meta- task, instead of task, procedures. Schraw (1998) suggested the use of explicit prompts to assist students in enhancing their regulating abilities. He recommended a checklist with entries that separately address planning, monitoring, and regulating, with subquestions for each entry that must be addressed as the instruction proceeds. Schraw argued that a checklist of this type helps students become more strategic and systematic during problem-solving activities. Similarly,

Kramarski and Mevarech (2003) gave students sets of metacognitive questions to be completed as they attempted a particular problem-solving task. Some of these questions were comprehension questions that were structured to motivate students to reflect on a problem before tackling its solution. Other questions were strategic questions that helped to encourage students

to think about which strategy might be most appropriate and useful for a particular task (while providing for themselves a supporting reason or rationale for that choice of strategy). The remaining questions were connection questions that motivated students to recognize and focus on deep-structure task attributes to activate their own background knowledge and strategic experience with similar tasks.

Another common recommendation is to use collaborative or cooperative learning strategies to help students develop their metacognitive skills (Cross & Paris, 1988; Hennessey,

1999; Kramarski & Mevarech, 2003; Martinez, 2006; Paris & Winograd, 1990; Schraw et al.,

2006). This recommendation arguably has its roots in the Piagetian and Vygotskyian traditions that stress the value of social interaction and involvement as a means of fostering the development of cognition (summarized in Dillenbourg, Baker, Blaye, & O’Malley, 1996). When students interact with each other, the result is an increase in potential for cognitive improvement

(Lai, 2011). For example, Cross and Paris (1988) used group discussions about the use of reading strategies as a critical component of their Informed Strategies for Learning curriculum.

Kramarski and Mevarech (2003) observed a higher quality of discourse among students who worked together, and found that students who engaged in cooperative learning expressed their mathematical ideas in writing more competently than students who worked alone. Kuhn and

Dean (2004) maintained that social discourse can help students to “interiorize” methods of explaining and elaborating, which have been linked to improvements in learning outcomes.

Some other pedagogical suggestions for developing metacognition include having students express their reasoning, beliefs, and concepts in a visible format (Hennessey, 1999), through the use of carefully constructed conceptual or mental models. Teachers should also promote their students’ general awareness of metacognition by modeling metacognitive abilities

during class. One possible way of doing this is to “think aloud” (Kramarski & Mevarech, 2003;

Martinez, 2006; Schraw, 1998). The affective and motivational aspects of metacognition, such as goal orientation, learning attribution, and self-efficacy, should also be stressed (Schraw, 1998).

As Schraw further articulated, students may already have sufficient metacognitive knowledge and skills, but fail to put them to use. Successful students generally have a higher level of self- efficacy, attribute their success to factors they can control (such as effort and selection of strategies), and refuse to give up in the face of obstacles and difficult circumstances (Schraw,

1998).

One of the challenges in assessing metacognition is that it is not directly observable in students (Sperling et al., 2002). Some investigators have argued that self-report assessment methods (such as rating scales or questionnaires that ask students to describe their strategies) rely excessively on verbal ability. To address this concern, such rating scales and questionnaires can be constructed in formats that do not presuppose major levels of verbal ability or allow the students to describe their strategies in their own words. Schraw and Moshman (1995) preferred verbal report methods because they arguably provide access to the very aspects of thinking that cannot be directly observed.

Because students are not always aware of their cognitive knowledge and self-monitoring, a “think-aloud” method of assessing metacognition might underestimate their metacognitive abilities (Lai, 2011). This further complicates efforts to assess metacognitive skills. Assessment of metacognition is also challenging because of the complexity of the metacognitive process itself, which involves both cognitive knowledge and cognitive regulation (Lai, 2011). Such assessment efforts are further confounded by the various forms of cognitive knowledge

(declarative, procedural, and conditional) and the multiple examples of cognitive regulation

(planning, regulating, monitoring, and evaluating).

One way of dealing with the complexity of metacognition is to focus on specific metacognitive skill sets during the assessment process. An example is the assessment of students’ self-monitoring by examining their allocation of study time. If students receive facts, rules, and other written information to memorize, and then allocate more study time to learning the more difficult materials, the students are displaying signs of effective self-monitoring skills

(Lai, 2011).

Notwithstanding the complexity of metacognition, some investigators have attempted to assess both general and specific aspects of the process. In this connection, a commonly used assessment technique is the self-report questionnaire or rating scale. Kramarski and Mevarech

(2003) utilized a metacognitive questionnaire that was designed to measure general metacognition and mathematics strategies (a form of domain-specific metacognition). They gave students a list of strategies and then asked them to state whether and how frequently they made use of the strategies, using a five-point Likert scale varying from “never” to “always.” The present study also used a questionnaire approach with a similar scale of responses.

Other studies have made the effort to assess metacognition in ways that are more linked to in-school learning activities. As an example, Hennessey (1999) examined metacognition in the realm of school science by having elementary school students work in collaborative groups. The students were required to represent their scientific ideas graphically and were expected to engage in various activities such as stating their own beliefs, looking for consistency and commonalities, and distinguishing between understanding an idea and believing in its truth. Also, Whitebread

et al. (2009) constructed an observational checklist to measure metacognition and self-regulation in kindergarten children. The checklist identified a variety of verbal and nonverbal student behaviors that the investigators viewed as representing metacognitive knowledge and regulation.

The teachers coded metacognitive events that they observed during individual and group learning activities by rating each student’s behavior on a four-point Likert scale varying from “always” to

“never.” This checklist resulted in rather high reliability, with agreement between raters ranging from 66% to 96%. Some investigators prefer observational techniques such as these because observational approaches record the learners’ actual behaviors and thus enable nonverbal behaviors to be noticed. In addition, observational methods give the investigators the chance to record social pressures and motivations that may help in the acquisition of metacognitive abilities (Whitebread et al., 2009).

In a recommendation that recalls the use of bibliotherapy to improve mathematics attitudes, Perry (1998) indicated that writing activities (such as planning, drafting, editing, and revising) all create substantial opportunities for improvements in self-regulated learning.

In addition to the studies referenced above, many other research results on metacognition in the general context of mathematics education can be found in the existing literature. It is useful to note some examples of these other results to gain a further perspective on how the present study can serve to augment or embellish the existing findings.

Efklides, Kiorpelidou, and Kiosseoglou (2006), in experiments with 381 junior high school mathematics students, found that the use of worked-out examples (i.e., examples presenting problems and models of how an expert would solve them) has the potential for affecting students’ estimates of their solution correctness and feelings of confidence. Based on

a problem-solving study involving 71 secondary school students, Veenman (2006) concluded that both intellectual and metacognitive skills are important in mathematics learning and that metacognition outweighed intelligence as a predictor of mathematics learning performance. In another study on mathematical problem solving, Focant, Grégoire, and Desoete (2006) showed that goal setting, planning, and control strategies are extensively used by fifth graders during the process of solving mathematics problems.

The importance of metacognitive instruction gains further support from the studies conducted by Mevarech, Tabuk, and Sinai (2006). Working with 100 eighth-grade algebra students, the investigators demonstrated that students who completed metacognitive training outperformed their counterparts on both algebraic word problems and mathematical modeling.

Further, the significance of belief systems forms the basis of the research of Eynde, De Corte, and Verschaffel (2006) that analyzed the nature and role of Flemish junior high school students’ beliefs regarding mathematical learning and problem solving. Said study of 365 students revealed that those who were aware of and confident in their mathematical thought processes and abilities were more likely to believe in the relevance of mathematics.

Taken as a whole, the research noted in the above literature review presents a portrait of the factors influencing attitudes towards mathematics, efforts made to introduce high school students to number theory, and the basic concepts and effects of metacognition and how to develop and assess metacognitive ability. A general limitation of the noted research is that it does not directly address whether or not high school students’ study of number theory (conducted independently or collaboratively) can have an effect on mathematical attitudes, mathematical curiosity, and metacognition. Students in high school may indeed have inherent metacognitive

skills that have hitherto remained dormant. They may also have the potential for developing more positive attitudes towards mathematics, as well as a healthy inquisitiveness about the subject. An exposure to some of the startling and intriguing concepts of number theory could be the spark that ignites a set of untapped skills and potentialities in the students.

Chapter III

METHODOLOGY

This chapter describes the methods and procedures used by the investigator to answer the four research questions listed in Chapter I. Each question lent itself to a similar research method that elicited qualitative information from subjects without relying very heavily on statistical computations.

The first site for the Main Study was a Catholic high school in New York City. The high school was selected with the help of a clergyman in the investigator’s neighborhood. After identifying the appropriate contact person at the high school, the investigator explained to him the nature of the research and sent him copies of the questionnaire and Interview Script. The mathematics department chairperson of the high school subsequently expressed interest in having the investigator conduct the research at the school, subject to all necessary approvals from the applicable Institutional Review Board.

The other site for the Main Study was a competitive high school in Dalian, China. This school is one of the most exclusive and highly ranked high schools in Dalian. Its student body consists of several thousand students, many of whom go on to prestigious colleges in China, the

United States, and other countries. The investigator secured contacts at this high school through professional networking. A Mandarin translation of the relevant research materials was used for the student participants from this high school.

The research study included a Preliminary Trial involving three high school students, and the Main Study for which 37 high school students (from the above-referenced high schools) formed the target group of subjects. The investigator used no control group. The Preliminary

Trial took place about four weeks before the start of the Main Study. For both the Preliminary

Trial and the Main Study, the subjects were male or female students in the 11th or 12th grade.

With the assistance of a local teacher, the investigator recruited the subjects for the Preliminary

Trial from groups of high school students living in the vicinity of the investigator’s principal residence, through discussions with these students and their parents and/or legal guardians outside of formal school settings. The subjects in the Preliminary Trial were students who attended schools other than the high schools selected for the Main Study. The subjects in the

Main Study were students from the high schools chosen as the research sites. Four of the students in the Main Study were from the New York City high school, and the remaining 33 students in the Main Study were from the high school in China. All student participants freely volunteered to participate in the research, with the approval of their parent(s) or legal guardian(s).

The investigator obtained the answer to Research Question 1 through the use of Part A of a three-part written questionnaire (which was administered to all subjects in the Preliminary Trial as well as the Main Study). On said questionnaire, students listed their age, gender, and class level, but not their names. Part A of the questionnaire served to measure the students’ then current metacognitive abilities as well as their existing attitudes towards mathematics and their propensities for inventing, discussing, and exploring mathematical problems and ideas. The first

30 questions (designed to gauge metacognitive abilities and tendencies) have as their primary basis a questionnaire from Panaoura, Philippou, and Christou (2001), but they also borrow from

related inventories of Fortunato, Hecht, Tittle, and Alvarez (1991), Schraw and Sperling-

Denisson (1994), and Sperling, Howard, and Murphy (2002). The remaining few questions in

Part A of the questionnaire served to measure the students’ existing attitudes towards mathematics and their levels of curiosity relative to mathematics in general. The investigator scored Part A of the questionnaire as follows: A = Always = 5; F = Frequently = 4;

S = Sometimes = 3; R = Rarely = 2; N = Never = 1.

Using this scoring method, the investigator obtained a Part A Metacognition Score based on questions 1-30, and a Part A Mathematics Attitude/Curiosity Score based on questions 31-35.

The highest possible Part A Metacognition Score would be 150 (i.e., 30 x 5), and the highest possible Part A Mathematics Attitude/Curiosity Score would be 25 (i.e., 5 x 5).

The investigator obtained the answer to Research Question 2 through the use of Part A of the three-part written questionnaire referenced and explained above.

The investigator obtained the answer to Research Question 3 through the use of Parts B and C of the three-part written questionnaire referenced above (and through the use of one-hour interviews with the three students in the Preliminary Trial and 10 selected students from the

Main Study). Part B of the three-part questionnaire consists of eight problems in the field of elementary number theory, which the students worked on independently, collaboratively, and/or using any research materials they chose. Part B presents a much wider exposure to number theory than one would find in New York State Regents examinations in mathematics. Some of these problems were purposefully written to prompt the students to evaluate their own thought processes, to challenge the investigator on what the problems were really asking, and to improve or rewrite the problems in a more meaningful way. The investigator had hoped that the students would rewrite the proof requirement of problem 6 as follows:

“Show that gcd (gcd (a,b), gcd (a,b) ) = gcd (a,b).”

The investigator had also hoped that the students would attempt to rewrite the definitions of perfect and multiperfect numbers in problem 1.

Part C of the three-part questionnaire is a revised version of the questions used in Part A.

The Part C questions measured the students’ metacognitive potential, their mathematical attitudes, and their levels of inquisitiveness in the context of the elementary number theory problems which they attempted while working on Part B. (Since it was preferred that the school staff not know which specific students were engaged in the study, the investigator instructed the students to approach question 30 on Part C of the questionnaire as if they were free to inform their teachers of their study participation.) The scoring for Part C of the questionnaire followed the same process used for Part A. Using this scoring method, the investigator obtained a Part C

Metacognition Score based on questions 1-30 from Part C, and a Part C Mathematics

Attitude/Curiosity Score based on questions 31-35 from Part C.

The investigator conducted interviews through the use of an Interview Script, which he designed to gather more information on the thought processes of the students relative to mathematical attitudes, mathematical curiosity, and metacognitive tendencies.

The investigator obtained the answer to Research Question 4 using Parts B and C of the above-referenced three-part written questionnaire and analyzing the interviews previously mentioned.

The students first worked on Parts A and B of the three-part questionnaire. They worked on Part C of the three-part questionnaire after they completed their responses to Parts A and B.

After receiving the completed questionnaire components from the students, the investigator reviewed the Part B responses, and then compared the Part A Metacognition Scores

with the Part C Metacognition Scores. Similarly, the investigator compared the Part A

Mathematics Attitude/Curiosity Scores with the Part C Mathematics Attitude/Curiosity Scores.

These comparisons enabled the investigator to determine if the Part B number theory exercises appeared to have caused some positive changes in metacognitive functioning, mathematical curiosity, and/or attitudes towards mathematics.

For the Preliminary Trial, the investigator interviewed all three students. For the Main

Study, if a student provided correct answers to at least five of the eight number theory problems in the problem set (Part B of the questionnaire), the investigator considered said student as having done very well on the problem set. If a student did not provide correct answers to at least five of the eight number theory problems, the investigator considered said student as not having done very well on the problem set. The investigator then attempted to select five of the students who did very well and five of the students who did not do very well on the problem set, and arranged interviews with these students after receiving the appropriate parental consents. If the above selection method became infeasible because not enough students did very well on the problem set (or for any other reason), the investigator chose to interview students whose written responses to the problem set indicated they made a strong effort to solve the problems. In any event, the investigator attempted to interview five male students and five female students. He conducted each interview at least one week after the interviewee completed and submitted his or her three-part questionnaire. The investigator prepared the Interview Script with questions that asked the students to provide, in their own words, details about their mathematics attitudes, beliefs, and thought processes. The context of the interview questions consists of the students’ work on the number theory problem set in Part B of the questionnaire, as well as the students’ mathematics activities before and after their work on the problem set. The interviews served to

obtain more information from the selected students on their experiences with the number theory problem set and any resultant changes in their mathematics attitudes, beliefs, and metacognitive processes.

All parts of the three-part questionnaire and the Interview Script are included in

Appendices A and B, respectively.

For subjects in the Main Study, each student participant earned a Certificate of

Participation, which the investigator filled out to evidence the student’s involvement in the research (listing the student’s name if he or she was interviewed, or the student’s questionnaire code number if the student was not interviewed and if the student’s parent(s) and/or legal guardian(s) so preferred). A separate Certificate of Participation applied to those students who completed both the three-part questionnaire and the interview sessions. The three Preliminary

Trial participants and the first four participants in the Main Study had the opportunity to use the name of the investigator (at their own discretion) as a reference for their college applications or for any work-related application processes. The investigator gave Consent for Reference Forms to the three students in the Preliminary Trial, to the first four students in the Main Study, and to their parents and/or legal guardians for this purpose. The Certificate of Participation Templates and Consent for Reference Form appear in Appendix C.

The answers to the questionnaires, as well as the anonymous interview responses, form a part of this dissertation. The investigator obtained such answers to the questionnaires (as well as the related signed forms) via the U.S. or international mail, or by delivery from the students’ parent(s) or legal guardian(s). None of the students appeared by name in any of the final written research results. The questionnaires had the codes Q1, Q2, Q3, ….. . Students did not provide their names on the questionnaires, but all students completing the questionnaires needed to

provide the phone numbers of their parent(s) or legal guardian(s). This information enabled the investigator to contact the students, through their parent(s) or legal guardian(s), to set up the interviews. The investigator has no intention to disclose the names, parent(s)’ or legal guardian(s)’ phone numbers, and name-specific interview results of the student interviewees to anyone, except to the extent necessary for the investigator to serve as a reference for the three students in the Preliminary Trial and the first four students in the Main Study (subject to advance receipt of all necessary consents). The investigator destroyed the records of such names and phone numbers at the conclusion of his final dissertation defense. However, he retained any such records that would enable him to serve as a reference for the three students in the Preliminary

Trial and the first four students in the Main Study (subject to advance receipt of all necessary consents for this disclosure using the Consent for Reference Form in Appendix C).

The investigator followed the data storage processes necessary to protect the confidentiality of the student responses, as more fully described in the Informed Consent and

Research Description Documentation for Parents included in Appendix D. No monetary remuneration or audio/videotaping constituted any part of the research study. For the Main

Study, the completion of the three-part questionnaire took place in the students’ classrooms or other facilities within the selected high schools, or at other locations as chosen by the students and their parent(s) and/or legal guardian(s). The three students in the Preliminary Trial worked on the three-part questionnaire at whatever locations they and their parent(s) and/or legal guardian(s) preferred. The interviews with the 10 selected students in the Main Study and the three students in the Preliminary Trial took place at locations approved beforehand by the parent(s) or legal guardian(s) of said students.

The investigator used the previously referenced Informed Consent template (with modifications reflecting the nature of the research study), Participant’s Rights Documentation,

Assent Form for Minors, and Researcher’s Verification of Explanation, to obtain the necessary informed consents and to describe the nature of the research and the subjects’ rights relative thereto. The investigator also expressed the willingness to answer any questions from the students, their teachers, administrators, parents and/or legal guardians, relative to the nature and potential risks and benefits of the research. The above-referenced templates and forms appear in

Appendix D.

The investigator analyzed the data using the following statistical and/or computational techniques:

The investigator determined the number of students in the Preliminary Trial for whom

the Part C Metacognition Score exceeded the Part A Metacognition Score and the

number of students in the Preliminary Trial for whom the Part C Mathematics

Attitude/Curiosity Score exceeded the Part A Mathematics Attitude/Curiosity Score.

He then determined the number of students in the Preliminary Trial whose interview

responses (or responses to specific corresponding questions on Part A and Part C)

otherwise revealed some positive changes in metacognitive functioning, mathematical

curiosity, and/or general attitudes towards mathematics. He also determined the

number of students in the Preliminary Trial whose interview responses and

questionnaire responses were not consistent relative to changes in metacognitive

functioning and changes in mathematical curiosity and/or general attitudes towards

mathematics. The investigator then compared each of said numbers to the total

number of students in the Preliminary Trial to arrive at percentages.

The investigator similarly determined the number of students in the Main Study for whom the Part C Metacognition Score exceeded the Part A Metacognition Score, and the number of students in the Main Study for whom the Part C Mathematics

Attitude/Curiosity Score exceeded the Part A Mathematics Attitude/Curiosity Score.

He then determined the number of student interviewees in the Main Study whose interview responses (or responses to specific corresponding questions on Part A and

Part C) otherwise revealed some positive changes in metacognitive functioning, mathematical curiosity, and/or general attitudes towards mathematics. He also determined the number of students in the Main Study whose interview responses and questionnaire responses were not consistent relative to changes in metacognitive functioning and changes in mathematical curiosity and/or general attitudes towards mathematics. The investigator then compared these numbers to the total number of students in the Main Study (or to the total number of student interviewees in the Main

Study, as applicable) to arrive at percentages.

The investigator reviewed all such percentages to measure the extent to which the students’ number theory study appeared to have made positive changes in their metacognitive functioning, mathematical curiosity, and/or general attitudes towards mathematics.

Chapter IV

FINDINGS AND RESULTS

Preliminary Trial Summary

The investigator began the research with a Preliminary Trial involving three high school students (one female 12th grader, one male 12th grader, and one male 11th grader). No control group was used in connection with the research. At the request of the investigator, a local teacher recruited the three subjects from groups of high school students living in the vicinity of the investigator’s principal residence, through discussions with these students and their parents and/or legal guardians outside of formal school settings. The investigator delivered all parts of the questionnaire to each of the three students in the Preliminary Trial, all of whom attended three different high schools in New York City. None of the students in the Preliminary Trial attended the high schools selected for the Main Study.

To assess the viability of the study methodology, the investigator spent a substantial amount of time on the Preliminary Trial. The students’ responses in the Preliminary Trial did not suggest that the methodology of the study would be too difficult or burdensome for high school students. The results of the Preliminary Trial revealed that none of the three participating students (or 0.00%) received a Part C Metacognition Score which exceeded said students’ Part A

Metacognition Score, and one (or 33.33%) of said students received a Part C Mathematics

Attitude/Curiosity Score which exceeded the student’s Part A Mathematics Attitude/Curiosity

Score. Two of the students in the Preliminary Trial (or 66.67%) provided responses to specific 46

corresponding questions on Part A and Part C of the questionnaire that revealed some positive changes in metacognitive functioning. One of the students in the Preliminary Trial (or 33.33%) provided responses to specific corresponding questions on Part A and Part C of the questionnaire that revealed some positive changes in mathematical curiosity and general attitudes towards mathematics. One of the students in the Preliminary Trial (or 33.33%) provided interview responses that revealed some positive changes in metacognitive functioning. All three of the students in the Preliminary Trial (or 100.00%) provided interview responses that revealed some positive changes in mathematical curiosity and general attitudes towards mathematics. For two of the students in the Preliminary Trial (or 66.67%), the interview responses and responses to the questionnaire were inconsistent relative to metacognitive enhancement. Those two students provided written responses that suggested metacognitive enhancement, while their interview responses did not support such enhancement. None of the students in the Preliminary Trial (or

0.00%) rewrote any parts of the problems or provided inconsistent interview and questionnaire responses relative to mathematical curiosity and general attitudes towards mathematics.

All three of the students in the Preliminary Trial (or 100.00%) presented some evidence of metacognitive enhancement, increased mathematical curiosity, and/or improvements in mathematics attitudes after working on the number theory problem set. This could be seen by reviewing the students’ responses to Part A and Part C of the questionnaire, their scores attained for said Parts A and C, and/or their interview responses.

A tabular summary of the results for the Preliminary Trial can be found in the

Preliminary Trial Summary in Appendix TT. The students’ scores for Parts A and C of the questionnaire appear in a table in Appendix VV, and the credits earned by the students for Part B problems correctly solved appear in the table in Appendix WW. The investigator provided

Student Q1 with a copy of the answer key at his request after the conclusion of the Preliminary

Trial. Students Q2 and Q3 did not request the answer key after they completed the Part B problems.

Preliminary Trial Interviews—Common Theme

Some particularly interesting interview responses are noted below:

Student Q1 stated: “But it seems that mathematics can be even more interesting than

what is taught in school.”

When asked if she wanted to look through a number theory textbook to find

information about other number theory concepts (after working on the number theory

problems), Student Q2 stated: “Yes, maybe.”

Student Q3 indicated that his attitudes towards mathematics changed since he worked

on the number theory problems. He stated: “There are many more aspects of

mathematics.”

The students’ interview responses, provided after completion of the number theory problem set, reflect a common theme of mathematics being seen as more varied and/or interesting.

Preliminary Trial Details

Student Q1. This student was a male 11th grader who was 16 years of age at the time he completed the questionnaire. His responses to Parts A and C of the questionnaire are included in

Tables Q1A and Q1C, respectively. Tables Q1A and Q1C appear in Appendix F.

Based on the results shown in Table Q1A, Student Q1 received a Part A Metacognition

Score of 112/150 (or 74.67%) and a Part A Mathematics Attitude/Curiosity Score of 14/25 (or

56.00%). These scores reflect a reasonably high level of metacognitive skills, but do not show very positive attitudes towards mathematics or high levels of mathematical curiosity, collaboration or inventiveness.

Of the eight problems included on Part B of the questionnaire, Student Q1 correctly answered four problems and part of another problem. He provided correct answers to problems

2, 5, 7, 8 and part of problem 1. He did not attempt to rewrite any of the problems.

Based on the results shown in Table Q1C, Student Q1 received a Part C Metacognition

Score of 108/150 (or 72.00%) and a Part C Mathematics Attitude/Curiosity Score of 9/25 (or

36.00%).

While Student Q1 performed reasonably well on the Part B problem set, his

Metacognition Score decreased by four points and his Mathematics Attitude/Curiosity Score decreased by five points after having completed the Part B problem set.

Table Q1A/C (also appearing in Appendix F) identifies the corresponding questions from

Parts A and C of the questionnaire, for which Student Q1’s responses evidenced an increase in scores. This table indicates some increase in the following metacognitive activities:

attempts at using study methods or techniques utilized by other successful

mathematics students;

deciding on clear goals;

trying different ways of approaching or learning mathematics;

focusing on data presented;

identifying parts of problems which the student does not understand;

trying to clear up confusion;

thinking about whether or not the main question is being addressed; and

finding different solutions to the same problem.

Student Q1 and his parent agreed to grant the investigator’s request for an interview with

Student Q1. Student Q1’s interview responses are included in Appendix F and are summarized in capital letters. Those of his responses which are particularly informative for purposes of this study are shown in bold. While some of Student Q1’s responses suggested metacognitive activity

(e.g., “was thinking about my thoughts”), his interview for the most part did not reveal the same level of increased metacognitive functioning as Table Q1 A/C might suggest. Upon further questioning, however, Student Q1 did indicate that for topics such as algebra and trigonometry

(which he encounters every day), he generally knows what to do. For the number theory problems in Part B of the questionnaire, he stated that he needed to do things on the spot. He also stated that one or two of these number theory problems made him think more and he needed a different thought process because most of the problems were so new to him. He further explained that he otherwise used similar approaches for the number theory problems as compared to his work on other mathematics problems.

While this was not evidenced in Table Q1 A/C, some of Student Q1’s interview responses revealed an increase in his level of interest in number theory and mathematics study in general (e.g., “ . . . mathematics can be even more interesting than what is taught in school”;

“Could also learn more from internet searches”; “To learn what number theory is and to look at it more to prepare for school”). As further evidence of an increase in his interest, at the conclusion of the interview (and after having received his signed Certificate of Participation and the investigator’s commitment to serve as one of his references), Student Q1 asked to stay and continue the discussion to go over his answers to the number theory problems in Part B of the questionnaire. He voluntarily continued to discuss the problems with his parent and the

investigator for at least 20 minutes after the conclusion of the scheduled interview, even though this was taking place on a sunny Saturday afternoon. This suggests a positive development in the student’s attitudes towards mathematics study.

The investigator’s overall reaction to Student Q1’s questionnaire and interview responses and behavior is that the number theory problems piqued his interest and developed his curiosity and attitudes towards mathematics. However, the extent to which the problems enhanced his metacognitive functioning remains questionable because some of his responses to Parts A and C of the questionnaire suggest such enhancement while his interview responses do not.

Student Q2. This student was a female 12th grader who was 17 years of age at the time she completed the questionnaire. Her responses to Parts A and C of the questionnaire appear in

Tables Q2A and Q2C, respectively. Both Tables Q2A and Q2C appear in Appendix G.

Based on her responses to Part A of the questionnaire, Student Q2 received a Part A

Metacognition Score of 102/150 (or 68.00%) and a Part A Mathematics Attitude/Curiosity Score of 5/25 (or 20.00%). These scores reflect a moderate level of metacognitive skills, but do not show positive attitudes towards mathematics or even moderate levels of mathematical curiosity, collaboration or inventiveness.

Of the eight problems included on Part B of the questionnaire, Student Q2 answered problem 2 correctly, received partial credit for problems 4 and 5, and no credit for problems 1, 3,

6, 7, and 8. She did not attempt to rewrite any of the problems.

Based on her responses to Part C of the questionnaire, Student Q2 received a Part C

Metacognition Score of 101/150 (or 67.33%) and a Part C Mathematics Attitude/Curiosity Score of 5/25 (or 20.00%).

Student Q2 had difficulty with the Part B problem set, and apparently did not expend much effort on problems 1, 6, 7, and 8. Her Metacognition Score decreased by only one point and her Mathematics Attitude/Curiosity Score remained the same after having completed the Part

B problem set.

Table Q2 A/C (also appearing in Appendix G) identifies the six corresponding questions from Parts A and C of the questionnaire, for which Student Q2’s responses evidenced an increase in scores. This table indicates some increase in the following metacognitive activities:

learning more about a topic if one already has some knowledge about it;

deciding on clear goals;

thinking of one’s own examples or illustrations of concepts;

thinking about whether or not the main question is being addressed;

finding different solutions to the same problem; and

knowing how one performed as soon as work on the problems was finished.

Student Q2 and her parent agreed to grant the investigator’s request for an interview of

Student Q2. This interview took place via telephone at the request of Student Q2’s parent.

Student Q2’s interview responses are included in Appendix G and are summarized in capital letters. Her responses that are particularly informative for this study are shown in bold.

Student Q2 indicated that she is not a “math person,” but in her interview she expressed some interest in looking at number theory textbooks to find out how to do the number theory problems. She also stated that the number theory problems made her a little curious. While her interview responses revealed that she analyzed and reflected on the number theory problems, planned her solution methods, and checked her work by re-reading, Student Q2 indicated that she engaged in more of these activities while in school. The number theory problems did not change

her view of mistakes as being part of learning, nor did the problems alter her approach to changing strategies in mathematics problem solving. Student Q2 also supplemented her responses to Part A of the questionnaire, when she indicated that she frequently evaluates her progress on new mathematics lessons (and did the same on the number theory problems).

Student Q2’s interview responses suggest that the number theory problems developed her curiosity to a small extent, but did not have a major effect on her attitudes towards mathematics.

As with Student Q1, the extent to which the problems enhanced Student Q2’s metacognitive and/or self-regulated learning abilities is questionable. Student Q2’s responses to Parts A and C of the questionnaire suggest such enhancement while her interview responses do not.

Student Q3. This student was a male 11th grader who was 16 years of age at the time he completed the questionnaire. His responses to Parts A and C of the questionnaire are included in

Tables Q3A and Q3C, respectively. Tables Q3A and Q3C appear in Appendix H.

Based on the results shown in Table Q3A, Student Q3 received a Part A Metacognition

Score of 105/150 (or 70.00%) and a Part A Mathematics Attitude/Curiosity Score of 7/25 (or

28.00%). These scores suggest a moderate level of metacognitive skills, but do not show positive attitudes towards mathematics or any appreciable levels of mathematical curiosity, collaboration or inventiveness.

Of the eight problems included on Part B of the questionnaire, Student Q3 answered one part of problem 5 correctly. He did not appear to have expended much effort on the remaining problems. He did not attempt to rewrite any of the problems, although he reported he had thought about doing so.

Based on the results shown in Table Q3C, Student Q3 received a Part C Metacognition

Score of 99/150 (or 66.00%) and a Part C Mathematics Attitude/Curiosity Score of 14/25 (or

56.00%).

Student Q3’s Metacognition Score decreased by six points and his Mathematics

Attitude/Curiosity Score doubled after having completed the Part B problem set. As noted above, he did not perform well on the Part B problem set.

Table Q3A/C (also appearing in Appendix H) identifies the corresponding questions from

Parts A and C of the questionnaire, for which Student Q3’s responses evidenced an increase in scores. This table indicates some increase in the following metacognitive activities:

learning more about a topic after having some existing knowledge of it;

drawing sketches to improve understanding;

deciding on clear goals;

evaluating progress;

questioning whether one learned anything new and valuable;

thinking about the most important elements of the problem;

trying different ways of approaching or learning mathematics;

asking for help in the face of confusion;

using solutions of similar problems;

thinking about whether or not the main question is being addressed; and

finding different solutions to the same problem.

Table Q3A/C also provides evidence of increases in mathematical curiosity and inventiveness, based on Student Q3’s responses for the following activities:

flipping through pages of mathematics books while browsing in a bookstore or

library;

thinking of new mathematics problems to work on;

preparing mathematics tests, quizzes, and puzzles; and

finding mathematics problems that have never been solved.

Student Q3 and his parent agreed to grant the investigator’s request for an interview with

Student Q3. The investigator conducted such interview via phone. Student Q3’s interview responses are included in Appendix H and are summarized in capital letters. His responses that are especially informative for this study are shown in Appendix H in bold.

Student Q3 appeared to be interested in the interview process. He indicated that (in contrast with other mathematics courses) he used a lot of “guess and check” strategies on the number theory problems. He also stated that since working on the number theory problems, he has seen there are many more aspects to mathematics (and he can improve in mathematics by studying more). Towards the end of the interview, he mentioned that the number theory problems made him curious. He reportedly explained to himself the procedures necessary for solving the number theory problems, but had not previously engaged in such a process to a great extent. He stated that while doing the number theory problems, he thought more about his own thought processes (but could not explain why).

The interview with Student Q3 suggests a small increase in levels of curiosity about mathematics (e.g., interview question 16), but not a major improvement in his attitudes towards mathematics. The doubling of his Mathematics Attitude/Curiosity Scores, however, does provide some evidence of such an attitude improvement. This student’s interview responses also suggest an increase in metacognitive functioning (e.g., interview question 13 and statements about

“thinking more about thought processes”), but his Metacognition Score comparisons in Table

Q3A/C suggest even greater levels of enhancement in this area.

Several months after Student Q3 submitted his responses to the questionnaire and completed his interview, his parent advised that Student Q3 wanted to discontinue participation in the study. His parent then clarified that Student Q3’s responses and interview results could be incorporated into this dissertation, but that additional participation on his part would not be possible.

Main Study Summary

The data from the Main Study revealed that 15 of the students (or 40.54%) received a

Part C Metacognition Score that exceeded their Part A Metacognition Score, and 21 of the students (or 56.76%) received a Part C Mathematics Attitude/Curiosity Score that exceeded their

Part A Mathematics Attitude/Curiosity Score. Thirty-six of the students (or 97.30%) provided responses to specific corresponding questions on Part A and Part C of the questionnaire, which revealed some positive changes in metacognitive functioning. Twenty-nine of the students (or

78.38%) provided responses to specific corresponding questions on Part A and Part C of the questionnaire, which revealed some positive changes in mathematical curiosity and general attitudes towards mathematics. Nine of the student interviewees (or 90.00% of said interviewees) provided interview responses that revealed some positive changes in metacognitive functioning.

Ten of the student interviewees (or 100.00%) provided interview responses that revealed some positive changes in mathematical curiosity and general attitudes towards mathematics. For six student interviewees (or 60.00% of said interviewees), the interview responses and responses to the questionnaire were inconsistent relative to metacognitive enhancement. Also, six student

interviewees (or 60.00% of the said interviewees) provided inconsistent interview and questionnaire responses relative to mathematical curiosity and general attitudes towards mathematics.

A significant percentage of the student interviewees provided interview responses that were inconsistent with their questionnaire responses regarding changes in metacognitive functioning, mathematical curiosity, and/or general attitudes towards mathematics. Most of those interviewees, however, provided interview responses that evidenced some positive changes in one or more of the indicated areas. All of the student interviewees appeared to give credible, honest responses to the investigator’s questions. Some students talked about the number theory problems making them think outside-the-box, find easier ways to solve the problems, and do more planning, analyzing, checking, and reflecting. They also mentioned that the problems made them think of the end result before starting their work. Other interview comments were to the effect that mathematics can be more interesting than what is taught in school, and that the number theory problems opened a student’s eyes to new topics. In addition, very high percentages of students provided questionnaire responses suggesting enhancements in specific metacognitive activities and mathematics attitudes.

All but one of the students presented some evidence of metacognitive growth, increased mathematical curiosity, and/or improvements in mathematics attitudes after completing work on the number theory problem set. This can be seen by analyzing the students’ responses to Part A and Part C of the questionnaire, their Part A and Part C scores, and/or their interview responses.

A tabular summary of the results for the Main Study can be found in the Main Study

Summary in Appendix UU. The students’ scores for Parts A and C of the questionnaire are tabulated in Appendix VV, and the credits earned by the students for Part B problems correctly

solved appear in the table in Appendix WW. Eight of the Main Study students (or 21.62%) rewrote one of the proof requirements from Part B, and none of them (or 0.00%) rewrote any of the Part B definitions.

Main Study Interviews—Common Themes

Some interview responses of particular note are as follows:

Student Q4 stated: “The problems were interesting. I want to look into them more.”

Student Q5 stated: “The number theory problems sort of caught my interest, . . . They

made me curious.” He also stated that after working on the number theory problems,

he had the idea of thinking more outside-the-box.

Student Q6 indicated that he enjoyed working on the number theory problems

because they were new and involved things he had never seen before. He also stated

that he did more analyzing, planning, checking, and reflecting while working on the

number theory problems.

Student Q7 responded: “I analyzed the problems more when I was working on the

number theory problems.” She also stated that before seeing the number theory

problems, she thought she knew a lot of math. She said she is curious to learn more

about number theory now.

Student Q8 stated: “I analyzed problems more when I was working on the number

theory problems.”

Student Q9 responded: “The number theory problems can change my thought process

and help me solve math problems more and more quickly.” He also stated that his

attitudes towards mathematics changed since he worked on the number theory

problems, he likes mathematics much more, and mathematics can be very fantastic. 58

Student Q10 stated that the number theory problems made him feel “always curious.”

Student Q11 stated that she did more analyzing, planning, and checking (and possibly

also more reflecting) while working on the number theory problems. She also

indicated that she felt interested in doing Internet searches to learn more about

number theory topics.

Student Q12 stated: “Math is boring in school, but the number theory problems were

different.” This student also stated that she did more analyzing, planning, checking,

and reflecting while working on the number theory problems.

Student Q13 indicated that he does not like math, but he evaluated his thoughts as he

worked on the number theory problems. He further stated that he did not perform that

evaluation in math class.

The common themes emerging from these interview responses are a greater level of curiosity about mathematics and more tendencies towards analysis, planning, checking one’s work, and reflecting.

Main Study Details

Students Q4, Q5, Q6, and Q7 attended the New York City Catholic high school selected as one of the sites for the Main Study. Their responses are presented in order.

Student Q4. This student was a female 11th grader who was 17 years of age when she completed the questionnaire. Her responses to Parts A and C of the questionnaire are included in

Tables Q4A and Q4C, respectively. Tables Q4A and Q4C appear in Appendix I.

Table Q4A reveals that Student Q4 received a Part A Metacognition Score of 100/150 (or

66.67%) and a Part A Mathematics Attitude/Curiosity Score of 10/25 (or 40.00%). These scores

reflect a reasonable level of metacognitive skills, but do not show very positive attitudes towards the study of mathematics or high levels of mathematical curiosity, collaboration or inventiveness.

Of the eight problems included on Part B of the questionnaire, Student Q4 answered almost two correctly. She solved problem 2 and part of problem 5, but otherwise made efforts to prove assertions using substitution. She did not attempt to rewrite any of the problems.

Table Q4C reveals that Student Q4 received a Part C Metacognition Score of 93/150 (or

62.00%) and a Part C Mathematics Attitude/Curiosity Score of 18/25 (or 72.00%).

Student Q4 worked on the Part B problem set, but was not able to solve most of the problems. Her Metacognition Score decreased by seven points, but her Mathematics

Attitude/Curiosity Score nearly doubled after having completed the Part B problem set. This increase in her Mathematics Attitude/Curiosity Score is very interesting within the context of this study.

Table Q4A/C (also appearing in Appendix I) sets forth the corresponding questions from

Parts A and C of the questionnaire, for which Student Q4’s responses evidenced some increase in scores. This table reveals increases in the following metacognitive activities:

wondering about finding easier or quicker ways to solve the problems;

knowing if one was able to solve the problems after reading them;

thinking about whether or not the main question is being addressed; and

finding different solutions to the same problem.

Table Q4A/C also provides evidence of increases in mathematical curiosity, collaboration, and inventiveness, based on Student Q4’s responses for the following activities:

flipping through pages of mathematics books in bookstores or libraries;

thinking of new mathematics problems to work on;

preparing one’s own mathematics tests, quizzes, and puzzles;

finding mathematics problems that have never been solved; and

when with friends, discussing new mathematics concepts learned from working on

the number theory problems.

Student Q4 and her parent granted the investigator’s request for an interview of Student

Q4. This interview was conducted via telephone. Student Q4’s interview responses are included in Appendix I and are summarized in capital letters for ease of reference. Her responses that are particularly informative for this study are shown in bold in Appendix I. (The investigator adopted this same manner of transcribing and highlighting interview responses for all other student-interviewees.)

Some of Student Q4’s interview responses revealed that the student enjoyed working on the number theory problems because they were different from the mathematics she had seen in school. The number theory problems made her slightly curious. She also became interested in doing Internet searches to learn more about number theory, since the number theory problems were interesting and involved concepts she had never heard of before.

Student Q4’s questionnaire and interview responses suggest that the number theory problems noticeably developed her interest and enhanced her curiosity (if not her attitudes) towards mathematics. This is further supported by the near doubling of her Mathematics

Attitude/Curiosity Score. Table Q4A/C seemed to show some enhancement of Student Q4’s metacognitive functioning, but this enhancement was not evidenced by her interview responses.

Student Q5. This student was a male 11th grader who was 16 years of age when he completed the questionnaire. His responses to Parts A and C of the questionnaire are included in

Tables Q5A and Q5C, respectively. Tables Q5A and Q5C appear in Appendix J.

Table Q5A indicates that Student Q5 received a Part A Metacognition Score of 102/150

(or 68.00%) and a Part A Mathematics Attitude/Curiosity Score of 12/25 (or 48.00%). These scores indicate a moderate level of metacognitive skills, but not very positive attitudes towards mathematics nor high levels of mathematical curiosity, collaboration or inventiveness.

Of the eight problems included on Part B of the questionnaire, Student Q5 provided correct answers to problem 2, problem 8, and parts of problems 1 and 5. He did not attempt to rewrite any of the problems.

Table Q5C shows that Student Q5 obtained a Part C Metacognition Score of 89/150 (or

59.33%) and a Part C Mathematics Attitude/Curiosity Score of 15/25 (or 60.00%).

Student Q5 attempted to prove assertions by using examples, and this prevented him from performing better on the Part B problem set. His Metacognition Score decreased by 13 points, and his Mathematics Attitude/Curiosity Score increased by three points after having completed the Part B problem set.

Table Q5A/C (also appearing in Appendix J) sets forth the corresponding questions from

Parts A and C of the questionnaire, for which Student Q5’s responses evidenced some increase in scores. This table reveals increases in the following metacognitive activities:

deciding on clear goals before trying to solve problems;

knowing how one performed on mathematics problems after finishing work on them;

and

believing that mathematics concepts vary in level of difficulty.

Table Q5A/C also provides evidence of increases in mathematical curiosity, collaboration, and inventiveness, based on Student Q5’s responses for the following activities:

flipping through mathematics books while browsing in bookstore or library;

thinking of new mathematics problems to work on; and

finding mathematics problems that have never been solved.

Student Q5 and his parent granted the investigator’s request for an interview with Student

Q5. This interview was conducted via telephone. Student Q5’s interview responses are included in Appendix J.

Some of Student Q5’s interview responses revealed that the number theory problems were more interesting than “normal problems with numbers.” The student found the number theory problems to be more abstract and more involved with theories. He had never seen some of the concepts presented by these problems. He indicated that as he worked on the number theory problems, he thought of a different way to reach the answer without using a formula, changed his thought processes and strategies for at least one of the problems, drew pictures to aid his understanding, and thought about procedures he needed to follow.

Student Q5 also indicated that after working on the number theory problems, he thought of “bouncing ideas off one of his friends” to improve his own mathematics ability. He further stated that he engaged in more analyzing, planning, checking, and reflecting as he worked on the number theory problems because they were different from the “normal math” he already knew.

The problems “opened his eyes to new topics” and made him want to learn more about perfect numbers and other topics he had never seen before in mathematics. He also expressed interest in doing Internet searches on perfect numbers.

Student Q5’s questionnaire and interview responses suggest that the number theory problems appreciably enhanced his interest and curiosity (if not his attitudes) towards mathematics. Student Q5’s work on the number theory problems also appeared to have enhanced his metacognitive functioning, with respect to the three metacognitive activities listed in Table

Q5A/C. This metacognitive enhancement is evidenced to some extent by the student’s interview responses.

Student Q6. This student was a male 11th grader who was 17 years of age when he completed the questionnaire. His responses to Parts A and C of the questionnaire are included in

Tables Q6A and Q6C, respectively. Tables Q6A and Q6C appear in Appendix K.

Table Q6A indicates that Student Q6 received a Part A Metacognition Score of 108/150

(or 72.00%) and a Part A Mathematics Attitude/Curiosity Score of 6/25 (or 24.00%). These scores indicate a moderate level of metacognitive skills, but not very positive attitudes towards mathematics nor high levels of mathematical curiosity, collaboration or inventiveness.

Of the eight problems included on Part B of the questionnaire, Student Q6 correctly answered problems 1, 2, and 5, and part of problem 4. He appeared to have expended considerable effort on the problems, but had a tendency to use examples as a method of proof.

He also did not attempt to rewrite any of the problems.

Table Q6C shows that Student Q6 obtained a Part C Metacognition Score of 112/150 (or

74.67%) and a Part C Mathematics Attitude/Curiosity Score of 11/25 (or 44.00%).

Student Q6 would likely have performed better on the Part B problem set if he had not tried to prove assertions via specific examples. His Metacognition Score increased by four points, and his Mathematics Attitude/Curiosity Score nearly doubled after having completed the

Part B problem set.

Table Q6A/C (also appearing in Appendix K) sets forth the corresponding questions from

Parts A and C of the questionnaire, for which Student Q6’s responses evidenced some increase in scores. This table reveals increases in the following metacognitive activities:

learning more about a mathematics topic if one has a real interest in it;

drawing sketches, pictures or diagrams to improve understanding of a mathematics

concept;

deciding on clear goals before trying to learn a new mathematics concept;

evaluating progress as one is studying a new mathematics lesson;

wondering if one could have found an easier or quicker way to solve a mathematics

problem;

trying different ways of learning mathematics;

thinking of one’s own examples or illustrations of a mathematics concept to help in

understanding it;

having one’s own ways of remembering what one has learned in mathematics;

reading a mathematics problem several times to make sure one understands it;

trying to find some other, different solutions to a mathematics problem after having

found one solution to it; and

knowing how one performed on a mathematics test before receiving teacher’s

feedback.

Table Q6A/C also provides evidence of increases in mathematical curiosity and inventiveness, based on Student Q6’s responses for the following activities:

thinking of new mathematics problems to work on;

enjoying preparing one’s own mathematics tests, quizzes, and puzzles; and

enjoying finding mathematics problems that have never been solved.

Student Q6 and his parent agreed to the investigator’s request for an interview with

Student Q6. This interview was conducted via telephone. Student Q6’s interview responses are included in Appendix K.

Some of Student Q6’s interview responses revealed that his attitudes towards mathematics remained the same after working on the number theory problems. He further stated that the number theory problems were interesting, opened his eyes to new topics, and prompted him to try to learn more about mathematics topics he had not previously seen.

Student Q6’s questionnaire and interview responses suggest that the number theory problems enhanced his interest and curiosity about mathematics to a major extent. The doubling of Student Q6’s Mathematics Attitude/Curiosity Score provides further support of this, as do his interview responses. Student Q6’s work on the number theory problems also appeared to have enhanced his metacognitive functioning, at least with respect to the 11 metacognitive activities listed in Table Q6A/C. Some of Student Q6’s interview responses also support this, especially his statements that he thought unconsciously of procedures for solving the problems and reflected more on the number theory problems because they were not “normal math.”

Student Q7. This student was a female 12th grader who was 17 years of age when she completed the questionnaire. Her responses to Parts A and C of the questionnaire are included in

Tables Q7A and Q7C, respectively. Tables Q7A and Q7C appear in Appendix L.

Table Q7A indicates that Student Q7 received a Part A Metacognition Score of 93/150

(or 62.00%) and a Part A Mathematics Attitude/Curiosity Score of 8/25 (or 32.00%). These scores indicate a moderate level of metacognitive skills, but not very positive attitudes towards mathematics study nor high levels of mathematical curiosity, collaboration or inventiveness.

Of the eight problems included on Part B of the questionnaire, Student Q7 provided correct responses for problem 8, most of problem 1, and part of problem 5. She tried to use examples as a method of proof for the remainder of the problems, and did not attempt to rewrite any parts of the problem set.

Table Q7C shows that Student Q7 obtained a Part C Metacognition Score of 100/150 (or

66.67%) and a Part C Mathematics Attitude/Curiosity Score of 5/25 (or 20.00%).

Student Q7 would likely have performed better on the Part B problem set if she had not tried to prove assertions by using examples. Her Metacognition Score increased by seven points and her Mathematics Attitude/Curiosity Score decreased by three points, after having completed the Part B problem set.

Table Q7A/C (also appearing in Appendix L) sets forth the corresponding questions from

Parts A and C of the questionnaire, for which Student Q7’s responses showed some increase in scores. This table reveals increases in the following metacognitive activities:

attempting to use study methods that have been used by successful mathematics

students while studying mathematics;

drawing sketches, pictures or diagrams to improve understanding of a mathematics

concept;

asking oneself if one has learned anything new and valuable after completing

mathematics homework;

having a better understanding of a problem by writing the data presented by the

problem;

wondering if one could have found an easier or quicker way to solve a mathematics

problem;

trying different ways of learning mathematics;

thinking of one’s own examples or illustrations of a mathematics concept to help in

understanding it;

trying to use solutions of similar mathematics problems that one encountered in the

past;

reading a mathematics problem several times to make sure one understands it;

knowing the nature of the difficulty one is experiencing on a mathematics problem;

and

believing that mathematical problem-solving techniques vary in their level of

difficulty.

Student Q7 and her parent agreed to the investigator’s request for an interview with

Student Q7. This interview was conducted via telephone. Student Q7’s interview responses are included in Appendix L.

Student Q7’s interview responses revealed that her attitudes towards mathematics did not change after she worked on the number theory problems. She further indicated that the number theory problems were more interesting than her other mathematics coursework, they required some different methods, and she had to analyze and think more about the number theory problems. After working on the number theory problems, she wanted to look through a number theory textbook.

Student Q7’s interview responses suggest that the number theory problems increased her level of curiosity about number theory and made her interested in learning more about it. This is not reflected in the comparison of her Mathematics Attitude/Curiosity Scores.

Student Q7’s work on the number theory problems also appeared to have enhanced her metacognitive functioning with respect to the 11 metacognitive activities listed in Table Q7A/C.

This is not reflected in the comparison of her Metacognition Scores that increased only slightly after she worked on the number theory problems. Some of her interview responses did indicate a

greater level of analysis during her work on the problems, which could indicate some metacognitive enhancement.

Students Q8 through Q40 attended a high school in Dalian, China; this school was also selected as one of the sites for the Main Study. These students’ responses are presented in order.

Student Q8. This student was a male 11th grader and was about 17 years of age when he completed the questionnaire. His responses to Parts A and C of the questionnaire are included in

Tables Q8A and Q8C, respectively. Tables Q8A and Q8C appear in Appendix M.

Table Q8A indicates that Student Q8 received a Part A Metacognition Score of 101/150

(or 67.33%) and a Part A Mathematics Attitude/Curiosity Score of 9/25 (or 36.00%). These scores indicate a moderate level of metacognitive skills, but not very positive attitudes towards mathematics nor high levels of mathematical curiosity, collaboration or inventiveness.

Student Q8 answered Part B problems 2, 3, 5, and 7 correctly. He did not rewrite the proof requirements for any of the problems.

Table Q8C shows that Student Q8 obtained a Part C Metacognition Score of 106/150 (or

70.67%) and a Part C Mathematics Attitude/Curiosity Score of 9/25 (or 36.00%).

Student Q8’s Metacognition Score increased by five points, and his Mathematics

Attitude/Curiosity Score did not change after the student completed the Part B problem set.

Table Q8A/C (also appearing in Appendix M) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q8’s responses evidenced some increase in scores. This table reveals increases in the following metacognitive activities:

using study methods used by successful mathematics students;

being able to learn more about a mathematics topic if one already knows something

about it;

being able to learn more about a mathematics topic if one has a real interest in it;

deciding on clear goals before trying to learn a new mathematics topic;

evaluating one’s progress while studying a new mathematics lesson;

asking oneself if one has learned anything new and valuable after completing

mathematics homework;

thinking about the most important elements of a mathematics problem to make sure

one really understands them;

having one’s own ways of remembering what one has learned in mathematics;

writing the data presented by a problem so as to have a better understanding of the

problem;

asking oneself questions while trying to solve a mathematics problem; and

knowing how one has performed on a mathematics test before receiving the teacher’s

feedback.

Table Q8A/C also provides evidence of increases in mathematical curiosity and inventiveness, based on Student Q8’s responses for the following activities:

flipping through the pages of mathematics books in a bookstore or library; and

enjoying preparing one’s own mathematics tests, quizzes, and puzzles.

Student Q8 and his parent agreed to the investigator’s request for an interview with

Student Q8. This interview was conducted via Skype. Student Q8 had sufficient knowledge of the English language to enable the interview to proceed as planned. His interview responses are included in Appendix M.

A few of Student Q8’s interview responses revealed that the number theory problems made him curious and his attitudes towards mathematics changed positively after he worked on

the problems. This is interesting given that his Mathematics Attitude/Curiosity Score remained the same after he completed the problem set.

Student Q8’s work on the number theory problems also appeared to have enhanced his metacognitive functioning with respect to the 11 metacognitive activities listed in Table Q8A/C.

Some of Student Q8’s interview responses support this, especially his statement that he engaged in more problem analysis while working on the number theory problems. It is also interesting to note that a comparison of Student Q8’s Part A and Part C Metacognition Scores indicates metacognitive enhancement for this student.

Student Q9. This student was a male 11th grader and was about 17 years of age when he completed the questionnaire. His responses to Parts A and C of the questionnaire are included in

Tables Q9A and Q9C, respectively. Tables Q9A and Q9C appear in Appendix N.

Table Q9A indicates that Student Q9 received a Part A Metacognition Score of 108/150

(or 72.00%) and a Part A Mathematics Attitude/Curiosity Score of 15/25 (or 60.00%). These scores indicate a reasonable level of metacognitive skills, somewhat positive attitudes towards mathematics, and moderate but not high levels of mathematical curiosity, collaboration, and inventiveness.

Student Q9 answered Part B problem 2 correctly. He also received partial credit for problem 1, but did not rewrite the proof requirements for any of the problems.

Table Q9C shows that Student Q9 obtained a Part C Metacognition Score of 96/150 (or

64.00%) and a Part C Mathematics Attitude/Curiosity Score of 14/25 (or 56.00%). His

Metacognition Score decreased by 12 points, and his Mathematics Attitude/Curiosity Score decreased by one point after he completed the Part B problem set.

Table Q9A/C (also appearing in Appendix N) sets forth the corresponding questions from

Parts A and C of the questionnaire, for which Student Q9’s responses evidenced some increase in scores. This table reveals increases in the following metacognitive activities:

using techniques that have been used by successful mathematics students;

deciding on clear goals before trying to learn a new mathematics concept;

thinking about the most important elements of a problem after working on it;

writing the data presented by a problem;

wondering if one could have found an easier or quicker way to solve a mathematics

problem;

reading a mathematics problem several times to make sure one understands it;

trying to find some other, different solutions to a mathematics problem after having

found one solution to it; and

knowing how one performed on a mathematics test before receiving teacher’s

feedback.

Table Q9A/C also provides evidence of increases in mathematical curiosity, based on

Student Q9’s responses for the following activities:

flipping through the pages of mathematics books while browsing in a bookstore or

library;

discussing new mathematics concepts with friends; and

enjoying finding mathematics problems that have never been solved.

Student Q9 and his parent agreed to the investigator’s request for an interview with

Student Q9. This interview was conducted via Skype. Student Q9 had sufficient knowledge of

the English language for purposes of the interview. Student Q9’s interview responses are included in Appendix N.

Some of Student Q9’s interview responses revealed that his attitudes towards mathematics reached more positive levels after he worked on the number theory problems. He stated that he likes mathematics much more after having worked on the number theory problems, the problems made him curious, and mathematics can be fantastic. He spent more time than usual discussing the number theory problems with his friends, and thought of some new ideas about the problems during those discussions.

Student Q9’s questionnaire responses also suggest that his work on the number theory problems enhanced his metacognitive functioning with respect to the eight metacognitive activities listed in Table Q9A/C. Some of Student Q9’s interview responses support this conclusion, especially his statements that the number theory problems changed his thought process and helped him solve math problems more quickly. The student further indicated that during this number theory exercise, he analyzed problems, planned solutions, checked his work, and reflected on the problems more so than during his other mathematics classes.

Student Q10. This student was a male 11th grader and was approximately 17 years of age when he participated in this study. His responses to Parts A and C of the questionnaire are included in Tables Q10A and Q10C, respectively. Tables Q10A and Q10C appear in Appendix

Table Q10A indicates that Student Q10 received a Part A Metacognition Score of

113/150 (or 75.33%) and a Part A Mathematics Attitude/Curiosity Score of 11/25 (or 44.00%).

These scores suggest a reasonable level of metacognitive abilities, but less than positive attitudes

towards mathematics and comparatively low levels of mathematical curiosity, collaboration, and inventiveness.

Student Q10 answered Part B problems 2 and 8 correctly, and received partial credit for problems 4 and 5. He did not rewrite the proof requirements for any of the problems.

Table Q10C shows that Student Q10 obtained a Part C Metacognition Score of 105/150

(or 70.00%) and a Part C Mathematics Attitude/Curiosity Score of 10/25 (or 40.00%). The Part C

Metacognition Score decreased by eight points and the Part C Mathematics Attitude/Curiosity

Score decreased by one point, as compared to this student’s corresponding Part A scores.

Table Q10A/C (also appearing in Appendix O) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q10’s responses evidenced some increase in scores. This table shows increases in the following metacognitive activities:

deciding on clear goals before trying to learn a new mathematics concept;

evaluating progress as one is studying a new mathematics lesson;

wondering if one could have found an easier or quicker way to solve a mathematics

problem;

trying different ways of learning mathematics;

asking someone else for help whenever one is confused in mathematics; and

knowing how one performed on a mathematics test before receiving teacher’s

feedback.

Table Q10A/C also provides evidence of increases in mathematical curiosity and inventiveness, based on Student Q10’s responses for the following activities:

flipping through the pages of mathematics books while browsing in a bookstore or

library; and

discussing new mathematics concepts with friends.

Student Q10 and his parent agreed to the investigator’s request for an interview with

Student Q10. This interview was conducted via Skype. Student Q10 understood the English language sufficiently well to proceed with the interview. Student Q10’s interview responses are included in Appendix O.

Student Q10’s interview responses revealed that the number theory problems changed his belief about the discovery of new mathematics concepts and problems. His interview also suggested that the difficulty of the number theory problems made him curious. No other interview responses indicated that the number theory problems enhanced Student Q10’s metacognitive abilities, mathematical curiosity or general attitudes towards mathematics. This is inconsistent with the data shown for this student in Table Q10A/C, which suggests greater metacognitive enhancement and more increases in mathematical curiosity in specified activities.

Student Q11. This student was a female 11th grader and about 17 years of age when she completed the questionnaire. Her responses to Parts A and C of the questionnaire are included in

Tables Q11A and Q11C, respectively. Tables Q11A and Q11C appear in Appendix P.

Table Q11A indicates that Student Q11 received a Part A Metacognition Score of

110/150 (or 73.33%) and a Part A Mathematics Attitude/Curiosity Score of 9/25 (or 36.00%).

These scores indicate a reasonably high level of metacognitive skills, but not very positive attitudes towards mathematics nor strong propensities for mathematical curiosity, collaboration or inventiveness.

Student Q11 answered Part B problem 2 correctly. She also received partial credit for problems 1 and 5, but did not rewrite the proof requirements for any of the problems.

Table Q11C shows that Student Q11 obtained a Part C Metacognition Score of 107/150

(or 71.33%) and a Part C Mathematics Attitude/Curiosity Score of 14/25 (or 56.00%).

This student’s Metacognition Score decreased by three points, and her Mathematics

Attitude/Curiosity Score increased by five points, after she completed the Part B problem set.

The increase in her Mathematics Attitude/Curiosity Score is particularly interesting.

Table Q11A/C (also appearing in Appendix P) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q11’s responses evidenced some increase in scores. This table reveals increases in the following metacognitive activities:

learning more about a mathematics topic if one has a real interest in it;

knowing if one is able to solve a mathematics problem after reading it;

having a better understanding of a problem if one writes the data presented by the

problem;

asking oneself questions to help focus attention on problems and the attempts to solve

them;

thinking about whether or not one is working on the main questions presented by the

problems;

trying to find some other, different solutions to a mathematics problem after having

found one solution to it;

knowing how one performed on a mathematics test before receiving teacher’s

feedback; and

believing that mathematical problem-solving techniques vary in their level of

difficulty.

Table Q11A/C also provides evidence of increases in mathematical curiosity and inventiveness, based on Student Q11’s responses for the following activities:

thinking of new mathematics problems to work on; and

discussing newly learned mathematics concepts with friends.

Student Q11 and her parent agreed to the investigator’s request for an interview with

Student Q11. This interview was conducted via Skype. Student Q11 understood the English language sufficiently well to enable the investigator to proceed with the interview. Student Q11’s interview responses are included in Appendix P.

Student Q11’s interview responses and Mathematics Attitude/Curiosity Scores revealed that her attitudes towards mathematics experienced a positive change after she worked on the number theory problems. She stated that she wanted to discuss the number theory problems with her friends and wanted to do Internet searches on number theory. She further stated that the number theory problems sometimes made her curious, she had some interest in looking through a number theory textbook, and she now believes that there are new discoveries in mathematics.

Student Q11’s work on the number theory problems also seemed to have enhanced her metacognitive functioning, at least with respect to the eight metacognitive activities listed in

Table Q11A/C. Some of her interview responses also support this, especially her statements that she had to understand the number theory problems as a first step, and that the number theory problems changed her thought process about mathematics and showed her that mathematics can be learned in new ways. She reportedly did more analyzing, planning, checking, and reflecting while working on the number theory problems. These interview results suggest metacognitive improvement, although the student’s Metacognition Score decreased slightly after she worked on the problems.

Student Q12. This student was a female 11th grader and about 16 years of age when she participated in the study. Her responses to Parts A and C of the questionnaire are included in

Tables Q12A and Q12C, respectively. Tables Q12A and Q12C appear in Appendix Q.

Table Q12A indicates that Student Q12 received a Part A Metacognition Score of

106/150 (or 70.67%) and a Part A Mathematics Attitude/Curiosity Score of 8/25 (or 32.00%).

These scores suggest a moderate level of metacognitive skills, but not very positive attitudes towards mathematics nor tendencies towards mathematical curiosity, collaboration or inventiveness.

Student Q12 answered Part B problem 2 correctly. She also received partial credit for problems 1 and 5, but did not rewrite the proof requirements for any of the problems. She specified that problems 6 and 7 were beyond her ability, and did not provide explanations for some of her answers.

Table Q12C shows that Student Q12 obtained a Part C Metacognition Score of 109/150

(or 72.67%) and a Part C Mathematics Attitude/Curiosity Score of 5/25 (or 20.00%). This student’s Metacognition Score increased by three points and her Mathematics Attitude/Curiosity

Score decreased by three points after she completed her work on the Part B problem set.

Table Q12A/C (also appearing in Appendix Q) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q12’s responses showed some increase in scores. This table reveals increases in the following metacognitive activities:

knowing how well one understands a mathematics lesson after having studied it;

learning more about a mathematics topic if one has a real interest in it;

asking oneself if one has learned anything new and valuable after completing

mathematics homework;

knowing if one is able to solve a mathematics problem after having read it;

asking oneself questions to help focus attention on mathematics problems and the

attempts to solve them;

trying to clear up one’s confusion in attempts to solve a mathematics problem;

thinking about what a mathematics problem is really asking; and

believing that mathematical problem-solving techniques vary in their level of

difficulty.

Table Q12A/C does not present evidence of increases or improvements in mathematical attitudes, curiosity or inventiveness.

Student Q12 and her parent agreed to the investigator’s request for an interview with

Student Q12. This interview was conducted via Skype. Student Q12 had sufficient knowledge of the English language for purposes of the interview process. Student Q12’s interview responses are included in Appendix Q.

Several of Student Q12’s interview responses indicated that her attitudes towards mathematics changed after she worked on the number theory problems. She stated that she felt good if she could solve the problems, the problems made her curious, and she thinks more about mathematics after having worked on the problems. She also indicated that she wants to do

Internet searches on number theory because she needs more knowledge about number theory topics. Her responses further indicted that mathematics is boring in school, but the number theory problems were different. These collective responses suggest a positive development of mathematical curiosity and attitudes towards mathematics, even though this student’s

Mathematics Attitude/Curiosity Scores indicate otherwise.

Student Q12’s work on the number theory problems also appeared to have improved her metacognitive functioning. The scores for the eight metacognitive activities listed in Table

Q12A/C support this conclusion, as does the increase in the student’s Metacognition Score.

Some of Student Q12’s interview responses also reveal metacognitive enhancement. She reportedly did more analyzing, planning, checking, and reflecting while working on the problems. She also stated that the number theory problems changed her thought process and made her try to find easier ways of solving problems.

Student Q13. This student was a male 11th grader who was about 17 years of age when he worked on the questionnaire. His responses to Parts A and C of the questionnaire are included in Tables Q13A and Q13C, respectively. Tables Q13A and Q13C appear in Appendix R.

Table Q13A indicates that Student Q13 received a Part A Metacognition Score of

115/150 (or 76.67%) and a Part A Mathematics Attitude/Curiosity Score of 14/25 (or 56.00%).

These scores suggest a relatively high level of metacognitive skills, somewhat neutral attitudes towards mathematics, and fair levels of mathematical curiosity, collaboration or inventiveness.

Student Q13 answered Part B problem 2 correctly. He also received partial credit for problems 1 and 5, but did not rewrite the proof requirements for any of the problems.

Table Q13C shows that Student Q13 obtained a Part C Metacognition Score of 120/150

(or 80.00%) and a Part C Mathematics Attitude/Curiosity Score of 13/25 (or 52.00%). His

Metacognition Score increased by five points, and his Mathematics Attitude/Curiosity Score decreased by one point after he completed the Part B problem set.

Table Q13A/C (also appearing in Appendix R) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q13’s responses evidenced some increase in scores. This table reveals increases in the following metacognitive activities:

knowing how well one understood a mathematics lesson after studying it;

deciding on clear goals before trying to learn a new mathematics concept;

trying to find some other, different solutions to a mathematics problem after having

found one solution to it; and

knowing how one performed on a mathematics test before receiving teacher’s

feedback.

Table Q13A/C also provides some evidence of an increase in mathematical curiosity, based on Student Q13’s responses for the following activity:

enjoying finding mathematics problems tha have never been solved.

Student Q13 and his parent agreed to the investigator’s request for an interview with

Student Q13. This student’s understanding of the English language was reasonably good and enabled the interview to proceed. The interview was done via Skype. Student Q13’s interview responses are included in Appendix R.

Student Q13’s interview responses did not show that the student had much interest in mathematics. His Mathematics Attitude/Curiosity Scores suggested that he had more interest in mathematics than he suggested during his interview.

Student Q13’s work on the number theory problems appeared to have enhanced his metacognitive functioning with respect to the four metacognitive activities listed in Table

Q13A/C. Several of Student Q13’s interview responses also provide evidence of metacognitive activity and growth. The student stated that he explained to himself the procedures he needed to follow while working on the number theory problems. He further explained that while working on the number theory problems, he evaluated his thoughts, was sometimes aware of what he was thinking about, and sometimes changed his thought processes and strategies if he was not getting

close to a solution. The student reported that he did not engage in the aforementioned three activities in his mathematics classes.

Student Q14. This student was a male 11th grader who was approximately 16 years of age when he completed the questionnaire. His responses to Parts A and C of the questionnaire are included in Tables Q14A and Q14C, respectively. Tables Q14A and Q14C appear in

Appendix S.

Table Q14A indicates that Student Q14 received a Part A Metacognition Score of

106/150 (or 70.67%) and a Part A Mathematics Attitude/Curiosity Score of 12/25 (or 48.00%).

These scores indicate a relatively high level of metacognitive skills, but rather negative attitudes towards mathematics and generally low levels of mathematical curiosity, collaboration, and inventiveness.

Of the eight problems included on Part B of the questionnaire, Student Q14 answered problem 7 correctly and received partial credit for problems 1 and 5. He did not rewrite the proof requirements for any of the problems.

Table Q14C shows that Student Q14 obtained a Part C Metacognition Score of 88/150 (or

58.67%) and a Part C Mathematics Attitude/Curiosity Score of 17/25 (or 68.00%). This student’s

Metacognition Score decreased by 18 points, and his Mathematics Attitude/Curiosity Score increased by five points after he completed the Part B problem set.

Table Q14A/C (also appearing in Appendix S) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q14’s responses evidenced some increase in scores. This table reveals increases in the following metacognitive activities:

trying to identify the parts of a problem which one does not understand;

trying to clear up one’s confusion if one becomes confused while trying to solve a

mathematics problem;

believing that mathematics topics vary in their level of difficulty;

believing that mathematical problem-solving techniques vary in their level of

difficulty; and

knowing how one performed on a mathematics test before receiving teacher’s

feedback.

Table Q14A/C also provides some evidence of enhancement in mathematical curiosity and inventiveness, based on Student Q14’s responses for the following activities:

flipping through pages of mathematics books while browsing in a bookstore or

library;

enjoying preparing one’s own mathematics tests, quizzes, and puzzles; and

enjoying finding mathematics problems that have never been solved.

Student Q14’s Metacognition Score decreased markedly after he completed the number theory problem set, but metacognitive enhancement in several areas is nonetheless suggested by

Table Q14A/C. This student’s Mathematics Attitude/Curiosity Score increased by a few points after he completed his work on the problem set, and this comports with the Table Q14A/C results summarized above.

Student Q15. This student was a male 11th grader who was about 16 years of age when he participated in the study His responses to Parts A and C of the questionnaire are included in

Tables Q15A and Q15C, respectively. Tables Q15A and Q15C appear in Appendix T.

Table Q15A indicates that Student Q15 received a Part A Metacognition Score of

112/150 (or 74.67%) and a Part A Mathematics Attitude/Curiosity Score of 16/25 (or 64.00%).

These scores indicate a rather high level of metacognitive skills, somewhat positive attitudes towards mathematics, and moderate but not high levels of mathematical curiosity, collaboration, and inventiveness.

Of the eight problems included on Part B of the questionnaire, Student Q15 answered problems 1, 2, 3, 6, and 7 correctly. He also received partial credit for problem 5. He rewrote the problem 6 proof requirement in a way similar to that contemplated by the investigator.

Table Q15C shows that Student Q15 obtained a Part C Metacognition Score of 88/150 (or

58.67%) and a Part C Mathematics Attitude/Curiosity Score of 21/25 (or 84.00%).

This student’s Metacognition Score decreased by 24 points, and his Mathematics

Attitude/Curiosity Score increased by five points having he completed the Part B problem set.

Table Q15A/C (also appearing in Appendix T) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q15’s responses evidenced some increase in scores. This table reveals increases in the following metacognitive activities:

evaluating progress as one is studying a new mathematics lesson;

focusing on whatever data are presented when asked to solve a mathematics problem;

trying to clear up one’s confusion when attempting to solve a mathematics problem;

knowing how one performed on a mathematics test before receiving teacher’s

feedback;

believing that mathematical problem-solving techniques vary in their level of

difficulty; and

not being embarrassed to ask a teacher for help in solving a mathematics problem.

Table Q15A/C also provides evidence of increases in mathematical curiosity and inventiveness, based on Student Q15’s responses for the following activities:

flipping through the pages of mathematics books while browsing in a bookstore or

library;

enjoying preparing one’s own mathematics tests, quizzes, and puzzles

enjoying finding mathematics problems that have never been solved; and

discussing recently learned mathematics concepts with friends.

Student Q15’s questionnaire responses suggest that the number theory problems enhanced his interest and curiosity about mathematics to a considerable extent. The data in Table

Q15A/C lend support for this, as does the increase in the student’s Mathematics

Attitude/Curiosity Score.

Student Q15’s work on the number theory problem set also seemed to have increased his tendencies to engage in the six metacognitive activities listed in Table Q15A/C. A comparison of his Metacognition Scores does not provide similar evidence in favor of metacognitive improvement for this student.

Student Q16. This student was a male 11th grader and was approximately 16 years of age when he completed the questionnaire. His responses to Parts A and C of the questionnaire are included in Tables Q16A and Q16C, respectively. Tables Q16A and Q16C appear in

Appendix U.

Table Q16A indicates that Student Q16 received a Part A Metacognition Score of

117/150 (or 78.00%) and a Part A Mathematics Attitude/Curiosity Score of 12/25 (or 48.00%).

These scores indicate a high level of metacognitive skills, but less than positive attitudes towards mathematics and low levels of mathematical curiosity, collaboration, and inventiveness.

Student Q16 answered all of the Part B problems correctly, except for problem 7. He also rewrote the proof requirement of problem 6 in a way similar to that contemplated by the investigator.

Table Q16C shows that Student Q16 obtained a Part C Metacognition Score of 111/150

(or 74.00%) and a Part C Mathematics Attitude/Curiosity Score of 18/25 (or 72.00%).

Student Q16’s Metacognition Score decreased by six points, and his Mathematics

Attitude/Curiosity Score increased by six points after he finished his work on the Part B problem set.

Table Q16A/C (also appearing in Appendix U) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q16’s responses evidenced some increase in scores. This table shows increases in the following metacognitive activities:

asking oneself if one has learned anything new and valuable after completing

mathematics homework;

asking someone for help when confused in mathematics class;

when attempting to solve a mathematics problem, trying to use solutions of similar

problems which one encountered in the past;

when attempting to solve a mathematics problem, trying to identify the parts of the

problem which one does not understand;

asking oneself questions to help focus one’s attention on a mathematics problem;

not being embarrassed to ask a teacher for help; and

thinking of one’s own examples or illustrations of a mathematics concept to help in

understanding it.

Table Q16A/C also provides evidence of increases in mathematical curiosity and inventiveness, based on Student Q16’s responses for the following activities:

flipping through the pages of mathematics books while browsing in a bookstore or

library;

enjoying thinking of new mathematics problems to work on;

enjoying preparing one’s own mathematics tests, quizzes, and puzzles; and

enjoying finding mathematics problems that have never been solved.

The increase in Student Q16’s Mathematics Attitude/Curiosity Score, as well as the data included in Table Q16A/C, suggest that the number theory problems significantly enhanced the student’s interest and curiosity about mathematics. Student Q16’s efforts on the number theory problems also appeared to have improved his metacognitive functioning, at least with respect to the seven metacognitive activities listed in Table Q16A/C. A comparison of this student’s

Metacognition Scores, however, does not provide supporting evidence in favor of metacognitive improvement.

Student Q17. This student was a male 11th grader who was about 17 years of age when he participated in this study. His responses to Parts A and C of the questionnaire are included in

Tables Q17A and Q17C, respectively. Tables Q17A and Q17C appear in Appendix V.

Table Q17A indicates that Student Q17 received a Part A Metacognition Score of

118/150 (or 78.67%) and a Part A Mathematics Attitude/Curiosity Score of 15/25 (or 60.00%).

These scores suggest a rather high level of metacognitive skills, marginally positive attitudes towards mathematics, and fair to low levels of mathematical curiosity, collaboration, and inventiveness.

Student Q17 answered all of the Part B problems correctly except for problems 3 and 6.

He did not rewrite the proof requirements for any of the problems.

Table Q17C shows that Student Q17 obtained a Part C Metacognition Score of 121/150

(or 80.67%) and a Part C Mathematics Attitude/Curiosity Score of 15/25 (or 60.00%).

This student’s Metacognition Score increased by three points, and his Mathematics

Attitude/Curiosity Score remained unchanged after he completed the Part B problem set.

Table Q17A/C (also appearing in Appendix V) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q17’s responses showed some increase in scores. This table reveals increases in the following metacognitive activities:

asking oneself if one has learned anything new and valuable after completing

mathematics homework;

trying different ways of learning mathematics;

asking someone else for help whenever one is confused in mathematics class;

knowing if one is able to solve a mathematics problem after reading it;

trying to clear up one’s confusion while attempting to solve a mathematics problem;

reading a mathematics problem several times to make sure one understands it;

trying to find some other, different solutions to a mathematics problem after having

found one solution to it; and

believing that mathematical problem-solving techniques vary in their level of

difficulty.

Table Q17A/C also provides some evidence of an increase in mathematical curiosity, based on Student Q17’s responses for the following activity:

flipping through the pages of mathematics books while browsing in a bookstore or

library.

Student Q17’s questionnaire responses suggest that the number theory problems slightly increased his curiosity level about mathematics. His work on the Part B problem set also seemed to have increased his metacognitive processes with respect to the eight metacognitive activities enumerated in Table Q17A/C. A comparison of this student’s Part A and Part C questionnaire responses further support this conclusion.

Student Q18. This student was a male 11th grader who was approximately 16 years of age when he completed the questionnaire. His responses to Parts A and C of the questionnaire are included in Tables Q18A and Q18C, respectively. Tables Q18A and Q18C appear in

Appendix W.

Table Q18A indicates that Student Q18 received a Part A Metacognition Score of

137/150 (or 91.33%) and a Part A Mathematics Attitude/Curiosity Score of 16/25 (or 64.00%).

These scores evidence a very high level of metacognitive skills, slightly positive attitudes towards mathematics, and a fair to low extent of mathematical curiosity, collaboration, and inventiveness.

Student Q18 completed the Part B problem set and answered problems 2, 4, 6, 7, and 8 correctly. He also received partial credit for problems 1 and 5, and rewrote the proof requirement of problem 6 in a way similar to that contemplated by the investigator.

Table Q18C shows that Student Q18 obtained a Part C Metacognition Score of 123/150

(or 82.00%) and a Part C Mathematics Attitude/Curiosity Score of 15/25 (or 60.00%).

This student’s Metacognition Score decreased by 14 points, and his Mathematics

Attitude/Curiosity Score decreased by one point after he completed his work on the Part B problem set. His Metacognition Score remained quite high.

Table Q18A/C (also appearing in Appendix W) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q18’s responses showed some increase in scores. This table reveals increases in the following metacognitive activities:

attempting to use study methods that have been used by successful mathematics

students;

having a better understanding of a problem if one writes the data presented by the

problem; and

when attempting to solve a mathematics problem, trying to use solutions of similar

problems which one encountered in the past.

Student Q18’s work on the Part B problem set does not seem to have enhanced his mathematical curiosity or attitudes towards mathematics. Table Q18A/C suggests metacognitive enhancement relative to three activities for this student, but the drop in the student’s

Metacognition Score does not support this enhancement.

Student Q19. This student was a male 11th grader who was approximately 16 years of age when he worked on the questionnaire. His responses to Parts A and C of the questionnaire are included in Tables Q19A and Q19C, respectively. Tables Q19A and Q19C appear in

Appendix X.

Table Q19A indicates that Student Q19 received a Part A Metacognition Score of

105/150 (or 70.00%) and a Part A Mathematics Attitude/Curiosity Score of 12/25 (or 48.00%).

These scores indicate a relatively high level of metacognitive skills, less than positive attitudes

towards mathematics, and comparatively low levels of mathematical curiosity, collaboration, and inventiveness.

Student Q19 completed the Part B problem set and answered problems 1, 2, 5, and 8 correctly. He did not rewrite the proof requirements for any of the problems.

Table Q19C shows that Student Q19 obtained a Part C Metacognition Score of 81/150 (or

54.00%) and a Part C Mathematics Attitude/Curiosity Score of 12/25 (or 48.00%).

Student Q19’s Metacognition Score decreased by 24 points, and his Mathematics

Attitude/Curiosity Score remained unchanged after he finished working on the Part B problem set. There was clearly a major drop in this student’s Metacognition Score.

Table Q19A/C (also appearing in Appendix X) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q19’s responses showed some increase in scores. This table reveals increases in the following metacognitive activities:

trying to understand the parts of a mathematics problem that one does not understand;

and

believing that mathematical problem-solving techniques vary in their level of

difficulty.

Table Q19A/C also provides slight evidence of an increase in mathematical curiosity, based on Student Q19’s responses for the following activity:

enjoying finding mathematics problems that have never been solved.

Student Q19’s questionnaire responses suggest that the number theory problems only slightly improved his mathematical curiosity. The student’s work on the number theory problems also seemed to have enhanced his metacognitive functioning with respect to two metacognitive

activities, but the very large drop in the student’s Metacognition Score does not support any such enhancement.

Student Q20. This student was a male 11th grader who was about 16 years of age when he completed the questionnaire. His responses to Parts A and C of the questionnaire are included in Tables Q20A and Q20C, respectively. Tables Q20A and Q20C appear in Appendix Y.

Table Q20A indicates that Student Q20 received a Part A Metacognition Score of

104/150 (or 69.33%) and a Part A Mathematics Attitude/Curiosity Score of 9/25 (or 36.00%).

These scores indicate a relatively high level of metacognitive skills, but do not provide evidence of positive attitudes towards mathematics nor high levels of mathematical curiosity, collaboration or inventiveness.

Student Q20 answered Part B problems 1, 4, 5, 6, 7 and 8 correctly. He also rewrote the proof requirement of problem 6 in a manner similar to that contemplated by the investigator.

Table Q20C shows that Student Q20 obtained a Part C Metacognition Score of 126/150

(or 84.00%) and a Part C Mathematics Attitude/Curiosity Score of 11/25 (or 44.00%).

This student’s Metacognition Score increased by 22 points, and his Mathematics

Attitude/Curiosity Score increased by two points after he completed the Part B problem set.

Table Q20A/C (also appearing in Appendix Y) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q20’s responses yielded some increase in scores. This table reveals increases in the following 18 metacognitive activities or factors:

realizing that one’s effort and persistence determine how well one performs on a

mathematics test;

attempting to use study methods that have been used by successful mathematics students; learning more about a mathematics topic if one already knows something about it; drawing sketches, pictures or diagrams to improve understanding of a mathematics concept; deciding on clear goals before trying to learn a new mathematics concept; evaluating progress as one is studying a new mathematics lesson; after completing work on a mathematics problem, thinking about the most important elements of the problem to make sure one really understands them; trying different ways of learning mathematics; thinking of one’s own examples or illustrations of a mathematics concept to help in understanding it; having one’s own ways of remembering what one has learned in mathematics; knowing if one is able to solve a mathematics problem after one has read it; focusing on the data presented by a mathematics problem; trying to clear up one’s confusion upon facing difficulty with a mathematics problem; knowing how one performed on a mathematics test before receiving teacher’s feedback; believing that mathematics topics vary in their level of difficulty; knowing the nature of the difficulty one is experiencing when one cannot find the solution to a mathematics problem; believing that mathematical problem-solving techniques vary in their level of difficulty; and 93

not being embarrassed to ask one’s teacher for help in solving a mathematics

problem.

Table Q20A/C also provides evidence of increases in mathematical curiosity, based on

Student Q20’s responses for the following two activities:

flipping through the pages of mathematics books while browsing in a bookstore or

library; and

enjoying finding mathematics problems that have never been solved.

Student Q20’s Metacognition Scores, as well as the information contained in Table

Q20A/C, suggest that the number theory problem set contributed to a major increase in his metacognitive behavior. The student’s Mathematics Attitude/Curiosity Scores and the final two cells in Table Q20A/C suggest an increase in mathematical curiosity for this student, though not nearly at the same level as the enhancement suggested for metacognition.

Student Q21. This student was a female 11th grader who was 16 years of age when she completed the questionnaire. Her responses to Parts A and C of the questionnaire are included in Tables Q21A and Q21C, respectively. Tables Q21A and Q21C appear in Appendix Z.

Table Q21A indicates that Student Q21 received a Part A Metacognition Score of

119/150 (or 79.33%) and a Part A Mathematics Attitude/Curiosity Score of 7/25 (or 28.00%).

These scores indicate a high level of metacognitive skills rather negative attitudes towards mathematics, and low levels of mathematical curiosity, collaboration, and inventiveness.

Student Q21 answered Part B problems 2, 3, 5, and 8 correctly. She also received partial credit for problem 4, but did not rewrite the proof requirements for any of the problems.

Table Q21C shows that Student Q21 obtained a Part C Metacognition Score of 122/150

(or 81.33%) and a Part C Mathematics Attitude/Curiosity Score of 8/25 (or 32.00%).

This student’s Metacognition Score increased by three points, and her Mathematics

Attitude/Curiosity Score increased by one point after she completed her work on the Part B problem set.

Table Q21A/C (also appearing in Appendix Z) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q21’s responses revealed some increase in scores. This table reveals increases in the following metacognitive activities:

drawing sketches, pictures or diagrams to improve understanding of a mathematics

concept;

deciding on clear goals before trying to learn a new mathematics concept;

evaluating progress as one is studying a new mathematics lesson;

asking oneself if one has learned anything new and valuable after completing

mathematics homework;

trying different ways of learning mathematics;

having a better understanding of a problem by writing the data presented by the

problem;

when attempting to solve a mathematics problem, trying to use solutions of similar

problems one encountered in the past;

asking oneself questions to help focus attention on a mathematics problem and one’s

attempts to solve it; and

trying to clear up one’s confusion while trying to solve a mathematics problem.

Table Q21A/C also provides limited evidence of an increase in mathematical curiosity and inventiveness, based on Student Q21’s responses for the following activity:

enjoying thinking of new mathematics problems to work on.

Student Q21’s questionnaire responses and the Table Q21A/C data suggest that the number theory problems led to metacognitive enhancement relative to at least nine activities. A comparison of Student Q21’s Metacognition Scores also suggests metacognitive growth, but not to the extent suggested by Table Q21A/C. This student’s Mathematics Attitude/Curiosity Scores, along with Table Q21A/C, suggest a minor improvement in mathematical attitudes and curiosity.

Student Q22. This student was a male 11th grader who was about 16 years of age when he participated in the study. His responses to Parts A and C of the questionnaire are included in

Tables Q22A and Q22C, respectively. Tables Q22A and Q22C appear in Appendix AA.

Table Q22A indicates that Student Q22 received a Part A Metacognition Score of

100/150 (or 66.67%) and a Part A Mathematics Attitude/Curiosity Score of 10/25 (or 40.00%).

These scores reflect a moderate level of metacognitive abilities, not very positive attitudes towards mathematics, and low levels of mathematical curiosity, collaboration, and inventiveness.

Student Q22 answered Part B problems 1, 2, 6, 7, and 8 correctly. He also received partial credit for problem 5, and rewrote the proof requirement of problem 6 in a way similar to that contemplated by the investigator.

Table Q22C shows that Student Q22 obtained a Part C Metacognition Score of 107/150

(or 71.33%) and a Part C Mathematics Attitude/Curiosity Score of 16/25 (or 64.00%).

Student Q22’s Metacognition Score increased by seven points, and his Mathematics

Attitude/Curiosity Score increased by six points after he completed work on the Part B problem set.

Table Q22A/C (also appearing in Appendix AA) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q22’s responses showed some increase in scores. This table reveals increases in the following 12 metacognitive activities:

attempting to use study methods that have been used by successful mathematics

students;

being able to learn more about a mathematics topic if one already knows something

about it;

being able to learn more about a mathematics topic if one has a real interest in it;

deciding on clear goals that one must reach as part of the learning process before one

starts trying to learn a new mathematics topic;

evaluating one’s progress in learning a new mathematics lesson;

asking oneself if one has learned anything new and valuable after completing

mathematics homework;

after completing work on a mathematics problem, thinking about the most important

elements of the problem to make sure one really understands them;

after one has read a mathematics problem, knowing if one is able to solve it;

when one is trying to solve a mathematics problem, asking oneself questions to help

focus attention on the problem and the attempts to solve it;

once one finds a solution to a mathematics problem, trying to find some other,

different solutions to the same problem;

believing that mathematical problem-solving techniques vary in their level of

difficulty; and

not being embarrassed to ask one’s teacher for help in solving a mathematics

problem.

Table Q22A/C also provides evidence of increases in mathematical curiosity and inventiveness, based on Student Q22’s responses for the following activities:

flipping through the pages of mathematics books while browsing in a bookstore or

library;

enjoying thinking of new mathematics problems to work on; and

enjoying finding mathematics problems that have never been solved.

Student Q22’s questionnaire responses suggest that the student’s work on the number theory problems improved his metacognitive functioning, mathematical curiosity, and attitudes towards mathematics. The numerical increases in his Metacognition Score and Mathematics

Attitude/Curiosity Score, as well as the Table Q22A/C data, lend support for this conclusion.

Student Q23. This student was a male 11th grader who was approximately 16 years of age when he completed the questionnaire. His responses to Parts A and C of the questionnaire are included in Tables Q23A and Q23C, respectively. Tables Q23A and Q23C appear in

Appendix BB.

Table Q23A indicates that Student Q23 received a Part A Metacognition Score of

135/150 (or 90.00%) and a Part A Mathematics Attitude/Curiosity Score of 23/25 (or 92.00%).

These scores are evidence of a very high level of metacognitive skills, very positive attitudes towards mathematics, and very high levels of mathematical curiosity, collaboration, and inventiveness.

Student Q23 answered Part B problems 2, 3, 4, 7, and 8 correctly. He also received partial credit for problems 1 and 5, but did not rewrite the proof requirements for any of the problems.

Table Q23C shows that Student Q23 obtained a Part C Metacognition Score of 150/150

(or 100.00%) and a Part C Mathematics Attitude/Curiosity Score of 25/25 (or 100.00%).

This student’s Metacognition Score increased by 15 points, and his Mathematics

Attitude/Curiosity Score increased by two points after he completed the Part B problem set. He demonstrated very high scores on the questionnaire, but his “perfect” scores on Part C were quite unusual.

Table Q23A/C (also appearing in Appendix BB) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q23’s responses demonstrated some increase in scores. This table reveals increases in the following 10 metacognitive activities:

attempting to use study methods that have been used by successful mathematics

students;

drawing sketches, pictures or diagrams to improve one’s understanding of a

mathematics concept;

trying different ways of learning mathematics;

having one’s own ways of remembering what one has learned in mathematics;

writing the data presented by a problem to have a better understanding of the

problem;

trying to use solutions of similar mathematics problems that one encountered in the

past;

when one is trying to solve a mathematics problem, asking oneself questions to help

focus attention on the problem and the attempts to solve it;

believing that mathematics topics vary in their level of difficulty;

believing that mathematical problem-solving techniques vary in their level of

difficulty; and

trying to find some other, different solutions to a mathematics problem after having

found one solution to it.

Table Q23A/C also provides some evidence of increases in mathematical curiosity, based on Student Q23’s responses for the following activities:

enjoying preparing one’s own mathematics tests, quizzes, and puzzles; and

enjoying finding mathematics problems which have never been solved.

The raw data from Student Q23’s questionnaire responses suggest that the number theory problems slightly increased the student’s curiosity about mathematics, and more extensively enhanced his metacognitive functioning. These results could be questioned after one reviews the student’s responses to the questions on Part C of the questionnaire, for which he answered

“Always” to each and every question.

Student Q24. This student was a male 11th grader who was 16 years of age when he completed his work on the questionnaire. His responses to Parts A and C of the questionnaire are included in Tables Q24A and Q24C, respectively. Tables Q24A and Q24C appear in Appendix

CC.

Table Q24A indicates that Student Q24 received a Part A Metacognition Score of

107/150 (or 71.33%) and a Part A Mathematics Attitude/Curiosity Score of 12/25 (or 48.00%).

These scores indicate a moderate level of metacognitive skills, but not very positive attitudes towards mathematics nor high levels of mathematical curiosity, collaboration or inventiveness.

Student Q24 answered Part B problems 1, 2, 7, and 8 correctly. He also received partial credit for problem 5, but did not rewrite the proof requirements for any of the problems.

Table Q24C shows that Student Q24 obtained a Part C Metacognition Score of 110/150

(or 73.33%) and a Part C Mathematics Attitude/Curiosity Score of 12/25 (or 48.00%).

100

This student’s Metacognition Score increased by three points, and his Mathematics

Attitude/Curiosity Score remained unchanged after he completed the Part B problem set.

Table Q24A/C (also appearing in Appendix CC) lists the corresponding questions from

Parts A and C of the questionnaire, for which Student Q24’s responses signified some increase in scores. This table reveals increases in the following four metacognitive activities:

asking someone else for help whenever one is confused in mathematics class;

knowing if one is able to solve a mathematics problem after reading it;

thinking about whether or not one is working on what a mathematics problem is

really asking; and

believing that mathematical problem-solving techniques vary in their level of

difficulty.

Student Q24’s questionnaire responses and the data in Table Q24A/C suggest that the number theory problems helped to enhance the student’s metacognitive functioning to a slight extent. His responses do not suggest any resultant increase in mathematical curiosity nor any improvements in mathematics attitudes.

Student Q25. This student was a female 11th grader who was approximately 16 years of age when she completed the questionnaire. Her responses to Parts A and C of the questionnaire are included in Tables Q25A and Q25C, respectively. Tables Q25A and Q25C appear in

Appendix DD.

Table Q25A indicates that Student Q25 received a Part A Metacognition Score of

109/150 (or 72.67%) and a Part A Mathematics Attitude/Curiosity Score of 8/25 (or 32.00%).

These scores suggest a reasonably high level of metacognitive skills, less than positive attitudes towards mathematics, and low levels of mathematical curiosity, collaboration, and inventiveness.

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Student Q25 answered Part B problems 2, 3, 5, and 8 correctly. She also received partial credit for problem 4, but did not rewrite the proof requirements for any of the problems.

Table Q25C shows that Student Q25 obtained a Part C Metacognition Score of 122/150

(or 81.33%) and a Part C Mathematics Attitude/Curiosity Score of 8/25 (or 32.00%).

Student Q25’s Mathematics Attitude/Curiosity Score remained unchanged, but her

Metacognition Score increased by 13 points after she completed her work on the Part B problem set.

Table Q25A/C (also appearing in Appendix DD) lists the corresponding questions from

Parts A and C of the questionnaire, for which Student Q25’s responses evidenced some increase in scores. This table reveals increases in the following eleven metacognitive activities:

attempting to use study methods that have been used by successful mathematics

students;

learning more about a mathematics topic if one already knows something about it;

learning more about a mathematics topic if one has a real interest in it;

deciding on clear goals before trying to learn a new mathematics concept;

asking oneself if one has learned anything new and valuable after completing

mathematics homework;

wondering if one could have found an easier or quicker way to solve a mathematics

problem;

after completing work on a mathematics problem, thinking about the most important

elements of the problem to make sure one really understands them;

while trying to solve a mathematics problem, asking oneself questions to help focus

attention on the problem and the attempts to solve it;

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thinking about whether or not one is working on what a mathematics problem is

really asking;

trying to find some other, different solutions to a mathematics problem after having

found one solution to it; and

knowing the nature of the difficulty one is experiencing when one cannot find the

solution to a mathematics problem.

Table Q25A/C also provides some evidence of a slight increase in mathematical curiosity and inventiveness, based on Student Q25’s responses for the following activity:

enjoying thinking of new mathematics problems to work on.

Student Q25’s questionnaire responses do not show much increase in mathematical curiosity nor enhancement of attitudes towards mathematics, apart from the data in the last row of Table Q25A/C. This student’s work on the number theory problems appeared to have enhanced her metacognitive functioning, based on a comparison of her Metacognition Scores and the relevant data in Table Q25A/C.

Student Q26. This student was a male 11th grader who was about 16 years of age when he participated in the study. His responses to Parts A and C of the questionnaire are included in

Tables Q26A and Q26C, respectively. Tables Q26A and Q26C appear in Appendix EE.

Table Q26A indicates that Student Q26 received a Part A Metacognition Score of

105/150 (or 70.00%) and a Part A Mathematics Attitude/Curiosity Score of 9/25 (or 36.00%).

These scores indicate a rather high level of metacognitive skills, but not very positive attitudes towards mathematics nor a high extent of mathematical curiosity, collaboration or inventiveness.

Student Q26 answered Part B problems 1, 2, 7 and 8 correctly. He also received partial credit for problem 5, but did not rewrite the proof requirements for any of the problems.

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Table Q26C shows that Student Q26 obtained a Part C Metacognition Score of 72/150 (or

48.00%) and a Part C Mathematics Attitude/Curiosity Score of 11/25 (or 44.00%).

This student’s Metacognition Score decreased by 33 points, and his Mathematics

Attitude/Curiosity Score increased by two points after he finished his work on the Part B problem set.

Table Q26A/C (also appearing in Appendix EE) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q26’s responses evidenced some increase in scores. This table reveals increases in the following metacognitive activities:

trying to clear up one’s confusion if one becomes confused in attempting to solve a

mathematics problem; and

believing that mathematical problem-solving techniques vary in their level of

difficulty.

Table Q26A/C also provides some evidence of increases in mathematical curiosity and inventiveness, based on Student Q26’s responses for the following activities:

flipping through the pages of mathematics books while browsing in a bookstore or

library;

enjoying thinking of new mathematics problems to work on; and

enjoying finding mathematics problems that have never been solved.

A comparison of Student Q26’s Metacognition Scores reveals a major drop in metacognitive functioning, at least in the context of the questionnaire. This result should be viewed in light of the slightly more encouraging data in the first two rows of Table Q26A/C.

This table also suggests that the number theory problems helped to enhance the student’s curiosity and inventiveness in the mathematical realm, at least for a few specified activities.

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Student Q27. This student was a male 11th grader who was approximately 16 years of age when he completed the questionnaire. His responses to Parts A and C of the questionnaire are included in Tables Q27A and Q27C, respectively. Tables Q27A and Q27C appear in

Appendix FF.

Table Q27A indicates that Student Q27 received a Part A Metacognition Score of

102/150 (or 68.00%) and a Part A Mathematics Attitude/Curiosity Score of 12/25 (or 48.00%).

These scores suggest a moderate level of metacognitive skills, less than positive attitudes towards mathematics, and rather low levels of mathematical curiosity, collaboration, and inventiveness.

Student Q27 answered Part B problems 1, 2, 4, 5, 6, and 8 correctly. He also rewrote the proof requirement of problem 6 in a manner similar to that contemplated by the investigator.

Table Q27C shows that Student Q27 obtained a Part C Metacognition Score of 96/150 (or

64.00%) and a Part C Mathematics Attitude/Curiosity Score of 10/25 (or 40.00%).

Student Q27’s Metacognition Score decreased by six points, and his Mathematics

Attitude/Curiosity Score decreased by two points after having completed the Part B problem set.

Table Q27A/C (also appearing in Appendix FF) lists the corresponding questions from

Parts A and C of the questionnaire, for which Student Q27’s responses showed some increase in scores. This table reveals increases in the following metacognitive activities:

attempting to use study methods that have been used by successful mathematics

students;

asking oneself if one has learned anything new and valuable after completing

mathematics homework;

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trying to use solutions of similar mathematics problems which one encountered in the

past;

trying to identify parts of a mathematics problem that one does not understand;

while trying to solve a mathematics problem, asking oneself questions to help focus

attention on the problem and the attempts to solve it; and

reading a mathematics problem several times to make sure one understands it.

Student Q27’s questionnaire responses and Mathematics Attitude/Curiosity Scores do not indicate that the number theory problems helped to improve his mathematical curiosity or attitudes towards mathematics. The student’s work on the number theory problems did not yield an increase in his Metacognition Score, but the data in Table Q27A/C provide some evidence of metacognitive enhancement with respect to the six specified metacognitive activities.

Student Q28. This student was a male 11th grader who was about 16 years of age when he completed his work on the questionnaire. His responses to Parts A and C of the questionnaire are included in Tables Q28A and Q28C, respectively. Tables Q28A and Q28C appear in

Appendix GG.

Table Q28A indicates that Student Q28 received a Part A Metacognition Score of

116/150 (or 77.33%) and a Part A Mathematics Attitude/Curiosity Score of 8/25 (or 32.00%).

These scores indicate a high level of metacognitive skills, negative attitudes towards mathematics, and generally low levels of mathematical curiosity, collaboration, and inventiveness.

Student Q28 answered Part B problems 1, 2, 3, 4, 5, and 8 correctly. He did not rewrite the proof requirements for any of the problems.

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Table Q28C shows that Student Q28 obtained a Part C Metacognition Score of 122/150

(or 81.33%) and a Part C Mathematics Attitude/Curiosity Score of 9/25 (or 36.00%).

This student’s Metacognition Score increased by six points, and his Mathematics

Attitude/Curiosity Score increased by one point after he having completed the Part B problem set.

Table Q28A/C (also appearing in Appendix GG) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q28’s responses evidenced an increase in scores. This table reveals increases in the following metacognitive activities and/or factors:

determining through one’s effort and persistence how well one performs on a

mathematics test;

learning even more about a mathematics topic if one already knows something about

it;

learning more about a mathematics topic if one has a real interest in it;

drawing sketches, pictures or diagrams to improve understanding of a mathematics

concept;

after completing work on a mathematics problem, thinking about the most important

elements of the problem to make sure one really understands them;

trying different ways of learning mathematics;

thinking of one’s own examples or illustrations of a mathematics concept to help in

understanding it; and

trying to find some other, different solutions to a mathematics problem after having

found one solution to it.

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Table Q28A/C also provides marginal evidence of an increase in mathematical curiosity and/or inventiveness, based on Student Q28’s responses for the following activity:

enjoying thinking of new mathematics problems to work on.

Student Q28’s questionnaire responses and Mathematics Attitude/Curiosity Scores suggest that the number theory problems enhanced his mathematical curiosity to a slight extent.

There is more persuasive evidence that his work on the number theory problem set improved his metacognitive functioning with respect to the eight metacognitive activities listed in Table

Q28A/C. The improvement in the student’s Metacognition Score also bolsters this conclusion.

Student Q29. This student was a female 11th grader who was approximately 16 years of age when she completed the questionnaire. Her responses to Parts A and C of the questionnaire are included in Tables Q29A and Q29C, respectively. Tables Q29A and Q29C appear in

Appendix HH.

Table Q29A indicates that Student Q29 received a Part A Metacognition Score of

121/150 (or 80.67%) and a Part A Mathematics Attitude/Curiosity Score of 10/25 (or 40.00%).

These scores suggest a high level of metacognitive skills, but rather negative attitudes towards mathematics and low levels of mathematical curiosity, collaboration, and inventiveness.

Student Q29 answered Part B problems 1, 3, and 5 correctly. She did not rewrite the proof requirements for any of the problems.

Table Q29C shows that Student Q29 obtained a Part C Metacognition Score of 119/150

(or 79.33%) and a Part C Mathematics Attitude/Curiosity Score of 20/25 (or 80.00%).

This student’s Metacognition Score decreased by two points, but her Mathematics

Attitude/Curiosity Score doubled after having she completed the Part B problem set.

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Table Q29A/C (also appearing in Appendix HH) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q29’s responses displayed some increase in scores. This table reveals increases in the following metacognitive activities and/or factors:

one’s effort and persistence determining how well one performs on a mathematics

test;

attempting to use study methods that have been used by successful mathematics

students;

learning even more about a mathematics topic if one already knows something about

it;

learning more about a mathematics topic if one has a real interest in it;

when one is trying to solve a mathematics problem, asking oneself questions to help

focus attention on the problem and the attempts to solve it;

wondering if one could have found an easier or quicker way to solve a mathematics

problem;

trying to find some other, different solutions to a mathematics problem after having

found one solution to it; and

knowing the nature of the difficulty one is experiencing when one cannot find the

solution to a mathematics problem.

Table Q29A/C also provides evidence of major increases in mathematical curiosity and inventiveness, based on Student Q29’s responses for the following activities:

flipping through the pages of mathematics books while browsing in a bookstore or

library;

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enjoying thinking of new mathematics problems to work on;

enjoying preparing one’s own mathematics tests, quizzes, and puzzles;

enjoying finding mathematics problems that have never been solved; and

discussing recently learned mathematics concepts with friends.

Student Q29’s questionnaire responses reveal that her attitudes towards mathematics improved substantially after she worked on the number theory problems. The student provided higher scores for all of the Mathematics Attitude/Curiosity questions on Part C. The doubling of her Mathematics Attitude/Curiosity Score further supports the finding of a more positive mathematics attitude.

Student Q29’s work on the number theory problems appears to have had positive effects on her metacognitive functioning, based on the data listed in Table Q29A/C for the specified eight activities. The slight decrease in her Metacognition Score does not, however, support this conclusion.

Student Q30. This student was a male 11th grader who was 16 years of age when he completed his work on the questionnaire. His responses to Parts A and C of the questionnaire are included in Tables Q30A and Q30C, respectively. Tables Q30A and Q30C appear in

Appendix II.

Table Q30A indicates that Student Q30 received a Part A Metacognition Score of

116/150 (or 77.33%) and a Part A Mathematics Attitude/Curiosity Score of 19/25 (or 76.00%).

These scores indicate a high level of metacognitive skills, relatively positive attitudes towards mathematics, and high levels of mathematical curiosity, collaboration, and/or inventiveness.

Student Q30 answered Part B problems 1, 2, 3, 5, and 8 correctly. He also received partial credit for problem 4, but did not rewrite the proof requirements for any of the problems.

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Table Q30C shows that Student Q30 obtained a Part C Metacognition Score of 88/150 (or

58.67%) and a Part C Mathematics Attitude/Curiosity Score of 12/25 (or 48.00%).

This student’s Metacognition Score decreased by 28 points, and his Mathematics

Attitude/Curiosity Score decreased by seven points after he completed the Part B problem set.

Table Q30A/C (also appearing in Appendix I-I) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q30’s responses evidenced some increase in scores. This table reveals increases in the following metacognitive activities:

deciding on clear goals before trying to learn a new mathematics concept;

knowing if one is able to solve a mathematics problem after reading it;

having a better understanding of a problem if one writes the data presented by the

problem;

knowing the nature of the difficulty one is experiencing when one cannot find the

solution to a mathematics problem; and

believing that mathematical problem-solving techniques vary in their level of

difficulty.

Student Q30’s questionnaire responses for several questions reveal some improvement in his metacognitive functioning, as shown by the data in Table Q30A/C. The major drop in this student’s Metacognition Score, as well as the decrease in his Mathematics Attitude/Curiosity

Score, together suggest that the number theory problems did not have a very positive effect on his metacognition or attitudes towards mathematics.

Student Q31. This student was a male 11th grader who was 16 years of age when he completed his work on the questionnaire. His responses to Parts A and C of the questionnaire

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are included in Tables Q31A and Q31C, respectively. Tables Q31A and Q31C appear in

Appendix JJ.

Table Q31A indicates that Student Q31 received a Part A Metacognition Score of

124/150 (or 82.67%) and a Part A Mathematics Attitude/Curiosity Score of 12/25 (or 48.00%).

These scores provide evidence of a very high level of metacognitive functioning, but rather negative attitudes towards mathematics and low levels of mathematical curiosity, collaboration, and inventiveness.

Student Q31 answered Part B problems 2, 3, 5, and 8 correctly. He also received partial credit for problem 1, but did not rewrite the proof requirements for any of the problems.

Table Q31C indicates that Student Q31 obtained a Part C Metacognition Score of

132/150 (or 88.00%) and a Part C Mathematics Attitude/Curiosity Score of 15/25 (or 60.00%).

This student’s Metacognition Score increased by eight points and remained at a very high level, and his Mathematics Attitude/Curiosity Score increased by three points after he finished the Part B problem set.

Table Q31A/C (also appearing in Appendix JJ) lists the corresponding questions from

Parts A and C of the questionnaire, for which Student Q31’s responses yielded some increase in scores. This table shows increases in the following metacognitive activities:

attempting to use study methods that have been used by successful mathematics

students;

learning even more about a mathematics topic if one already knows something about

it;

learning more about a mathematics topic if one has a real interest in it;

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wondering if one could have found an easier or quicker way to solve a mathematics

problem;

knowing if one is able to solve a mathematics problem after one has read the

problem;

having a better understanding of a problem if one writes the data presented by the

problem;

trying to find some other, different solutions to a mathematics problem after having

found one solution to it; and

believing that mathematical problem-solving techniques vary in their level of

difficulty.

Table Q31A/C also provides some evidence of increases in mathematical curiosity and inventiveness, based on Student Q31’s responses for the following activities:

flipping through the pages of mathematics books while browsing in a bookstore or

library; and

enjoying preparing one’s own mathematics tests, quizzes, and puzzles.

The increase in Student Q31’s Metacognition Score, as well as the data presented in

Table Q31A/C, support the conclusion that the student’s efforts on the number theory problems developed his metacognition to some extent. One can see improvements in this student’s mathematical curiosity and attitudes (based his Mathematics Attitude/Curiosity Scores and the

Table Q31A/C data), but these improvements do not appear to be major.

Student Q32. This student was a male 11th grader who was about 16 years of age when he participated in the study. His responses to Parts A and C of the questionnaire are included in

Tables Q32A and Q32C, respectively. Tables Q32A and Q32C appear in Appendix KK.

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Table Q32A indicates that Student Q32 received a Part A Metacognition Score of

120/150 (or 80.00%) and a Part A Mathematics Attitude/Curiosity Score of 13/25 (or 52.00%).

These scores show a high level of metacognitive skills, slightly less than positive attitudes towards mathematics, and marginal levels of mathematical curiosity, collaboration, and inventiveness.

Student Q32 answered Part B problems 1, 2, 3, 4, 5, 6, and 8 correctly. He also rewrote the proof requirement of problem 6 in a way similar to that contemplated by the investigator.

Table Q32C reveals that Student Q32 obtained a Part C Metacognition Score of 118/150

(or 78.67%) and a Part C Mathematics Attitude/Curiosity Score of 12/25 (or 48.00%).

Student Q32’s Metacognition Score decreased by two points, and his Mathematics

Attitude/Curiosity Score decreased by one point after he worked on the Part B problem set.

Table Q32A/C (also appearing in Appendix KK) shows the corresponding questions from

Parts A and C of the questionnaire, for which Student Q32’s responses evidenced some increase in scores. This table reveals increases in the following metacognitive activities:

learning even more about a mathematics topic if one already knows something about

it;

evaluating one’s progress in learning a new mathematics lesson while studying it;

wondering if one could have found an easier or quicker way to solve a mathematics

problem;

trying different ways of learning mathematics;

asking someone else for help whenever one is confused in mathematics class;

trying to use solutions of similar mathematics problems that one encountered in the

past;

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trying to clear up one’s confusion while attempting to solve a mathematics problem;

thinking about whether or not one is working on what a mathematics problem is

really asking; and

knowing how one performed on a mathematics test before receiving feedback from

one’s teacher.

Student Q32’s work on the number theory problems does not appear to have improved his attitudes towards mathematics nor his mathematical curiosity. The problem set seems to have enhanced his metacognition with respect to the nine metacognitive activities listed in Table

Q32A/C, but the drop in the student’s Metacognition Score does not bolster this conclusion. His

Part A Metacognition Score suggests he already had a high level of metacognitive tendencies before he attempted the number theory problem set.

Student Q33. This student was a male 11th grader who was about 16 years of age when he participated in the study. His responses to Parts A and C of the questionnaire are included in

Tables Q33A and Q33C, respectively. Tables Q33A and Q33C appear in Appendix LL.

Table Q33A indicates that Student Q33 received a Part A Metacognition Score of

123/150 (or 82.00%) and a Part A Mathematics Attitude/Curiosity Score of 20/25 (or 80.00%).

These scores indicate a high level of metacognitive skills, positive attitudes towards mathematics, and high levels of mathematical curiosity, collaboration, and/or inventiveness.

Student Q33 answered Part B problem 1 correctly. He also received partial credit for problems 1 and 5, but did not rewrite the proof requirements for any of the problems.

Table Q33C shows that Student Q33 obtained a Part C Metacognition Score of 123/150

(or 82.00%) and a Part C Mathematics Attitude/Curiosity Score of 16/25 (or 64.00%).

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This student’s Metacognition Score remained the same, and his Mathematics

Attitude/Curiosity Score decreased by four points after he finished his work on the Part B problem set.

Table Q33A/C (also appearing in Appendix LL) lists the corresponding questions from

Parts A and C of the questionnaire, for which Student Q33’s responses evidenced some increase in scores. This table reveals increases in the following metacognitive activities:

learning more about a mathematics topic if one has a real interest in it;

drawing sketches, pictures or diagrams to improve understanding of a mathematics

concept;

evaluating progress as one is studying a new mathematics lesson; and

knowing how one performed on a mathematics test before receiving teacher’s

feedback.

Student Q33’s questionnaire responses suggest that the number theory problems improved his metacognitive functioning, but only with respect to the four activities referenced in

Table Q33A/C. His Metacognition Scores reveal no overall growth in his metacognitive tendencies. The slight decrease in the student’s Mathematics Attitude/Curiosity Score leads one to the conclusion that the number theory problems did not enhance his mathematical curiosity or general attitudes towards mathematics.

Student Q34. This student was a female 11th grader who was approximately 16 years of age when she completed the questionnaire. Her responses to Parts A and C of the questionnaire are included in Tables Q34A and Q34C, respectively. Tables Q34A and Q34C appear in

Appendix MM.

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Table Q34A indicates that Student Q34 received a Part A Metacognition Score of

105/150 (or 70.00%) and a Part A Mathematics Attitude/Curiosity Score of 9/25 (or 36.00%).

These scores suggest a reasonable level of metacognitive skills, rather negative attitudes towards mathematics, and low levels of mathematical curiosity, collaboration, and inventiveness.

Student Q34 answered Part B problems 2, 3, and 5 correctly. She did not rewrite the proof requirements for any of the problems.

Table Q34C shows that Student Q34 obtained a Part C Metacognition Score of 96/150 (or

64.00%) and a Part C Mathematics Attitude/Curiosity Score of 10/25 (or 40.00%).

This student’s Metacognition Score decreased by nine points, and her Mathematics

Attitude/Curiosity Score nearly increased by one point after she completed her efforts on the Part

B problem set.

Table Q34A/C (also appearing in Appendix MM) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q34’s responses showed some increase in scores. This table reveals increases in the following metacognitive activities:

learning more about a mathematics topic if one already knows something about it;

deciding on clear goals before trying to learn a new mathematics concept;

evaluating progress as one is studying a new mathematics lesson;

having a better understanding of a problem if one writes the data presented by the

problem;

trying to identify the parts of a mathematics problem which one does not understand,

when attempting to solve the problem; and

trying to clear up one’s confusion in attempting to solve a mathematics problem.

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Table Q34A/C also provides evidence of some improvement in mathematical curiosity and mathematics attitudes, based on Student Q34’s responses for the following activities:

flipping through the pages of mathematics books while browsing in a bookstore or

library; and

enjoying preparing one’s own mathematics tests, quizzes, and puzzles.

Student Q34’s questionnaire responses evidence some metacognitive improvement with respect to the activities specified in Q34A/C. The significant drop in the student’s Metacognition

Score suggests, however, that the number theory problems did not cause an overall enhancement of his metacognitive functioning. A very slight improvement in the student’s mathematical curiosity and mathematics attitudes can be seen from the data.

Student Q35. This student was a female 11th grader who was about 16 years of age when she participated in the study. Her responses to Parts A and C of the questionnaire are included in Tables Q35A and Q35C, respectively. Tables Q35A and Q35C appear in Appendix

NN.

Table Q35A indicates that Student Q35 received a Part A Metacognition Score of

116/150 (or 77.33%) and a Part A Mathematics Attitude/Curiosity Score of 8/25 (or 32.00%).

These scores indicate a reasonably high level of metacognitive skills, but rather negative attitudes towards mathematics and low levels of mathematical curiosity, collaboration, and inventiveness.

Student Q35 answered Part B problem 5 correctly. She did not rewrite the proof requirements for any of the problems.

Table Q35C indicates that Student Q35 obtained a Part C Metacognition Score of

112/150 (or 74.67%) and a Part C Mathematics Attitude/Curiosity Score of 16/25 (or 64.00%).

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This student’s Metacognition Score decreased by four points, and her Mathematics

Attitude/Curiosity Score doubled after she completed her work on the Part B problem set.

Table Q35A/C (also appearing in Appendix NN) lists the corresponding questions from

Parts A and C of the questionnaire, for which Student Q35’s responses evidenced some increase in scores. This table shows increases in the following metacognitive activities:

attempting to use study methods that have been used by successful mathematics

students;

learning more about a mathematics topic if one has a real interest in it;

wondering if one could have found an easier or quicker way to solve a mathematics

problem;

trying different ways of learning mathematics;

asking someone else for help if one is confused in mathematics class;

reading a mathematics problem several times to make sure one understands it;

trying to clear up one’s confusion if one becomes confused in attempting to solve a

mathematics problem;

trying to find some other, different solutions to a mathematics problem after having

found one solution to it;

knowing how one performed on a mathematics test before receiving teacher’s

feedback;

knowing the nature of the difficulty one is experiencing when one cannot find the

solution to a mathematics problem; and

not being embarrassed to ask one’s teacher for help if one encounters difficulty in

solving a mathematics problem.

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Table Q35A/C also provides evidence of improvement in mathematical curiosity and mathematics attitudes, based on Student Q35’s responses for the following activities:

flipping through the pages of mathematics books while browsing in a bookstore or

library;

enjoying thinking of new mathematics problems to work on;

enjoying preparing one’s own mathematics tests, quizzes, and puzzles;

enjoying finding mathematics problems that have never been solved; and

discussing newly learned mathematics concepts with friends.

Student Q35’s questionnaire responses suggest that the number theory problems enhanced her metacognitive functioning with respect to the eleven activities listed in Table

Q35A/C. This result is surprising in light of the slight drop in the student’s Metacognition Score.

Student Q35’s questionnaire responses and Mathematics Attitude/Curiosity Scores reveal that the number theory problems developed her interest and curiosity about mathematics to a significant extent. The doubling of her Mathematics Attitude/Curiosity Score is particularly interesting.

Student Q36. This student was a female 11th grader who was about 16 years of age when she completed the questionnaire. Her responses to Parts A and C of the questionnaire are included in Tables Q36A and Q36C, respectively. Tables Q36A and Q36C appear in

Appendix OO.

Table Q36A indicates that Student Q36 received a Part A Metacognition Score of

120/150 (or 80.00%) and a Part A Mathematics Attitude/Curiosity Score of 13/25 (or 52.00%).

These scores indicate a high level of metacognitive skills, marginal attitudes towards

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mathematics, and somewhat moderate levels of mathematical curiosity, collaboration, and inventiveness.

Student Q36 answered Part B problems 1 and 2 correctly. She also received partial credit for problem 5, but did not rewrite the proof requirements for any of the problems.

Table Q36C shows that Student Q36 obtained a Part C Metacognition Score of 129/150

(or 86.00%) and a Part C Mathematics Attitude/Curiosity Score of 21/25 (or 84.00%).

This student’s Metacognition Score increased by nine points, and her Mathematics

Attitude/Curiosity Score increased by eight points after she completed the Part B problem set.

Table Q36A/C (also appearing in Appendix OO) sets forth the corresponding questions from Parts A and C of the questionnaire, for which Student Q36’s responses evidenced some increase in scores. This table reveals increases in the following metacognitive activities and/or factors:

determining through one’s effort and persistence how well one performs on a

mathematics test;

learning more about a mathematics topic if one already knows something about it;

learning more about a mathematics topic if one has a real interest in it;

asking oneself if one has learned anything new and valuable after completing

mathematics homework;

having one’s own ways of remembering what one has learned in mathematics;

knowing if one is able to solve a mathematics problem after reading it;

focusing on whatever data is presented by a mathematics problem when trying to

solve the problem;

trying to identify the parts of a mathematics problem which one does not understand;

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asking oneself questions to help focus attention on the problem and one’s attempts to

solve it;

thinking about whether or not one is working on what a mathematics problem is

really asking; and

knowing how one performed on a mathematics test before receiving teacher’s

feedback.

Table Q36A/C also provides evidence of improvements in mathematical curiosity and mathematics attitudes, based on Student Q36’s responses for the following activities:

flipping through the pages of mathematics books while browsing in a bookstore or

library;

enjoying thinking of new mathematics problems to work on;

enjoying preparing one’s own mathematics tests, quizzes, and puzzles; and

discussing newly learned mathematics concepts with friends.

Student Q36’s Mathematics Attitude/Curiosity Scores and questionnaire responses provide strong evidence that the number theory problems enhanced her attitudes and curiosity about mathematics to a significant extent. The data also clearly show metacognitive enhancement, by virtue of the increase in the student’s Metacognition Score and the results provided in Table Q36A/C.

Student Q37. This student was a female 11th grader who was approximately 16 years of age when she completed her work on the questionnaire. Her responses to Parts A and C of the questionnaire are included in Tables Q37A and Q37C, respectively. Tables Q37A and Q37C appear in Appendix PP.

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Table Q37A indicates that Student Q37 received a Part A Metacognition Score of

124/150 (or 82.67%) and a Part A Mathematics Attitude/Curiosity Score of 8/25 (or 32.00%).

These scores suggest a high level of metacognitive functioning, negative attitudes towards mathematics, and low tendencies toward mathematical curiosity, collaboration, and inventiveness.

Student Q37 answered Part B problems 5 and 8 correctly. She also received partial credit for problem 1, but did not rewrite the proof requirements for any of the problems.

Table Q37C shows that Student Q37 obtained a Part C Metacognition Score of 121/150

(or 80.67%) and a Part C Mathematics Attitude/Curiosity Score of 11/25 (or 44.00%).

This student’s Metacognition Score decreased by three points, and her Mathematics

Attitude/Curiosity Score increased by three points after she finished working on the Part B problem set.

Table Q37A/C (also included in Appendix PP) lists the corresponding questions from

Parts A and C of the questionnaire, for which Student Q37’s responses indicated some increase in scores. This table reveals increases in the following metacognitive activities:

knowing how well one understands a mathematics lesson after studying it;

learning more about a mathematics topic if one already knows something about it;

drawing sketches, pictures or diagrams to improve understanding of a mathematics

concept;

evaluating progress as one is studying a new mathematics lesson;

asking someone else for help whenever one is confused in mathematics class;

trying to identify the parts of a mathematics problem that one does not understand;

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trying to clear up one’s confusion if one becomes confused in attempting to solve a

mathematics problem; and

trying to find some other, different solutions to a mathematics problem after having

found one solution to it.

Table Q37A/C also provides evidence of improvement in mathematical curiosity and mathematics attitudes, based on Student Q37’s responses for the following activities:

flipping through the pages of mathematics books while browsing in a bookstore or

library; and

enjoying thinking of new mathematics problems to work on.

Student Q37’s Metacognition Score decreased slightly, but the data in Table Q37A/C suggest that the number theory problem set led to metacognitive enhancement with respect to eight separate activities. The student’s questionnaire responses and Mathematics

Attitude/Curiosity Scores evidence improvement in the student’s attitudes and curiosity about mathematics, but this improvement appears modest.

Student Q38. This student was a male 11th grader who was about 16 years old when he completed his work on the questionnaire. His responses to Parts A and C of the questionnaire are included in Tables Q38A and Q38C, respectively. Tables Q38A and Q38C appear in

Appendix QQ.

Table Q38A indicates that Student Q38 received a Part A Metacognition Score of

126/150 (or 84.00%) and a Part A Mathematics Attitude/Curiosity Score of 14/25 (or 56.00%).

These scores reflect a very high level of metacognitive skills, less than positive attitudes towards mathematics, and marginal levels of mathematical curiosity, collaboration, and inventiveness.

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Of the eight problems included on Part B of the questionnaire, Student Q38 answered problems 2, 3, 4, 6, 7, and 8 correctly. He received partial credit for problems 1 and 5, and rewrote the problem 6 proof requirement in a way similar to that contemplated by the investigator.

Table Q38C reveals that Student Q38 obtained a Part C Metacognition Score of 126/150

(or 84.00%) and a Part C Mathematics Attitude/Curiosity Score of 14/25 (or 56.00%). His

Metacognition Score and Mathematics Attitude/Curiosity Score both remained unchanged after he worked on the Part B problem set. This student’s response to each Part C question was the same as his response to the corresponding Part A question.

Table Q38A/C (appearing in Appendix QQ) also documents the fact that Student Q38’s responses did not yield any increase in scores.

Student Q38’s questionnaire responses do not provide any evidence that the number theory problems improved his metacognitive functioning, mathematical curiosity or attitudes towards mathematics. The reason for his Part C responses being a “mirror image” of his Part A responses is not readily apparent in the context of this study.

Student Q39. This student was a male 11th grader who was approximately 17 years old when he completed the questionnaire. His responses to Parts A and C of the questionnaire are included in Tables Q39A and Q39C, respectively. Tables Q39A and Q39C appear in Appendix

RR.

Table Q39A indicates that Student Q39 received a Part A Metacognition Score of

115/150 (or 76.67%) and a Part A Mathematics Attitude/Curiosity Score of 11/25 (or 44.00%).

These scores suggest a rather high level of metacognitive functioning, negative attitudes towards

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mathematics, and relatively low tendencies toward mathematical curiosity, collaboration, and inventiveness.

Student Q39 answered Part B problems 3 and 4 correctly. He did not rewrite the proof requirements for any of the problems.

Table Q39C shows that Student Q39 obtained a Part C Metacognition Score of 100/150

(or 66.67%) and a Part C Mathematics Attitude/Curiosity Score of 16/25 (or 64.00%).

This student’s Metacognition Score decreased by 15 points, and his Mathematics

Attitude/Curiosity Score increased by five points after he completed his efforts on the Part B problem set.

Table Q39A/C (also included in Appendix RR) lists the corresponding questions from

Parts A and C of the questionnaire, for which Student Q39’s responses yielded some increase in scores. This table reveals increases in the following metacognitive activities:

using study methods that have been used by successful mathematics students;

learning more about a mathematics topic if one has a real interest in it;

deciding on clear goals that one must reach before starting to learn a new

mathematics concept;

asking someone else for help whenever one is confused in mathematics class;

reading a mathematics problem several times to make sure one understands it;

thinking about whether or not one is working on what a mathematics problem is

really asking; and

not being embarrassed to ask one’s teacher for help if one is encountering difficulty in

solving a mathematics problem.

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Table Q39A/C also provides evidence of improvement in mathematical curiosity and mathematics attitudes, based on Student Q39’s responses for the following activities:

enjoying thinking of new mathematics problems to work on;

enjoying preparing one’s own mathematics tests, quizzes, and puzzles;

enjoying finding mathematics problems that have never been solved; and

discussing newly learned mathematics concepts with friends.

The significant drop in Student Q39’s Metacognition Score does not suggest an overall enhancement of metacognitive tendencies arising from the student’s work on the Part B problem set. The number theory problems, however, appear to have led to some metacognitive growth with respect to the seven activities listed in Table Q39A/C. The student’s questionnaire responses and Mathematics Attitude/Curiosity Scores also show noticeable improvement in his attitudes and curiosity about the subject of mathematics. This conclusion finds further support in the data appearing in the last four rows of Table Q39A/C.

Student Q40. This student was a female 11th grader who was approximately 15 years of age when she completed her work on the questionnaire. Her responses to Parts A and C of the questionnaire are included in Tables Q40A and Q40C, respectively. Tables Q40A and Q40C appear in Appendix SS.

Table Q40A indicates that Student Q40 received a Part A Metacognition Score of

121/150 (or 80.67%) and a Part A Mathematics Attitude/Curiosity Score of 10/25 (or 40.00%).

These scores indicate a high level of metacognitive functioning, less than positive attitudes towards mathematics, and relatively low levels of mathematical curiosity, collaboration, and inventiveness.

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Student Q40 answered Part B problems 3, 5 and 8 correctly. She did not rewrite the proof requirements for any of the problems.

Table Q40C shows that Student Q40 obtained a Part C Metacognition Score of 121/150

(or 80.67%) and a Part C Mathematics Attitude/Curiosity Score of 13/25 (or 52.00%).

This student’s Metacognition Score remained unchanged, and her Mathematics

Attitude/Curiosity Score increased by three points after she finished her work on the Part B problem set.

Table Q40A/C (also found in Appendix SS) lists the corresponding questions from Parts

A and C of the questionnaire, for which Student Q40’s responses yielded some increase in scores. This table reveals increases in the following metacognitive activities and factors:

knowing how well one understands a mathematics lesson after studying it;

determining through one’s effort and persistence how well one performs on a

mathematics test;

attempting to use study methods that have been used by successful mathematics

students;

deciding on clear goals that one must reach as part of learning a new mathematics

concept;

asking oneself if one has learned anything new and valuable after completing

mathematics homework;

asking oneself questions to help focus attention on a mathematics problem and one’s

attempts to solve it;

trying to clear up one’s confusion if one becomes confused in attempting to solve a

mathematics problem; and

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believing that mathematical problem solving techniques vary in their level of

difficulty.

Table Q40A/C also provides evidence of enhancement in mathematical curiosity and mathematics attitudes, based on Student Q40’s responses for the following activities:

enjoying thinking of new mathematics problems to work on;

enjoying preparing one’s own mathematics tests, quizzes, and puzzles; and

enjoying finding mathematics problems which have never been solved.

Student Q40’s Metacognition Score did not change after the student worked on the number theory problems, but Table Q40A/C suggests some amount of metacognitive enhancement with respect to eight activities and/or factors. The student’s Mathematics

Attitude/Curiosity Scores evidence improvement in her attitudes and curiosity levels concerning mathematics, and the Table Q40A/C data further support this improvement.

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Chapter V

SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS

Summary

This study aimed to determine the extent to which the study of number theory might enhance high school students’ metacognitive functioning, mathematical curiosity, and/or attitudes towards mathematics. The subjects for the study were 40 high school students in the

11th or 12th grade. Seven of the students attended high schools in New York City, and the remaining 33 students attended a high school in Dalian, China. Both male and female students participated in the study.

The investigator used Part A of a three-part written questionnaire (administered to all student participants) to measure the students’ then current metacognitive abilities, as well as their attitudes towards mathematics and levels of mathematical curiosity. After the students completed

Part A, the investigator asked them to work on Part B of the questionnaire. Part B consisted of eight problems in the field of elementary number theory, which the students worked on independently, collaboratively, and/or using any books or resources they chose. Some of the students worked on the problems independently, while others worked on them in small groups.

Once the students completed the Part B problem set, they completed Part C of the questionnaire. Part C was a revised version of the questions used in Part A. The Part C questions were designed to measure the students’ metacognitive tendencies, their mathematical attitudes,

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and their levels of curiosity towards mathematics in the context of the Part B number theory problems they had just attempted to solve.

The investigator compared the Part A scores and the Part C scores of each student to determine if the scores increased or decreased after the students had worked on the number theory problems. He also compared the students’ answers to specific, corresponding questions on

Part A and Part C to determine if there were any changes in such answers relative to the metacognitive or mathematical curiosity factor(s) being measured.

The investigator also interviewed 13 of the students, using an interview script designed to measure the extent to which the students’ metacognitive functioning and/or mathematics attitudes changed after they had worked on the number theory problems.

The study consisted of a Preliminary Trial with three high school students from New

York City, and a Main Study with four other New York City high school students and 33 high school students from Dalian, China. The study materials included Mandarin translations of the questionnaire and all other required forms for use by the Chinese-speaking students. The investigator used no control group. The Preliminary Trial took place about four weeks before the start of the Main Study.

Conclusions

Question 1: What are the levels of the participant students’ metacognitive skills at the start of this study?

The students’ Part A Metacognition Scores ranged from low to high percentage values, as measured in the context of this study. These scores suggested that at the start of this study, slightly more than half of the participating students had reasonably high levels of metacognitive

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skills, and one-quarter of the participating students had very high levels of metacognitive skills.

Slightly less than one-quarter of the participating students had rather low metacognitive skill levels when they began their work as study participants.

The interviews with the participating students also provided information on their then- current metacognitive skills. Ten of the 13 student-interviewees stated that they evaluated their thoughts while working in their mathematics classes, tried to change their strategies in such classes, and/or analyzed and reflected on mathematics problems while in school.

The data suggested that most of the student participants had at least reasonable levels of metacognitive functioning at the start of this study, while some of them did not appear to be very metacognitive.

Question 2: What are the participant students’ attitudes towards mathematics at the start of this study, and to what extent do the students then exhibit curiosity, collaboration, and/or inventiveness in mathematics?

The students’ Part A Mathematics Attitude/Curiosity Scores ranged from very low to high percentage values, as measured in the context of this study. These scores suggested that at the start of this study, less than one-tenth of the participating students had high levels of mathematical curiosity and positive mathematics attitudes. The scores of the rest of the participating students evidenced lower levels of mathematical curiosity and less positive mathematics attitudes when they began their work as study participants. The scores further suggested that the participating students initially had a very wide variety of mathematics attitudes, mathematical curiosity levels, and/or tendencies toward collaboration and/or inventiveness in mathematics. In this respect, the participating students turned out to be a diverse group.

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The interviews with the participating students also provided relevant information on their then-current mathematics attitudes and levels of mathematical curiosity, collaboration, and/or inventiveness. Six of the 13 student-interviewees reported positive attitudes towards mathematics and/or high levels of mathematical curiosity, collaboration, and/or inventiveness, while seven of the 13 student-interviewees reported a dislike for mathematics and/or low levels of mathematical curiosity, collaboration, and/or inventiveness.

Nine of the participating students (the above six student-interviewees and three other students with high Part A Mathematics Attitude/Curiosity Scores) thus displayed very positive mathematics attitudes and/or high levels of mathematical curiosity, collaboration, and/or inventiveness at the start of this study.

Question 3: How do the participant students describe their metacognitive functioning and general attitudes towards mathematics after having worked on a number theory problem set?

The results of this study indicated that all but one of the 40 participating students described or presented some evidence of metacognitive enhancement, greater mathematical curiosity, and/or improved attitudes towards mathematics, after the students had the new experience of working on the Part B number theory problem set. A number of the findings are especially interesting in this context.

All three of the interviewees in the Preliminary Trial gave interview responses suggesting positive changes in mathematical curiosity and general attitudes towards mathematics. None of their interview responses conflicted with their written responses in that respect. One of the interviewees in the Preliminary Trial gave interview responses suggesting positive changes in metacognitive functioning, and that the student’s written responses did not conflict with said result. One of the Preliminary Trial students had an increased Mathematics Attitude/Curiosity

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Score. Two Preliminary Trial students gave inconsistent responses relative to changes in metacognitive functioning.

Several aspects of the Main Study are also of particular note. Ten of the interviewees in the Main Study gave interview responses suggesting positive changes in mathematical curiosity and general attitudes towards mathematics, but there were some inconsistencies in the responses of six of those students. Nine of the interviewees in the Main Study gave interview responses suggesting positive changes in metacognitive functioning, but again there were some inconsistencies in the responses of six of those students. Twenty-one of the Main Study students had an increased Mathematics Attitude/Curiosity Score, and 15 of the Main Study students had an increased Metacognition Score. For very high percentages of the Main Study students (i.e.,

97.30% and 78.38%), responses to specific corresponding Part A and Part C questions suggested positive changes in metacognitive functioning or in mathematical curiosity and general attitudes towards mathematics.

While the interview responses and questionnaire responses of 10 students were inconsistent, those students amounted to only 25% of the total number of participating students in this study.

Question 4: Can number theory study improve high school students’ metacognitive functioning, mathematical curiosity, and/or attitudes towards mathematics?

Based on the data presented, the investigator concluded that the study of number theory has at least some potential for improving high school students’ metacognitive functioning, mathematical curiosity, and/or attitudes towards mathematics. Only one of the participating students did not provide responses in support of such a conclusion. In fact, while some inconsistent responses were provided by the students, 100% of the Main Study student-

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interviewees gave interview responses revealing some positive changes in mathematical curiosity and general attitudes towards mathematics after they had worked on the number theory problems. Further support for the investigator’s conclusion can be based on the interview responses of 90% of the Main Study student-interviewees, whose responses revealed some positive changes in metacognitive functioning after they had completed the number theory problem set.

Recommendations

The results of this study suggested several recommendations for other researchers in this area. The investigator recommends that researchers should begin seeking participating students very early in the research process. Consideration should also be given to providing some monetary compensation for participating students. Such an approach might facilitate the collection of a sufficient number of subjects within a shorter period of time. The investigator also suggests seeking participating students from a wide variety of states and/or countries, in the event that international comparisons might be desired. This could help quicken the data collection process and might lead to an even more varied set of student responses. The problems in the Part B problem set could be made slightly easier to lessen the chance that participating students might become discouraged as they proceed through the study.

The investigator further recommends that high school mathematics teachers consider efforts to broaden their students’ familiarity with elementary number theory concepts. Some of the participating students in this study reported that mathematics can be boring, but that the number theory problems were interesting because they were new and different. This is some indication that high school students might appreciate an early exposure to number theory.

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Teachers should also consider some special, extra-credit assignments for their mathematics students, such as asking students to find five definitions prevalent in the number theory literature.

The students could then present their findings to the class and respond to questions from their fellow students on the material presented. The novelty of this approach might help to engage the students. Another relevant pedagogical approach is to provide high school students with formal training in metacognition.

The present study also suggests a number of possibilities for further research. It would be interesting to replicate the study with one group of students working on number theory problems, and a second group of students working on problems in another area of mathematics. The second group of students could work on trigonometry problems, or problems in a subject which might not appear in the then-current high school mathematics curriculum (e.g., mathematical modeling or statistics). One could then compare the questionnaire responses of the first group of students with the questionnaire responses of the second group of students. Another possibility is to select a group of high school students, provide them with some formal training in metacognitive thinking, and then ask them to complete questionnaires similar to those used in the present study.

This study did not aim to compare U.S. and Chinese high school students in terms of their academic interest or performance. It is, however, curious to note that it was much easier to recruit student participants from the high school in China than from schools in the local metropolitan areas in which the investigator concentrated the search for subjects. As one can see from the data in this study, the set of students Q8 through Q40 (i.e., the students from China) included a number of students who performed much better on the Part B problem set than students Q1 through Q7 (i.e., the students from New York City). This leads one to question the factors which may have caused this performance differential. Another study could very well

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investigate the reasons for these major differences in response rates and levels of mathematical performance, as between the Chinese students and the local students. The educational systems, high school curricula, and cultural patterns of the relevant jurisdictions would likely be relevant to any such investigation.

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Schraw, G., & Sperling-Dennison, R. (1994). Assessing metacognitive awareness, Contemporary Educational Psychology, 19, 460-470.

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Schunk, D. (1987). Peer models and children’s behavioral change. Review of Educational Research, 57(2), 149-174.

Shirvani, H. (2007). Effects of teacher communication on parents’ attitudes and their children’s behaviors at school. Education, 128(1), 34-47. Retrieved from ProQuest Education Journals database. (Document ID: 1451352881).

Silver, E. A. (1992). Referential mappings and the solution of division story problems involving remainders. Focus on Learning Problems in Mathematics, 14(3), 29-39.

Slavson, S. R. (1950). Analytic group psychotherapy with children, adolescents, and adults. New York, NY: Columbia University Press.

Sperling, R. A., Howard, B. C., Miller, L. A., & Murphy, C. (2002). Measures of children’s knowledge and regulation of cognition. Contemporary Educational Psychology, 27, 51- 79.

Sperling, R., Howard, L., & Murphy, C. (2002). Measures of children’s knowledge and regulation of cognition, Contemporary Educational Psychology, 27, 51-79.

Sridhar, D., & Vaughn, S. (2000). Bibliotherapy for all: enhancing reading comprehension. Teaching Exceptional Children, 33, 74-82.

Srinivasan, V. K. (2010). Promoting number theory in high schools or birthday problem and number theory. International Journal of Mathematical Education in Science and Technology, 41(3), 387-398.

Steinberg, L., Lamborn, S., Dornbusch, S., & Darling, N. (1992). Impact of parenting practices on adolescent achievement: Authoritative parenting, school involvement, and encouragement to succeed. Child Development, 63(5), 1266-81. Retrieved from ERIC database.

Stiff, L. V. (1999). Developing mathematical reasoning in grades K-12: 1999 yearbook. Reston, VA: National Council of Teachers of Mathematics.

Sullivan, P., Tobias, S., & McDonough, A. (2006). Perhaps the decision of some students not to engage in learning mathematics in school is deliberate. Educational Studies in Mathematics, 62(1), 81-99. doi:10.1007/s10649-006-1348-8.

Trends in International Mathematics and Science Study (TIMSS) (2003).

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Trusty, J., & Lampe, R. (1997). Relationship of high-school seniors’ perceptions of parental involvement and control to seniors’ locus of control. Journal of Counseling and Development, 75(5), 375-84. Retrieved from ERIC database.

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Veenman, M. (2006). The role of intellectual and metacognitive skills in math problem solving. In A. Desoete & M. Veenman (Eds.), Metacognition in mathematics education (pp. 35- 50). New York, NY: Nova.

Whitebread, D., Coltman, P., Pasternak, D. P., Sangster, C., Grau, V., Bingham, S., Almeqdad, Q., & Demetriou, D. (2009). The development of two observational tools for assessing metacognition and self-regulated learning in young children. Metacognition and Learning, 4(1), 63-85.

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Yara, P. O. (2009). Relationship between teachers’ attitude and students’ academic achievement in mathematics in some selected senior secondary schools in southwestern Nigeria. European Journal of Social Sciences, 11(3), 364-369.

Internet Resources edina.k12.mn.us/teach/curriconline/guide/la/secondary/STRP/metacog/index.htm http://www.math.cornell.edu/~mec/2008-2009/Anema/numbertheory/numbers.html http://www.math.osu.edu/ross/index.html http://www.math.utah.edu/hsp/desc.html http://www.nysedregents.org/IntegratedAlgebra/ http://www.realcty.org/mw/index.php?title=Number_Theory www.corestandards.org/the-standards www.nctm.org www.promys.org 145

Appendix A

Questionnaire Parts A, B, and C

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Questionnaire (Q_)

Part A

Age: ______Gender: ______Class Level: ______Parents’ Phone: ______

Mark each of the statements below with one of the following responses:

A = Always; F = Frequently; S = Sometimes; R = Rarely; N = Never

1. After I have studied a mathematics lesson, I know how well I understand it.

2. My effort and persistence determine how well I perform on a mathematics test.

3. When I am studying mathematics, I attempt to use study methods that have been used by successful mathematics students.

4. If I already know something about a mathematics topic, I can learn even more about it.

5. If I have a real interest in a mathematics topic, I can learn more about it.

6. When I want to improve my understanding of a mathematics concept, I draw sketches, pictures or diagrams relating to the concept.

7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I must reach as part of the learning process.

8. As I am studying a new mathematics lesson, I evaluate my progress in learning the material.

9. After completing my mathematics homework, I ask myself if I have learned anything new and valuable.

10. As soon as I solve a mathematics problem, I wonder if I could have found an easier or quicker way to solve it.

11. After I complete my work on a mathematics problem, I think about the most important elements of the problem to make sure I really understand them.

12. I try different ways of learning mathematics, depending on the type of mathematics involved.

13. Whenever I am confused in mathematics class, I ask someone else for help.

14. In order to help myself understand a mathematics concept, I think of my own examples or illustrations of the concept.

15. I have my own ways of remembering what I have learned in mathematics.

16. After I have read a mathematics problem, I know if I am able to solve it.

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17. When I am asked to solve a mathematics problem, I focus on whatever data are presented.

18. If I write the data presented by a problem, I have a better understanding of the problem.

19. When I am trying to solve a mathematics problem, I try to use solutions of similar problems which I encountered in the past.

20. When I am trying to solve a mathematics problem, I try to identify the parts of the problem which I do not understand.

21. When I am trying to solve a mathematics problem, I ask myself questions to help focus my attention on the problem and my attempts to solve it.

22. If I am having difficulty with a mathematics problem, I read the problem several times to make sure I understand it.

23. If I become confused in trying to solve a mathematics problem, I try to clear up my confusion.

24. When I am working on a mathematics problem, I think about whether or not I am working on what the problem is really asking.

25. Once I find a solution to a mathematics problem, I try to find some other, different solutions to the same problem.

26. I know how I performed on my mathematics test before I receive any feedback from my teacher.

27. I believe that mathematics topics vary in their level of difficulty.

28. When I cannot find the solution to a mathematics problem, I know the nature of the difficulty I am experiencing.

29. I believe that mathematical problem solving techniques vary in their level of difficulty.

30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to ask my teacher for help.

31. When I am browsing in a bookstore or library, I flip through the pages of mathematics books.

32. I enjoy thinking of new mathematics problems to work on.

33. I enjoy preparing my own mathematics tests, quizzes, and puzzles.

34. I enjoy finding mathematics problems which have never been solved.

35. When I am with my friends, I discuss new mathematics concepts which I have recently learned.

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Part B (Q_)

Age: ______Gender: ______Class Level: ______Parents’ Phone ______

Elementary Number Theory Problems

1. A Proper Divisor d of a positive integer n is any whole number less than n, such that when n is divided by d, the result is an integer.

A Perfect Number is an integer n such that n is the sum of its proper divisors. Example: 28 is a Perfect Number because 28 = 1+2+4+7+14.

A Multiperfect Number is an integer n such that the sum of all the positive divisors of n (including n itself) add up to an integral multiple of n. Example: 120 is a Multiperfect Number because (1+2+3+4+5+6+8+10+12+15+20+24+30+40+60+120) = 3x120.

Are all Perfect Numbers also Multiperfect Numbers? Are all Multiperfect Numbers also Perfect Numbers? Explain your answers.

2. A palindrome is an integer that reads the same backwards as it does forwards. (Some examples of palindromes are 44, 2662, and 935,787,539.)

How many two-digit palindromes exist? (Hint: “00” is not an integer.)

3. Let a and b be integers. We say that a divides b if and only if there exists an integer d such that ad = b. (For example, 2 divides 6 because 2∙3 = 6; 5 divides 10 because 5∙2 = 10; 4 divides 28 because 4∙7 = 28).

Show that if a divides b, and a divides c, then a2 divides bc.

4. A prime is an integer that is greater than 1 and that has no positive divisors other than 1 and itself. (Some examples of primes are 2, 3, 5, 11 and 17.) Show that the formula f(n) = n2 + n + 17 produces primes for n = 0, 1, 2, and 3. Find the smallest whole number n for which f(n) = n2 + n + 17 yields a number that is not a prime.

5. An integer greater than 1 that is not prime is called composite. For example, 6, 8, 15 and 121 are composite.

Mark True or False, and explain your answers:

A positive integer that is not prime must be composite.

A positive integer that is not composite must be prime.

There exists an infinite number of composite numbers.

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6. We say that d is the greatest common divisor of integers a and b if and only if:

d is a divisor of a and d is a divisor of b; and

if c is a divisor of a and c is a divisor of b, then c ≤ d

(Note: 2 is a divisor of 8 because 8 = 2∙4).

When d is the greatest common divisor of integers a and b, we write d = gcd (a,b). (For example, 4 = gcd (8, 20); 5 = gcd (5, 10)).

Show that gcd (gcd (a,b)) = gcd (a,b).

7. Let a, b and m be integers. We say that a is congruent to b modulo m, and write a ≡ b (mod m), if and only if a – b = mk for some integer k. (For example, 20 ≡ 6 (mod 7) because 20 – 6 = 14 = 7∙2. Also, 8 ≡ 2 (mod 3) because 8 – 2 = 6 = 3∙2.

Show that if a ≡ b (mod m), then a2 ≡ b2 (mod m).

8. Let d(n) be the number of positive divisors of an integer n (including 1 and n). (For example, d (10) = 4 since there exist four positive divisors of 10; namely, 1, 2, 5, and 10. Also, d (7) = 2 since there exist 2 positive divisors of 7; namely, 1 and 7.)

Let p be a prime. What is d (p2)?

Part C (Q_)

Age: ______Gender: ______Class Level: ______Parents’ Phone:___

Mark each of the statements below with one of the following responses:

A = Always; F = Frequently; S = Sometimes; R = Rarely; N = Never

(As used in the statements below, the “number theory problems” refer to the elementary number theory problems in Part B of this Questionnaire.)

1. After I worked on the number theory problems, I knew how well I understood the concepts.

2. My effort and persistence determined how well I was able to do on the number theory problems.

3. When I worked on the number theory problems, I attempted to use techniques that have been used by successful mathematics students.

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4. If I already know something about the concepts referenced in the number theory problems, I can learn even more about them.

5. If I have a real interest in the number theory problems and the concepts to which they relate, I can learn more about them.

6. When I wanted to improve my understanding of the concepts referenced by the number theory problems, I drew sketches, pictures or diagrams relating to the concepts.

7. Before I started trying to solve one or more of the number theory problems, I decided on clear goals that I needed to reach as part of the problem solving process.

8. As I was working on the number theory problems, I evaluated my progress.

9. After completing my work on the number theory problems, I asked myself if I learned anything new and valuable.

10. As soon as I solved one or more of the number theory problems, I wondered if I could have found easier or quicker ways to arrive at the solution(s).

11. After I completed my work on the number theory problems, I thought about the most important elements of the problems to make sure I really understood them.

12. I tried different ways of approaching the number theory problems, depending on the type of concepts involved.

13. Whenever I was confused by the number theory problems, I asked someone else for help.

14. In order to help myself understand the concepts relating to the number theory problems, I thought of my own examples or illustrations of the concepts.

15. I have my own ways of remembering what I have learned from the number theory problems.

16. After I finished reading the number theory problems, I knew if I was able to solve them.

17. In attempting to solve the number theory problems, I focused on whatever data had been presented.

18. If I wrote the data presented by the number theory problems, I had a better understanding of the problems.

19. In attempting to solve the number theory problems, I tried to use solutions of similar problems which I had encountered in the past.

20. In attempting to solve the number theory problems, I tried to identify the parts of the problems which I did not understand.

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21. When I was trying to solve the number theory problems, I asked myself questions to help focus my attention on the problems and on my attempts to solve them.

22. If I had difficulty with the number theory problems, I read the problems several times.

23. If I became confused in trying to solve the number theory problems, I tried to clear up my confusion.

24. When I was working on the number theory problems, I thought about whether or not I was working on the main questions presented by the problems.

25. Once I found a solution to the number theory problems, I tried to find some other, different solutions to the same problems.

26. I knew how I performed on the number theory problems as soon as I finished my work on the problems.

27. I believe that the concepts referenced by the number theory problems vary in their level of difficulty.

28. When I could not find solutions to the number theory problems, I knew the nature of the difficulty I was experiencing.

29. I believe that the number theory problems require problem solving techniques which vary in their level of difficulty.

30. If I encountered difficulty in solving the number theory problems, I was not embarrassed to ask my teacher for help.

31. Having completed my work on the number theory problems, I flip through the pages of mathematics books while I am browsing in a bookstore or library.

32. Having completed my work on the number theory problems, I enjoy thinking of new mathematics problems to work on.

33. Having completed my work on the number theory problems, I enjoy preparing my own mathematics tests, quizzes and puzzles.

34. Having completed my work on the number theory problems, I enjoy finding mathematics problems which have never been solved.

35. When I am with my friends, I discuss new mathematics concepts which I have learned from working on the number theory problems.

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Appendix B

Interview Script

153

Interview Script

(As used in the questions below, the “number theory problems” refer to the elementary number theory problems in Part B of the Questionnaire used in this study.)

1. What types of strategies or research methods did you use as you tried to solve the number theory problems? Were any of these strategies or methods different from the strategies you previously used in your mathematics courses? If so, explain how they were different. Was there one number theory problem that you liked the most? If so, explain the thinking process and strategy you used in your attempt to solve that particular problem.

2. Would you say that the number theory problems were more interesting or less interesting than your other mathematics coursework? Explain your answer.

3. As you worked on the number theory problems, did you feel bored or discouraged? If so, explain why you think you felt that way.

4. Did you enjoy working on the number theory problems? Why or why not?

5. Do you usually discuss mathematics problems with your friends? Is there anything about the number theory problems that made you want to discuss them with your friends? If so, explain. If you discussed the number theory problems with your friends, about how much time did you spend discussing the problems? Was this discussion longer than other discussions you may have previously had with your friends on mathematics topics?

6. After completing your work on the number theory problems, did you want to look through a number theory textbook to find information about other number theory concepts? If so, explain. Did you think about changing any of the number theory problems? If so, explain.

7. Before working on the number theory problems, what were your attitudes towards mathematics? Have your attitudes towards mathematics changed since you worked on the number theory problems? Explain.

8. As you worked on the number theory problems, were you aware of what you were thinking about? As you worked on the problems, were you evaluating your thoughts? For any of the problems, did you decide to change your thought processes and strategies if you found that you were not getting closer to a solution? Did you go through any of these steps in any of your mathematics classes before you started working on the number theory problems? Explain further.

9. As you were working on the number theory problems, did you prefer to think on your own about how to find the solutions? Why or why not?

10. Were you able to solve the number theory problems more easily if you worked on them with your fellow students? If so, how did the group work help you?

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11. Before starting your work on the number theory problems, to what extent, if any, did you believe that new mathematical concepts and problems continue to be discovered? Do you have that same belief after having completed your work on the number theory problems?

12. How do you feel about making mistakes in mathematics? Do you see mistakes as something to be ashamed of, or as a part of learning? Do you feel the same way about mistakes which you may have made on the number theory problems? Explain.

13. As you were working on the number theory problems, did you quietly explain to yourself the procedures which you needed to follow in order to solve the problems? If not, what exactly were you thinking as you worked on the problems?

14. In your past mathematics courses, did you ever think about how you could improve your mathematics ability? If so, what ideas did you have? Did you think of any new ideas for improving your mathematics ability after working on the number theory problems? If so, what new ideas do you think of?

15. While working on the number theory problems, which of the following activities did you engage in: analyzing the problem, planning the method of the solution, checking the work, reflecting on the problem. Did you engage in those activities to a greater extent while working on the number theory problems, as compared to your work in your prior mathematics classes?

16. Had you ever previously studied or read about the topics covered by the number theory problems? If so, how did that prior learning affect your work on the number theory problems? If the number theory problems presented topics which you had never heard of before, did the problems make you feel surprised, anxious, frustrated or curious?

17. Having worked on the number theory problems, do you feel interested in doing Internet searches to learn more about number theory topics? Why or why not?

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Appendix C

Certificate of Participation Templates and Consent for Reference Form

156

Certificate of Participation Date: ______

This is to certify that during the (Fall 2012) (Spring 2013) (Fall 2013) (Spring 2014) semester,

______successfully participated as a subject in a PhD dissertation research study relating to number theory, metacognition and mathematics attitudes in high school. Participation included completion of a questionnaire and a number theory problem set.

______

Anthony M. Miele

Certificate of Participation Date: ______

This is to certify that during the (Fall 2012) (Spring 2013) (Fall 2013) (Spring 2014) semester,

______

Anthony M. Miele

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Consent for Reference Form Date: ______

I hereby give Anthony M. Miele the authority to disclose the name of ______, as well as any other identifying information relating to ______, for the sole purpose of serving as a reference for ______, in connection with his/her college applications or any work-related application processes.

______

Parent/Legal Guardian

______

Student

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Appendix D

Institutional Review Board Required Forms

159

Teachers College, Columbia University 525 West 120th Street New York NY 10027 212 678 3000 www.tc.edu

INFORMED CONSENT AND RESEARCH DESCRIPTION DOCUMENTATION FOR PARENTS

DESCRIPTION OF THE RESEARCH: Your child is invited to participate in a research study which is being conducted to determine the effects of Number Theory study on high school students’ thought processes and attitudes towards mathematics. Your child will be asked to spend a few hours filling out a three-part questionnaire about the study of mathematics. One part of the questionnaire will include several number theory problems which will need to be attempted. Other questions will attempt to determine what types of attitudes your child has towards mathematics, and to what extent your child reflects upon his or her thought patterns while working on mathematics problems. The work on the questionnaire can be done at any time that is convenient for your child, in school or at home. Your child is free to work with other students on the questionnaire, and can consult any relevant books or other resources, but otherwise no staff member from your child’s school is to have knowledge of the specific students who are participating in the study (and so participation in the study [as designed by the researcher] will in no way affect your child’s academic standing at his or her school (positively or negatively)). The research will be conducted by Anthony M. Miele, who will obtain the completed questionnaires for review and analysis about one week after the questionnaires are distributed. In addition, three (3) students in a Pilot Study, and ten (10) students in the Main Study, will be selected for follow-up interviews on the responses to the questionnaires, but participation in these interviews is voluntary. Students’ names will be obtained during the said interviews, but names will not be included as part of any of the written research results, and no recording or videotaping will be done as part of the research.

RISKS AND BENEFITS: The risks and possible benefits associated with this study are as follows: Risks: Boredom, fatigue and possible interference with your child’s regularly scheduled study and homework time. Benefits: There are no direct benefits to participants for participation in the study. If your child does not want to participate in all aspects of the study, or if he or she wishes to withdraw from participation in the research at any time, such will be acceptable.

PAYMENTS: There will be no payments or other remuneration due from the researcher to any of the student participants.

DATA STORAGE TO PROTECT CONFIDENTIALITY: Answers to the questionnaires, as well as the interview responses, will be incorporated into the researcher’s PhD dissertation. None of the students will be identified by name in any of the final written research results. The questionnaires will be coded as Q1, Q2, Q3, ….. Students who are not interviewed will not be required to provide their names on the questionnaires, but all students completing the questionnaires will need to provide their parent(s)’/legal guardian(s)’ phone numbers. All student interviewees will need to provide their names and their parent(s)’/ legal guardian(s)’ phone numbers to the researcher, so as to enable the researcher to contact the students and their parent(s) or legal guardian(s) to set up the interviews. The names of the participating students, the parent(s)’ and legal guardian(s)’ phone numbers, and the name-specific interview results of the participating students will not be disclosed by the researcher to anyone. Records of such names and phone numbers obtained as part of the research will be destroyed at the conclusion of the researcher’s final dissertation defense.

On the questionnaires, students will also list their age, gender, and class level.

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All completed questionnaires and interview responses will be stored by the researcher at his home office in Little Neck, New York.

TIME INVOLVEMENT: Your child’s participation will include a few hours on the questionnaire. If your child is selected for a follow-up interview, such interview will last approximately one hour.

HOW WILL RESULTS BE USED: The results of the study will be used as part of the researcher’s PhD dissertation, which will eventually be available (in whole or in part) to the academic community.

161

Teachers College, Columbia University 525 West 120th Street New York NY 10027 212 678 3000 www.tc.edu

PARTICIPANT’S RIGHTS DOCUMENTATION

Principal Investigator: Anthony M. Miele.

Research Title: The Effects of Number Theory Study on High School Students’ Metacognition and Mathematics Attitudes

I have read and discussed the Research Description with the researcher. I have had the opportunity to ask questions about the purposes and procedures regarding this study. My participation in this research is voluntary. I may refuse to participate or withdraw from participation at any time without jeopardy to future medical care, employment, student status or other entitlements. The researcher may withdraw me from the research at his professional discretion. If, during the course of the study, significant new information that has been developed becomes available which may relate to my willingness to continue to participate, the researcher will provide this information to me. Any information derived from the research project that personally identifies me will not be voluntarily released or disclosed without my separate consent, except as specifically required by law. If at any time I have any questions regarding the research or my participation, I can contact the researcher, who will answer my questions. The researcher’s phone number is (917) 741-8504. If at any time I have comments, or concerns regarding the conduct of the research or questions about my rights as a research subject, I should contact the Teachers College, Columbia University Institutional Review Board /IRB. The phone number for the IRB is (212) 678-4105. Or, I can write to the IRB at Teachers College, Columbia University, 525 W. 120th Street, New York, NY, 10027, Box 151. I should receive a copy of the Research Description and this Participant’s Rights document. I understand that no video and/or audio taping is part of this research. Written materials relating to this research may be viewed in an educational setting outside the research site. My signature means that I agree to participate in this research.

Participant’s signature: ______Date:____/____/____

Name: ______

If necessary:

Parent/Legal Guardian’s Signature/consent: ______Date:____/____/____

Name: ______

162

Teachers College, Columbia University 525 West 120th Street New York NY 10027 212 678 3000 www.tc.edu

Assent Form for Minors (8-17 years-old)

I ______(child’s name) agree to participate in the research entitled:

The Effects of Number Theory Study on High School Students’ Metacognition and Mathematics Attitudes

The purpose and nature of the research has been fully explained to me by Anthony M. Miele. I understand what is being asked of me, and should I have any questions, I know that I can contact Anthony M. Miele at any time. I also understand that I can quit the study any time I want to.

Name of Participant: ______

Signature of Participant: ______

Witness: (Parent or Legal Guardian): ______

Date: ______

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Researcher’s Verification of Explanation

I certify that I have carefully explained the purpose and nature of this research to ______(participant’s name) in age-appropriate language. He/She has had the opportunity to discuss it with me in detail. I have answered all his/her questions and he/she provided the affirmative agreement (i.e. assent) to participate in this research.

Researcher’s / Principal Investigator’s Signature: ______

Date: ______

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Appendix E

Mandarin Translations of the Cover Memo to the High School in China

(with English Version), Parts A, B, and C of the Questionnaire, the Certificate of

Participation Templates and the Institutional Review Board Required Forms

165

Re: Teachers College, Columbia University Institutional Review Board Protocol # 13-033

Consent Form approved until 11/25/2013

To: Parents of Participating Students

The plan would be to have the students first work on Parts A and B of the questionnaire. The students would then complete Part C. Once Parts A, B and C are completed, they can be scanned and E-mailed back to me. I would also arrange to interview 5 or 6 students (via phone) after obtaining written parental permission. I will utilize an interview script that has been approved as part of my research (pages 26-28 of the materials, not included with the materials provided). I will also number the completed questionnaires after I receive them.

It has been preferred that students not discuss the questionnaires or their research participation with the staff at their respective school(s). This is to make it clear to the students that the students’ participation in the research has no bearing on their grades. (Hence, students would approach question 30 in Part C as only applying to situations in which they might have been free to ask their teachers.) Also, the questionnaires are designed to be completed during students’ spare time and not during school hours.

I would need to complete the forms on pages 29, 30 and 36 to the extent applicable. The Informed Consent form on pages 32 and 33 would need to be retained by each student and his/her parent(s). The forms on pages 34 and 35 are intended to be completed by each student and his/her parent(s) and then returned to me. Before I return any signed Certificates of Participation to the students and their parent(s), I will use the then current form.

Anthony M. Miele

[email protected]

December __, 2013

Mandarin Translations:

Teachers College IRB Protocol No. 13-033 Mandarin Translation 11.27.2013.doc

166

Appendix F

Student Q1 Responses

167

Table Q1A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 5 2. My effort and persistence determine how well I perform on a mathematics test. 4 3. When I am studying mathematics, I attempt to use study methods that have been 2 used by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 4 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 3 6. When I want to improve my understanding of a mathematics concept, I draw 5 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that 2 I must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 3 material. 9. After completing my mathematics homework, I ask myself if I have learned 1 anything new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 4 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 4 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 3 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 2 14. In order to help myself understand a mathematics concept, I think of my own 4 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 4 16. After I have read a mathematics problem, I know if I am able to solve it. 5 17. When I am asked to solve a mathematics problem, I focus on whatever data are 4 presented. 18. If I write the data presented by a problem, I have a better understanding of the 5 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 5 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 2 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 4 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 3 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 3 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 4 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback 5 from my teacher.

168

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 5 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 4 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 3 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 1 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 4 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 4 34. I enjoy finding mathematics problems which have never been solved. 4 35. When I am with my friends, I discuss new mathematics concepts which I have 1 recently learned.

Notes: i.) Student Q1’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 112/150 = 74.67% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 14/25 = 56.00%

169

Table Q1C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 5 concepts. 2. My effort and persistence determined how well I was able to do on the number theory 4 problems. 3. When I worked on the number theory problems, I attempted to use techniques that 4 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 2 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 3 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 1 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 4 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 1 9. After completing my work on the number theory problems, I asked myself if I learned 3 anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 3 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the most 2 important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 5 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 1 help. 14. In order to help myself understand the concepts relating to the number theory 3 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 3 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 5 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 5 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 5 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 5 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 3 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 5 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear up 5 my confusion. 24. When I was working on the number theory problems, I thought about whether or not 5 I was working on the main questions presented by the problems.

170

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 5 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 4 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 5 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 3 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 3 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 1 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 1 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 3 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 1 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 3 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 1 learned from working on the number theory problems.

Notes: i.) Student Q1’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 108/150 = 72.00% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 9/25 = 36.00%

171

Table Q1A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 3. When I am studying mathematics, I attempt 2 3. When I worked on the number 4 to use study methods that have been used by theory problems, I attempted to use successful mathematics students. techniques that have been used by successful mathematics students. 7. Before I start trying to learn a new 2 7. Before I started trying to solve one 4 mathematics concept, I decide on clear goals or more of the number theory that I must reach as part of the learning problems, I decided on clear goals that process. I needed to reach as part of the problem solving process. 12. I try different ways of learning 4 12. I tried different ways of 5 mathematics, depending on the type of approaching the number theory mathematics involved. problems, depending on the type of concepts involved. 17. When I am asked to solve a mathematics 4 17. In attempting to solve the number 5 problem, I focus on whatever data are theory problems, I focused on whatever presented. data had been presented. 20. When I am trying to solve a mathematics 2 20. In attempting to solve the number 3 problem, I try to identify the parts of the theory problems, I tried to identify the problem which I do not understand. parts of the problems which I did not understand. 23. If I become confused in trying to solve a 3 23. If I became confused in trying to 5 mathematics problem, I try to clear up my solve the number theory problems, I confusion. tried to clear up my confusion. 24. When I am working on a mathematics 3 24. When I was working on the number 5 problem, I think about whether or not I am theory problems, I thought about working on what the problem is really asking. whether or not I was working on the main questions presented by the problems. 25. Once I find a solution to a mathematics 4 25. Once I found a solution to the 5 problem, I try to find some other, different number theory problems, I tried to find solutions to the same problem. some other, different solutions to the same problems.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q1’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

172

Student Q1 Interview Responses

(As used in the questions below, the “number theory problems” refer to the elementary number theory problems in Part B of the questionnaire used in this study.)

1. What types of strategies or research methods did you use as you tried to solve the number theory problems? NO SPECIAL STRATEGIES. LOOKED AT PROBLEMS AND SOLVED THEM. Were any of these strategies or methods different from the strategies you previously used in your mathematics courses? NO. If so, explain how they were different. Was there one number theory problem that you liked the most? PROBLEM 4. If so, explain the thinking process and strategy you used in your attempt to solve that particular problem. SAW IT IN SCHOOL AND USED SAME APPROACH FROM SCHOOL.

2. Would you say that the number theory problems were more interesting or less interesting than your other mathematics coursework? Explain your answer. SAME LEVEL OF INTEREST.

3. As you worked on the number theory problems, did you feel bored or discouraged? If so, explain why you think you felt that way. SOMETIMES BORED IF DOING THEM IN ONE SITTING; NEVER DISCOURAGED.

4. Did you enjoy working on the number theory problems? YES. Why or why not? NEVER REALLY THOUGHT ABOUT MANY NUMBER THEORY CONCEPTS BEFORE. NEVER TOOK A NUMBER THEORY COURSE IN HIGH SCHOOL.

5. Do you usually discuss mathematics problems with your friends? SOMETIMES. Is there anything about the number theory problems that made you want to discuss them with your friends? NO. If so, explain. If you discussed the number theory problems with your friends, about how much time did you spend discussing the problems? Was this discussion longer than other discussions you may have previously had with your friends on mathematics topics? DID NOT COME UP AS A TOPIC.

6. After completing your work on the number theory problems, did you want to look through a number theory textbook to find information about other number theory concepts? NO, BUT LOOKED AT INTERNET SITES FOR INFORMATION ON THE NUMBER THEORY PROBLEMS. If so, explain. Did you think about changing any of the number theory problems? NO. If so, explain. LOOKED FOR WAYS TO MAKE WORK SHORTER. DID THIS BEFORE ON OTHER PROBLEMS.

7. Before working on the number theory problems, what were your attitudes towards mathematics? LIKED IT. INTERESTING SUBJECT SINCE CHILDHOOD. Have your attitudes towards mathematics changed since you worked on the number theory problems? NO. STILL INTERESTING. BUT IT SEEMS THAT MATHEMATICS CAN BE EVEN MORE INTERESTING THAN WHAT IS TAUGHT IN SCHOOL. Explain. 173

8. As you worked on the number theory problems, were you aware of what you were thinking about? YES. As you worked on the problems, were you evaluating your thoughts? YES. WAS THINKING ABOUT MY THOUGHTS. For any of the problems, did you decide to change your thought processes and strategies if you found that you were not getting closer to a solution? YES. REVIEWED MY WORK TO SEE IF I MAY HAVE GONE WRONG. Did you go through any of these steps in any of your mathematics classes before you started working on the number theory problems? YES. Explain further.

9. As you were working on the number theory problems, did you prefer to think on your own about how to find the solutions? YES. Why or why not? TO BUILD MYSELF UP BY DOING THE PROBLEMS ALONE.

10. Were you able to solve the number theory problems more easily if you worked on them with your fellow students? If so, how did the group work help you? NO GROUP WORK.

11. Before starting your work on the number theory problems, to what extent, if any, did you believe that new mathematical concepts and problems continue to be discovered? THE TEXTBOOK MAY DO PROBLEMS ONE WAY, TEACHERS MAY FIND A BETTER AND MORE EFFICIENT WAY TO DO THEM. Do you have that same belief after having completed your work on the number theory problems? YES.

12. How do you feel about making mistakes in mathematics? USUALLY MY MISTAKES ARE CARELESS MISTAKES. Do you see mistakes as something to be ashamed of, or as a part of learning? DEPENDS ON THE CALIBER OF STUDENT. FOR SOME STUDENTS, MISTAKES MAY BE GOOD FOR LEARNING. BUT FOR EXCEPTIONAL STUDENTS, MISTAKES CAN BE SOMETHING TO LOOK DOWN ON. Do you feel the same way about mistakes which you may have made on the number theory problems? YES. Explain.

13. As you were working on the number theory problems, did you quietly explain to yourself the procedures which you needed to follow in order to solve the problems? YES. REREAD THE PROBLEMS. If not, what exactly were you thinking as you worked on the problems?

14. In your past mathematics courses, did you ever think about how you could improve your mathematics ability? YES. If so, what ideas did you have? CONTINUE LEARNING IN PRE-CALC AND AP CALC. Did you think of any new ideas for improving your mathematics ability after working on the number theory problems? YES. If so, what new ideas do you think of? COULD ALWAYS LEARN MORE FROM INTERNET SEARCHES.

15. While working on the number theory problems, which of the following activities did you engage in: analyzing the problem YES, TO SEE WHAT THE QUESTION IS, WHAT IS GIVEN, AND WHAT NEEDS TO BE FOUND., planning the method of the solution YES, I THINK SO., checking the work YES., reflecting on the problem 174

YES. RETRACED STEPS FOR PROBLEMS I WAS UNSURE ABOUT. Did you engage in those activities to a greater extent while working on the number theory problems, as compared to your work in your prior mathematics classes? NO.

16. Had you ever previously studied or read about the topics covered by the number theory problems? NO, NOT THESE SPECIFIC PROBLEMS. BUT SAW PROBLEM 4 IN HIGH SCHOOL CLASS AND ATTENDED A COLLEGE NUMBER THEORY COURSE. If so, how did that prior learning affect your work on the number theory problems? USED MY KNOWLEDGE OF PROBLEM 4 TO APPROACH IT. If the number theory problems presented topics which you had never heard of before, did the problems make you feel surprised YES, anxious MAYBE, frustrated MAYBE or curiousYES ?

17. Having worked on the number theory problems, do you feel interested in doing Internet searches to learn more about number theory topics? YES Why or why not? TO LEARN WHAT NUMBER THEORY IS AND TO LOOK AT IT MORE TO PREPARE FOR SCHOOL.

175

Appendix G

Student Q2 Responses

176

Table Q2A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 4 3. When I am studying mathematics, I attempt to use study methods that have been used 4 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 4 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 5 6. When I want to improve my understanding of a mathematics concept, I draw 3 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 1 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the No response material. 9. After completing my mathematics homework, I ask myself if I have learned anything 2 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 2 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 3 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 4 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 4 14. In order to help myself understand a mathematics concept, I think of my own 2 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 4 16. After I have read a mathematics problem, I know if I am able to solve it. 5 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 5 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 3 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 4 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 3 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 1 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 1 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 4 my teacher.

177

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 5 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 5 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 1 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 1 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 1 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 1 34. I enjoy finding mathematics problems which have never been solved. 1 35. When I am with my friends, I discuss new mathematics concepts which I have 1 recently learned.

Notes: i.) Student Q2’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 102/150 = 68.00% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 5/25 = 20.00%

178

Table Q2C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 2 concepts. 2. My effort and persistence determined how well I was able to do on the number 3 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 4 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 5 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 5 relate, I can learn more about them. 6. When I wanted to improve my understanding of the concepts referenced by the 3 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 2 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I 2 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 2 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 2 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 4 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 1 help. 14. In order to help myself understand the concepts relating to the number theory 3 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 3 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 5 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 5 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 5 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 3 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 3 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 3 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I become confused in trying to solve the number theory problems, I tried to clear 3 up my confusion. 24. When I was working on the number theory problems, I thought about whether or not 4 I was working on the main questions presented by the problems.

179

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 3 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 5 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 4 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 4 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 4 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 1 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 1 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of 1 new mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 1 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 1 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 1 learned from working on the number theory problems.

Notes: i.) Student Q2’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 101/150 = 67.33% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 5/25 = 20.00%

180

Table Q2A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 4. If I already know something about a 4 4. If I already know something about 5 mathematics topic, I can learn even more the concepts referenced in the number about it. theory problems, I can learn even more about them. 7. Before I start trying to learn a new 1 7. Before I started trying to solve one 2 mathematics concept, I decide on clear goals or more of the number theory that I must reach as part of the learning problems, I decided on clear goals that process. I needed to reach as part of the problem solving process. 14. In order to help myself understand a 2 14. In order to help myself understand 3 mathematics concept, I think of my own the concepts relating to the number examples or illustrations of the concept. theory problems, I thought of my own examples or illustrations of the concepts. 24. When I am working on a mathematics 1 24. When I was working on the number 4 problem, I think about whether or not I am theory problems, I thought about working on what the problem is really asking. whether or not I was working on the main questions presented by the problems. 25. Once I find a solution to a mathematics 1 25. Once I found a solution to the 3 problem, I try to find some other, different number theory problems, I tried to find solutions to the same problem. some other, different solutions to the same problems. 26. I know how I performed on my 4 26. I knew how I performed on the 5 mathematics test before I receive any feedback number theory problems as soon as I from my teacher. finished my work on the problems

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q2’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

181

Student Q2 Interview Responses

(As used in the questions below, the “number theory problems” refer to the elementary number theory problems in Part B of the questionnaire used in this study.)

1. What types of strategies or research methods did you use as you tried to solve the number theory problems? I DON’T REMEMBER. APPLIED WHAT I LEARNED IN THE PAST. Were any of these strategies or methods different from the strategies you previously used in your mathematics courses? NO. If so, explain how they were different. Was there one number theory problem that you liked the most? NO. I’M NOT A MATH PERSON. If so, explain the thinking process and strategy you used in your attempt to solve that particular problem.

2. Would you say that the number theory problems were more interesting or less interesting than your other mathematics coursework? THE SAME. Explain your answer. NOT MORE INTERESTING, NOT LESS INTERESTING.

3. As you worked on the number theory problems, did you feel bored or discouraged? BORED, NO. DISCOURAGED, MAYBE. If so, explain why you think you felt that way. IF I COULD NOT ANSWER THE PROBLEM, IT DIDN’T FEEL LIKE AN ACCOMPLISHMENT.

4. Did you enjoy working on the number theory problems? NO. Why or why not? DON’T LIKE MATH.

5. Do you usually discuss mathematics problems with your friends? NO. Is there anything about the number theory problems that made you want to discuss them with your friends? NO. If so, explain. If you discussed the number theory problems with your friends, about how much time did you spend discussing the problems? Was this discussion longer than other discussions you may have previously had with your friends on mathematics topics? DO NOT DISCUSS MATH WITH FRIENDS.

6. After completing your work on the number theory problems, did you want to look through a number theory textbook to find information about other number theory concepts? YES, MAYBE. If so, explain. IF I DIDN’T KNOW HOW TO DO THE PROBLEMS, I WANTED TO KNOW HOW. Did you think about changing any of the number theory problems? NO. If so, explain.

7. Before working on the number theory problems, what were your attitudes towards mathematics? DON’T LIKE MATH. Have your attitudes towards mathematics changed since you worked on the number theory problems? NO. Explain. STILL DON’T LIKE MATH.

8. As you worked on the number theory problems, were you aware of what you were thinking about? YES. As you worked on the problems, were you evaluating your thoughts? NOT SURE. For any of the problems, did you decide to change your thought 182

processes and strategies if you found that you were not getting closer to a solution? NO. Did you go through any of these steps in any of your mathematics classes before you started working on the number theory problems? YES. Explain further. TRIED TO CHANGE STRATEGY IN OTHER MATH CLASSES.

9. As you were working on the number theory problems, did you prefer to think on your own about how to find the solutions? YES. Why or why not? THOUGHT THIS WAS AN INDEPENDENT THING. DID NOT SHOW PROBLEMS TO FRIENDS.

10. Were you able to solve the number theory problems more easily if you worked on them with your fellow students? If so, how did the group work help you? DID NOT SHOW PROBLEMS TO FRIENDS.

11. Before starting your work on the number theory problems, to what extent, if any, did you believe that new mathematical concepts and problems continue to be discovered? YES, TO SOME EXTENT. Do you have that same belief after having completed your work on the number theory problems? YES.

12. How do you feel about making mistakes in mathematics? NOT TOO FRUSTRATED. DON’T REALLY FEEL ANYTHING. Do you see mistakes as something to be ashamed of, or as a part of learning? SOMETHING TO BE ASHAMED OF, NO. PART OF LEARNING, YES. Do you feel the same way about mistakes which you may have made on the number theory problems? Explain. YES. PART OF LEARNING.

13. As you were working on the number theory problems, did you quietly explain to yourself the procedures which you needed to follow in order to solve the problems? YES. If not, what exactly were you thinking as you worked on the problems?

14. In your past mathematics courses, did you ever think about how you could improve your mathematics ability? YES. If so, what ideas did you have? ASK FOR EXTRA HELP. Did you think of any new ideas for improving your mathematics ability after working on the number theory problems? NO. If so, what new ideas do you think of?

15. While working on the number theory problems, which of the following activities did you engage in: analyzing the problem YES, planning the method of the solution YES, checking the work YES. WOULD RE-READ QUESTION IF I THOUGHT I KNEW THE ANSWER, reflecting on the problem YES. Did you engage in those activities to a greater extent while working on the number theory problems, as compared to your work in your prior mathematics classes? DID MORE OF THIS WHILE IN SCHOOL.

16. Had you ever previously studied or read about the topics covered by the number theory problems? NO. If so, how did that prior learning affect your work on the number theory problems? If the number theory problems presented topics which you had never heard of before, did the problems make you feel surprised NO, anxious YES, frustrated A 183

LITTLE, or curious? A LITTLE BIT. FOUND IT CONFUSING. MADE ME THINK “WHAT IS THIS?”

17. Having worked on the number theory problems, do you feel interested in doing Internet searches to learn more about number theory topics? NO. Why or why not? MATH DOESN’T INTEREST ME. IT IS MY WEAK SUBJECT.

184

Appendix H

Student Q3 Responses

185

Table Q3A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 3 2. My effort and persistence determine how well I perform on a mathematics test. 4 3. When I am studying mathematics, I attempt to use study methods that have been used 5 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about it. 2 5. If I have a real interest in a mathematics topic, I can learn more about it. 5 6. When I want to improve my understanding of a mathematics concept, I draw sketches, 3 pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 2 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 2 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 1 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 3 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 3 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 2 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 4 14. In order to help myself understand a mathematics concept, I think of my own 4 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 5 16. After I have read a mathematics problem, I know if I am able to solve it. 3 17. When I am asked to solve a mathematics problem, I focus on whatever data are 4 presented. 18. If I write the data presented by a problem, I have a better understanding of the 4 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 3 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 3 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 4 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 4 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 3 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 2 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 4 my teacher.

186

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 4 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 5 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 4 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 1 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 1 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 1 34. I enjoy finding mathematics problems which have never been solved. 3 35. When I am with my friends, I discuss new mathematics concepts which I have 1 recently learned.

Notes: i.) Student Q3’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 105/150 = 70.00% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 7/25 = 28.00%

187

Table Q3C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 3 concepts. 2. My effort and persistence determined how well I was able to do on the number theory 3 problems. 3. When I worked on the number theory problems, I attempted to use techniques that 4 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 5 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 3 relate, I can learn more about them. 6. When I wanted to improve my understanding of the concepts referenced by the 4 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 3 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I learned 3 anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 3 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the most 4 important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 3 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 5 help. 14. In order to help myself understand the concepts relating to the number theory 3 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 4 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 3 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 3 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 1 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 4 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 3 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 3 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 3 times. 23. If I become confused in trying to solve the number theory problems, I tried to clear 3 up my confusion. 24. When I was working on the number theory problems, I thought about whether or not 4 I was working on the main questions presented by the problems.

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Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 3 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 3 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 3 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 4 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 3 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 3 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 3 of mathematics books while I am browsing in a bookstore or library, 32. Having completed my work on the number theory problems, I enjoy thinking of 3 new mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 3 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 4 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 1 learned from working on the number theory problems.

Notes: i.) Student Q3’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 99/150 = 66.00% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 14/25 = 56.00%

189

Table Q3A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 4. If I already know something about a mathematics 2 4. If I already know something about the 5 topic, I can learn even more about it. concepts referenced in the number theory problems, I can learn even more about them. 6. When I want to improve my understanding of a 3 6. When I wanted to improve my 4 mathematics concept, I draw sketches, pictures or understanding of the concepts referenced by diagrams relating to the concept. the number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I start trying to learn a new mathematics 2 7. Before I started trying to solve one or more 3 concept, I decide on clear goals that I must reach as of the number theory problems, I decided on part of the learning process. clear goals that I needed to reach as part of the problem solving process. 8. As I am studying a new mathematics lesson, I 2 8. As I was working on the number theory 3 evaluate my progress in learning the material. problems, I evaluated my progress. 9. After completing my mathematics homework, I ask 1 9. After completing my work on the number 3 myself if I have learned anything new and valuable. theory problems, I asked myself if I learned anything new and valuable. 11. After I complete my work on a mathematics 3 11. After I completed my work on the number 4 problem, I think about the most important elements of theory problems, I thought about the most the problem to make sure I really understand them. important elements of the problems to make sure I really understood them. 12. I try different ways of learning mathematics, 2 12. I tried different ways of approaching the 3 depending on the type of mathematics involved. number theory problems, depending on the type of concepts involved. 13. Whenever I am confused in mathematics class, I 4 13. Whenever I was confused by the number 5 ask someone else for help. theory problems, I asked someone else for help. 19. When I am trying to solve a mathematics problem, 3 19. In attempting to solve the number theory 4 I try to use solutions of similar problems which I problems, I tried to use solutions of similar encountered in the past. problems which I had encountered in the past. 24. When I am working on a mathematics problem, I 3 24. When I was working on the number theory 4 think about whether or not I am working on what the problems, I thought about whether or not I was problem is really asking. working on the main questions presented by the problems. 25. Once I find a solution to a mathematics problem, I 2 25. Once I found a solution to the number 3 try to find some other, different solutions to the same theory problems, I tried to find some other, problem. different solutions to the same problems. 31. When I am browsing in a bookstore or library, I 1 31. Having completed my work on the number 3 flip through the pages of mathematics books. theory problems, I flip through the pages of mathematics books while I am browsing in a bookstore or library. 32. I enjoy thinking of new mathematics problems to 1 32. Having completed my work on the number 3 work on. theory problems, I enjoy thinking of new mathematics problems to work on. 33. I enjoy preparing my own mathematics tests, 1 33. Having completed my work on the number 3 quizzes and puzzles. theory problems, I enjoy preparing my own mathematics tests, quizzes and puzzles. 34. I enjoy finding mathematics problems which have 3 34. Having completed my work on the number 4 never been solved. theory problems, I enjoy finding mathematics problems which have never been solved.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q3’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

190

Student Q3 Interview Responses

(As used in the questions below, the “number theory problems” refer to the elementary number theory problems in Part B of the questionnaire used in this study.)

1. What types of strategies or research methods did you use as you tried to solve the number theory problems? USED A LOT OF SCRAP PAPER. Were any of these strategies or methods different from the strategies you previously used in your mathematics courses? YES. If so, explain how they were different. USED A LOT OF “GUESS AND CHECK”. Was there one number theory problem that you liked the most? YES. ANYTHING THAT WAS A PLUG-IN TYPE PROBLEM. If so, explain the thinking process and strategy you used in your attempt to solve that particular problem. JUST PLUGGED-IN THE NUMBERS.

2. Would you say that the number theory problems were more interesting or less interesting than your other mathematics coursework? THEY WERE ALRIGHT. Explain your answer. LIKED REGULAR MATH COURSES BETTER BECAUSE THEY WERE EASIER.

3. As you worked on the number theory problems, did you feel bored or discouraged? BORED, YES. DISCOURAGED, NO. If so, explain why you think you felt that way. NEEDED MORE TIME TO WORK ON THEM.

4. Did you enjoy working on the number theory problems? NOT REALLY. Why or why not? THEY REQUIRED A LOT OF TIME.

5. Do you usually discuss mathematics problems with your friends? RIGHT AFTER MATH CLASS, ONLY. Is there anything about the number theory problems that made you want to discuss them with your friends? NO. If so, explain. If you discussed the number theory problems with your friends, about how much time did you spend discussing the problems? Was this discussion longer than other discussions you may have previously had with your friends on mathematics topics? WE USUALLY DON’T TALK ABOUT MATH.

6. After completing your work on the number theory problems, did you want to look through a number theory textbook to find information about other number theory concepts? NO. NOT REALLY INTERESTED. If so, explain. Did you think about changing any of the number theory problems? YES. If so, explain. NOT REALLY SURE, BUT WANTED TO MAKE THE PROBLEMS EASIER.

7. Before working on the number theory problems, what were your attitudes towards mathematics? INTERESTING. INVOLVES A LOT OF READING. Have your attitudes towards mathematics changed since you worked on the number theory problems? YES. Explain. THERE ARE MANY MORE ASPECTS OF MATHEMATICS.

191

8. As you worked on the number theory problems, were you aware of what you were thinking about? NO. As you worked on the problems, were you evaluating your thoughts? NO. For any of the problems, did you decide to change your thought processes and strategies if you found that you were not getting closer to a solution? YES. Did you go through any of these steps in any of your mathematics classes before you started working on the number theory problems? YES. Explain further. TRIED TO CHANGE STRATEGIES TO GET A SOLUTION.

9. As you were working on the number theory problems, did you prefer to think on your own about how to find the solutions? YES. AT FIRST. Why or why not? WANTED TO DO THEM ALONE AT FIRST, THEN ASKED PARENTS FOR HELP.

10. Were you able to solve the number theory problems more easily if you worked on them with your fellow students? NO. If so, how did the group work help you?

11. Before starting your work on the number theory problems, to what extent, if any, did you believe that new mathematical concepts and problems continue to be discovered? YES, I BELIEVED THIS. Do you have that same belief after having completed your work on the number theory problems? YES.

12. How do you feel about making mistakes in mathematics? IF IT IS A LONG PROBLEM, THEN A MISTAKE EARLY ON COULD MESS UP THE WHOLE PROBLEM. Do you see mistakes as something to be ashamed of, or as a part of learning? PART OF LEARNING. Do you feel the same way about mistakes which you may have made on the number theory problems? YES. Explain. PART OF LEARNING.

13. As you were working on the number theory problems, did you quietly explain to yourself the procedures which you needed to follow in order to solve the problems? YES. BEFORE, DID NOT DO THIS THAT MUCH. If not, what exactly were you thinking as you worked on the problems?

14. In your past mathematics courses, did you ever think about how you could improve your mathematics ability? YES. If so, what ideas did you have? USE REPETITION. Did you think of any new ideas for improving your mathematics ability after working on the number theory problems? YES. If so, what new ideas do you think of? STUDY MORE.

15. While working on the number theory problems, which of the following activities did you engage in: analyzing the problem, planning the method of the solution, checking the work, reflecting on the problem. ANALYZING, YES. BROKE DOWN PROBLEM AND TRIED TO SOLVE IT. PLANNING METHOD, NO. CHECKING WORK, YES. REFLECTING ON THE PROBLEM, NO. Did you engage in those activities to a greater extent while working on the number theory problems, as compared to your work in your prior mathematics classes? DID THESE MORE IN OTHER MATH CLASSES.

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17. Having worked on the number theory problems, do you feel interested in doing Internet searches to learn more about number theory topics? NO. Why or why not? NOT INTERESTED IN NUMBER THEORY.

193

Appendix I

Student Q4 Responses

194

Table Q4A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 4 3. When I am studying mathematics, I attempt to use study methods that have been used 4 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 4 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 4 6. When I want to improve my understanding of a mathematics concept, I draw 2 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 2 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 2 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 3 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 2 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 4 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 3 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 4 14. In order to help myself understand a mathematics concept, I think of my own 3 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 3 16. After I have read a mathematics problem, I know if I am able to solve it. 3 17. When I am asked to solve a mathematics problem, I focus on whatever data are 4 presented. 18. If I write the data presented by a problem, I have a better understanding of the 4 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 3 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 3 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 4 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 4 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 3 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 2 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 3 my teacher.

195

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 4 28. When I cannot find the solution to a mathematics problem, I know the nature of the 3 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 4 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 4 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 2 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 2 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 2 34. I enjoy finding mathematics problems which have never been solved. 2 35. When I am with my friends, I discuss new mathematics concepts which I have 2 recently learned.

Notes: i.) Student Q4’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 100/150 = 66.67% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 10/25 = 40.00%

196

Table Q4C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 3 concepts. 2. My effort and persistence determined how well I was able to do on the number 4 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 4 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 4 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which 3 they relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 2 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I 2 decided on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 2 9. After completing my work on the number theory problems, I asked myself if I 3 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 4 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 3 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on 3 the type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else 4 for help. 14. In order to help myself understand the concepts relating to the number theory 3 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 3 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 4 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 4 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 2 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of 2 similar problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of 3 the problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions 2 to help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 4 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear 3 up my confusion.

197

Question from Questionnaire Part C Score 24. When I was working on the number theory problems, I thought about whether or 4 not I was working on the main questions presented by the problems. 25. Once I found a solution to the number theory problems, I tried to find some other, 3 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 3 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 2 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature 3 of the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques 3 which vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 4 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the 3 pages of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of 3 new mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 4 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 4 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 4 learned from working on the number theory problems.

Notes: i.) Student Q4’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 93/150 = 62.00% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 18/25 = 72.00%

198

Table Q4A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 10. As soon as I solve a mathematics problem, 2 10. As soon as I solved one or more of 4 I wonder if I could have found an easier or the number theory problems, I quicker way to solve it. wondered if I could have found easier or quicker ways to arrive at the solution(s). 16. After I have read a mathematics problem, I 3 16. After I finished reading the number 4 know if I am able to solve it. theory problems, I knew if I was able to solve them. 24. When I am working on a mathematics 3 24. When I was working on the number 4 problem, I think about whether or not I am theory problems, I thought about working on what the problem is really asking. whether or not I was working on the main questions presented by the problems. 25. Once I find a solution to a mathematics 2 25. Once I found a solution to the 3 problem, I try to find some other, different number theory problems, I tried to find solutions to the same problem. some other, different solutions to the same problems. 31. When I am browsing in a bookstore or 2 31. Having completed my work on the 3 library, I flip through the pages of number theory problems, I flip through mathematics books. the pages of mathematics books while I am browsing in a bookstore or library. 32. I enjoy thinking of new mathematics 2 32. Having completed my work on the 3 problems to work on. number theory problems, I enjoy thinking of new mathematics problems to work on. 33. I enjoy preparing my own mathematics 2 33. Having completed my work on the 4 tests, quizzes and puzzles. number theory problems, I enjoy preparing my own mathematics tests, quizzes and puzzles. 34. I enjoy finding mathematics problems 2 34. Having completed my work on the 4 which have never been solved. number theory problems, I enjoy finding mathematics problems which have never been solved. 35. When I am with my friends, I discuss new 2 35. When I am with my friends, I 4 mathematics concepts which I have recently discuss new mathematics concepts learned. which I have learned from working on the number theory problems.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q4’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

199

Student Q4 Interview Responses

(As used in the questions below, the “number theory problems” refer to the elementary number theory problems in Part B of the questionnaire used in this study.)

1. What types of strategies or research methods did you use as you tried to solve the number theory problems? SUBSTITUTION RATHER THAN USE OF VARIABLES. Were any of these strategies or methods different from the strategies you previously used in your mathematics courses? NO. If so, explain how they were different. Was there one number theory problem that you liked the most? NO. MATH IS NOT MY FAVORITE SUBJECT. If so, explain the thinking process and strategy you used in your attempt to solve that particular problem.

2. Would you say that the number theory problems were more interesting or less interesting than your other mathematics coursework? MORE INTERESTING. Explain your answer. THE NUMBER THEORY PROBLEMS WERE LONGER. I PREFER PROBLEMS WITH LONGER INFORMATION. I ENJOY READING LONGER PROBLEMS.

3. As you worked on the number theory problems, did you feel bored or discouraged? BORED, NOT REALLY. DISCOURAGED, NO. If so, explain why you think you felt that way.

4. Did you enjoy working on the number theory problems? YES. Why or why not? THEY WERE DIFFERENT FROM THE REGULAR MATH IN SCHOOL. THEY WERE MORE LIKE WORD PROBLEMS.

6. After completing your work on the number theory problems, did you want to look through a number theory textbook to find information about other number theory concepts? NO. If so, explain. Did you think about changing any of the number theory problems? NO. If so, explain.

7. Before working on the number theory problems, what were your attitudes towards mathematics? MATH IS SLIGHTLY INTERESTING, BUT HARD. Have your attitudes towards mathematics changed since you worked on the number theory problems? NOT REALLY. Explain.

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8. As you worked on the number theory problems, were you aware of what you were thinking about? YES, I WAS THINKING ABOUT THE PROBLEMS. As you worked on the problems, were you evaluating your thoughts? NOT REALLY. For any of the problems, did you decide to change your thought processes and strategies if you found that you were not getting closer to a solution? NO. Did you go through any of these steps in any of your mathematics classes before you started working on the number theory problems? NOT REALLY. Explain further.

9. As you were working on the number theory problems, did you prefer to think on your own about how to find the solutions? YES. Why or why not? THINGS BECOME MORE CONFUSING WHEN SOMEONE ELSE TRIES TO HELP ME WITH MATH.

10. Were you able to solve the number theory problems more easily if you worked on them with your fellow students? NO. If so, how did the group work help you?

11. Before starting your work on the number theory problems, to what extent, if any, did you believe that new mathematical concepts and problems continue to be discovered? I BELIEVED THIS. Do you have that same belief after having completed your work on the number theory problems? YES.

12. How do you feel about making mistakes in mathematics? THERE IS ALWAYS A REASON WHY A MISTAKE WAS MADE. Do you see mistakes as something to be ashamed of, or as a part of learning? PART OF LEARNING. Do you feel the same way about mistakes which you may have made on the number theory problems? YES. Explain. THERE WAS A REASON WHY I MADE MISTAKES.

14. In your past mathematics courses, did you ever think about how you could improve your mathematics ability? YES. If so, what ideas did you have? WORKING ON MORE PROBLEMS OUTSIDE OF CLASS TO BETTER MY SKILLS. Did you think of any new ideas for improving your mathematics ability after working on the number theory problems? NOT REALLY. If so, what new ideas do you think of?

15. While working on the number theory problems, which of the following activities did you engage in: analyzing the problem YES, planning the method of the solution YES, checking the work YES, reflecting on the problem NO. Did you engage in those activities to a greater extent while working on the number theory problems, as compared to your work in your prior mathematics classes? NO.

17. Having worked on the number theory problems, do you feel interested in doing Internet searches to learn more about number theory topics? YES. Why or why not? THE NUMBER THEORY PROBLEMS INVOLVED CONCEPTS THAT I HAD NEVER HEARD OF BEFORE. I WANT TO LEARN HOW TO USE THESE CONCEPTS. THE PROBLEMS WERE INTERESTING. I WANT TO LOOK INTO THEM MORE.

202

Appendix J

Student Q5 Responses

203

Table Q5A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 4 3. When I am studying mathematics, I attempt to use study methods that have been 2 used by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 3 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 5 6. When I want to improve my understanding of a mathematics concept, I draw 3 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that 1 I must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 3 material. 9. After completing my mathematics homework, I ask myself if I have learned 3 anything new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 5 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 4 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 5 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 4 14. In order to help myself understand a mathematics concept, I think of my own 5 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 4 16. After I have read a mathematics problem, I know if I am able to solve it. 3 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 4 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 3 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 2 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 4 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 3 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 4 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 3 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 1 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback 3 from my teacher.

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Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 2 28. When I cannot find the solution to a mathematics problem, I know the nature of the 4 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 3 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 3 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 3 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 3 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 1 34. I enjoy finding mathematics problems which have never been solved. 4 35. When I am with my friends, I discuss new mathematics concepts which I have 1 recently learned.

Notes: i.) Student Q5’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 102/150 = 68.00% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 12/25 = 48.00%

205

Table Q5C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 3 concepts. 2. My effort and persistence determined how well I was able to do on the number 3 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 1 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 3 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 4 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 1 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 3 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I 1 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 5 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 3 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 4 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 2 help. 14. In order to help myself understand the concepts relating to the number theory 5 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 3 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 2 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 4 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 2 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 3 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 3 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 4 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 2 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear 3 up my confusion. 24. When I was working on the number theory problems, I thought about whether or not 3 I was working on the main questions presented by the problems.

206

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 1 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 4 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 4 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 4 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 3 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 3 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 4 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 4 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 1 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 5 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 1 learned from working on the number theory problems.

Notes: i.) Student Q5’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 89/150 = 59.33% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 15/25 = 60.00%

207

Table Q5A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 7. Before I start trying to learn a new 1 7. Before I started trying to solve one 3 mathematics concept, I decide on clear goals or more of the number theory that I must reach as part of the learning problems, I decided on clear goals that process. I needed to reach as part of the problem solving process. 26. I know how I performed on my 3 26. I knew how I performed on the 4 mathematics test before I receive any feedback number theory problems as soon as I from my teacher. finished my work on the problems. 27. I believe that mathematics topics vary in 2 27. I believe that the concepts 4 their level of difficulty. referenced by the number theory problems vary in their level of difficulty. 31. When I am browsing in a bookstore or 3 31. Having completed my work on the 4 library, I flip through the pages of number theory problems, I flip through mathematics books. the pages of mathematics books while I am browsing in a bookstore or library. 32. I enjoy thinking of new mathematics 3 32. Having completed my work on the 4 problems to work on. number theory problems, I enjoy thinking of new mathematics problems to work on. 34. I enjoy finding mathematics problems 4 34. Having completed my work on the 5 which have never been solved. number theory problems, I enjoy finding mathematics problems which have never been solved.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q5’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

208

Student Q5 Interview Responses

(As used in the questions below, the “number theory problems” refer to the elementary number theory problems in Part B of the questionnaire used in this study.)

1. What types of strategies or research methods did you use as you tried to solve the number theory problems? FOR SOME OF THE PROBLEMS, I SUBSTITUTED NUMBERS. FOR OTHER PROBLEMS, I USED VARIABLES. Were any of these strategies or methods different from the strategies you previously used in your mathematics courses? YES. WE NEVER COVERED NUMBER THEORY IN MY MATH CLASSES. If so, explain how they were different. THE NUMBER THEORY PROBLEMS INVOLVED SOME OPERATIONS THAT I NEVER SAW BEFORE IN PROBLEMS – LIKE QUESTION 7 ON MODS. I TRIED TO USE DIFFERENT NUMBERS TO PROVE THAT ONE. IN SCHOOL, WE STUDY ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION, AND I CAN SUBSTITUTE NUMBERS TO CHECK THOSE OPERATIONS. Was there one number theory problem that you liked the most? YES. THE ONE ON PERFECT NUMBERS. If so, explain the thinking process and strategy you used in your attempt to solve that particular problem. THOUGHT ABOUT THE DEFINITIONS.

2. Would you say that the number theory problems were more interesting or less interesting than your other mathematics coursework? I LIKED MY NORMAL MATH COURSES BETTER. Explain your answer. MY NORMAL MATH COURSES HAVE MORE APPLICATIONS IN SCIENCE. THE NUMBER THEORY PROBLEMS WERE LIKE PUZZLES WHICH DID NOT HAVE ANY APPLICATIONS.

3. As you worked on the number theory problems, did you feel bored or discouraged? NOT BORED AND NOT DISCOURAGED. I WOULD JUST REREAD THE PROBLEMS IF I NEEDED TO. If so, explain why you think you felt that way.

4. Did you enjoy working on the number theory problems? YES, A FEW OF THEM. Why or why not? I ENJOYED THE ONES I WAS ABLE TO DO WITHOUT SUBSTITUTING NUMBERS.

5. Do you usually discuss mathematics problems with your friends? NO. Is there anything about the number theory problems that made you want to discuss them with your friends? YES. If so, explain. I WAS CURIOUS TO SEE HOW ONE OF MY FRIENDS DID THEM. If you discussed the number theory problems with your friends, about how much time did you spend discussing the problems? ABOUT 15 MINUTES. Was this discussion longer than other discussions you may have previously had with your friends on mathematics topics? YES. I USUALLY TAKE ABOUT 5 MINUTES TO DISCUSS MATH WITH MY FRIENDS BEFORE CLASS.

6. After completing your work on the number theory problems, did you want to look through a number theory textbook to find information about other number theory concepts? YES. If so, explain. I WAS CURIOUS TO READ MORE ABOUT PROOFS OF ABSTRACT CONCEPTS. THERE ARE BOOKS ABOUT THIS 209

SORT OF THING. Did you think about changing any of the number theory problems? NO. If so, explain.

7. Before working on the number theory problems, what were your attitudes towards mathematics? I LIKE MATH. IT IS PRETTY STRAIGHTFORWARD, AND YOU ALWAYS KNOW WHEN YOU HAVE THE RIGHT ANSWER. Have your attitudes towards mathematics changed since you worked on the number theory problems? NO. Explain. I STILL LIKE MATH, BUT FOR THE NUMBER THEORY PROBLEMS THERE WERE DIFFERENT WAYS TO REACH THE RIGHT ANSWER.

8. As you worked on the number theory problems, were you aware of what you were thinking about? YES. As you worked on the problems, were you evaluating your thoughts? YES. For any of the problems, did you decide to change your thought processes and strategies if you found that you were not getting closer to a solution? YES. Did you go through any of these steps in any of your mathematics classes before you started working on the number theory problems? YES. Explain further. FOR ALL MY MATH CLASSES, THERE ARE TOPICS WHERE THERE IS SOME SHORT- CUT WAY OR PATTERN FOR DOING PROBLEMS MORE EASILY.

9. As you were working on the number theory problems, did you prefer to think on your own about how to find the solutions? YES. Why or why not? FOR MATH, I PREFER TO DO THINGS MYSELF. I LIKE TO STICK TO MY OWN WAY OF DOING MATH.

10. Were you able to solve the number theory problems more easily if you worked on them with your fellow students? NO. If so, how did the group work help you?

11. Before starting your work on the number theory problems, to what extent, if any, did you believe that new mathematical concepts and problems continue to be discovered? I KNEW THAT THERE ARE CONJECTURES ABOUT MATH, LIKE PRIME NUMBERS AND THE INFINITE NUMBER OF PRIMES. Do you have that same belief after having completed your work on the number theory problems? YES.

12. How do you feel about making mistakes in mathematics? IT GIVES ME A CHANCE TO REALIZE MY MISTAKE, START FRESH AND DO THE WHOLE PROBLEM OVER AND RETHINK EACH STEP. Do you see mistakes as something to be ashamed of, or as a part of learning? PART OF LEARNING. Do you feel the same way about mistakes which you may have made on the number theory problems? YES. Explain. IT WAS PART OF LEARNING.

14. In your past mathematics courses, did you ever think about how you could improve your mathematics ability? YES. If so, what ideas did you have? PRACTICE MORE 210

AND DO MORE QUESTIONS. IT THEN BECOMES CLEARER. Did you think of any new ideas for improving your mathematics ability after working on the number theory problems? YES. If so, what new ideas do you think of? THINK OUTSIDE- THE-BOX MORE. THE NUMBER THEORY PROBLEMS DIFFERED FROM THE MATH PROBLEMS IN SCHOOL. THE NUMBER THEORY PROBLEMS COULD HELP ME USE ABSTRACT PRINCIPLES FOR MY OTHER MATH COURSES.

15. While working on the number theory problems, which of the following activities did you engage in: analyzing the problem, planning the method of the solution, checking the work, reflecting on the problem. ANALYZING? YES. PLANNING? YES. CHECKING? YES. REFLECTING? A LITTLE, BUT NOT A LOT. FOR SOME PROBLEMS, NONE. Did you engage in those activities to a greater extent while working on the number theory problems, as compared to your work in your prior mathematics classes? EQUALLY FOR BOTH.

16. Had you ever previously studied or read about the topics covered by the number theory problems? HEARD SOMETHING ABOUT MOD M BEFORE. SAW PALINDROMES BEFORE. If so, how did that prior learning affect your work on the number theory problems? IT DIDN’T HELP. If the number theory problems presented topics which you had never heard of before, did the problems make you feel surprised, anxious, frustrated, or curious? SURPRISED? NO. ANXIOUS? NO. FRUSTRATED? NO. CURIOUS? YES. THERE WERE NEW TOPICS FOR ME TO FIGURE OUT.

17. Having worked on the number theory problems, do you feel interested in doing Internet searches to learn more about number theory topics? YES. Why or why not? THE NUMBER THEORY PROBLEMS SORT OF CAUGHT MY INTEREST, BECAUSE OF THE WAY THEY WERE PRESENTED. I WANT TO RESEARCH THEM FURTHER. THEY MADE ME CURIOUS.

FOR THE NUMBER THEORY PROBLEMS, I HAD TO THINK OF THE END- RESULT BEFORE STARTING TO WORK ON THEM. THESE PROBLEMS MADE ME THINK ABOUT MATH AS “PURE” AND NOT AS A TOOL. THEY MADE ME THINK OF MATH ITSELF. SOME MATH CAN BE DONE IN SMALL STEPS, LIKE THE PALINDROME PROBLEM. SOME OF THE NUMBER THEORY PROBLEMS INVOLVED TOO MANY STEPS AND WERE DIFFICULT IN THEMSELVES.

211

Appendix K

Student Q6 Responses

212

Table Q6A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 5 2. My effort and persistence determine how well I perform on a mathematics test. 5 3. When I am studying mathematics, I attempt to use study methods that have been used 2 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 4 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 4 6. When I want to improve my understanding of a mathematics concept, I draw 1 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 1 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 2 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 3 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 3 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 3 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 3 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 5 14. In order to help myself understand a mathematics concept, I think of my own 2 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 4 16. After I have read a mathematics problem, I know if I am able to solve it. 4 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 4 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 5 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 4 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 3 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 4 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 5 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 5 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 2 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 3 my teacher.

213

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 3 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 4 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 5 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 1 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 1 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 1 34. I enjoy finding mathematics problems which have never been solved. 2 35. When I am with my friends, I discuss new mathematics concepts which I have 1 recently learned.

Notes: i.) Student Q6’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 108/150 = 72.00% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 6/25 = 24.00%

214

Table Q6C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 4 concepts. 2. My effort and persistence determined how well I was able to do on the number theory 5 problems. 3. When I worked on the number theory problems, I attempted to use techniques that 1 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 4 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 5 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 2 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 2 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I learned 3 anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 4 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the most 3 important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 4 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 5 help. 14. In order to help myself understand the concepts relating to the number theory 5 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 5 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 3 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 5 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 4 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 4 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 4 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 3 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear up 5 my confusion. 24. When I was working on the number theory problems, I thought about whether or not 4 I was working on the main questions presented by the problems.

215

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 3 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 4 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 3 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 3 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 3 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 4 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 1 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 3 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 2 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 4 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 1 learned from working on the number theory problems.

Notes: i.) Student Q6’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 112/150 = 74.67% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 11/25 = 44.00%

216

Table Q6A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 5. If I have a real interest in a mathematics topic, I 4 5. If I have a real interest in the number 5 can learn more about it. theory problems and the concepts to which they relate, I can learn more about it. 6. When I want to improve my understanding of a 1 6. When I wanted to improve my 2 mathematics concept, I draw sketches, pictures or understanding of the concepts referenced by diagrams relating to the concept. the number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I start trying to learn a new mathematics 1 7. Before I started trying to solve one or 2 concept, I decide on clear goals that I must reach as more of the number theory problems, I part of the learning process. decided on clear goals that I needed to reach as part of the problem solving process. 8. As I am studying a new mathematics lesson, I 2 8. As I was working on the number theory 3 evaluate my progress in learning the material. problems, I evaluated my progress. 10. As soon as I solve a mathematics problem, I 3 10. As soon as I solved one or more of the 4 wonder if I could have found an easier or quicker number theory problems, I wondered if I way to solve it. could have found easier or quicker ways to arrive at the solution(s). 12. I try different ways of learning mathematics, 3 12. I tried different ways of approaching the 4 depending on the type of mathematics involved. number theory problems, depending on the type of concepts involved. 14. In order to help myself understand a 2 14. In order to help myself understand the 5 mathematics concept, I think of my own examples concepts relating to the number theory or illustrations of the concept. problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I 4 15. I have my own ways of remembering 5 have learned in mathematics what I have learned from the number theory problems. 22. If I am having difficulty with a mathematics 4 22. If I had difficulty with the number 5 problem, I read the problem several times to make theory problems, I read the problems sure I understand it. several times. 25. Once I find a solution to a mathematics 2 25. Once I found a solution to the number 3 problem, I try to find some other, different solutions theory problems, I tried to find some other, to the same problem. different solutions to the same problems. 26. I know how I performed on my mathematics 3 26. I knew how I performed on the number 4 test before I receive any feedback from my teacher. theory problems as soon as I finished my work on the problems. 32. I enjoy thinking of new mathematics problems 1 32. Having completed my work on the 3 to work on. number theory problems, I enjoy thinking of new mathematics problems to work on. 33. I enjoy preparing my own mathematics tests, 1 33. Having completed my work on the 2 quizzes and puzzles. number theory problems, I enjoy preparing my own mathematics tests, quizzes and puzzles. 34. I enjoy finding mathematics problems which 2 34. Having completed my work on the 4 have never been solved. number theory problems, I enjoy finding mathematics problems which have never been solved.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q6’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

217

Student Q6 Interview Responses

(As used in the questions below, the “number theory problems” refer to the elementary number theory problems in Part B of the questionnaire used in this study.)

1. What types of strategies or research methods did you use as you tried to solve the number theory problems? IDENTIFIED WHAT WAS GIVEN AND WHAT I NEEDED TO FIND. USED PRIOR KNOWLEDGE OF MATHEMATICS. Were any of these strategies or methods different from the strategies you previously used in your mathematics courses? NO. If so, explain how they were different. Was there one number theory problem that you liked the most? DO NOT REMEMBER. If so, explain the thinking process and strategy you used in your attempt to solve that particular problem.

2. Would you say that the number theory problems were more interesting or less interesting than your other mathematics coursework? MORE INTERESTING. Explain your answer. PRIOR MATHEMATICS COURSES INVOLVED NORMAL PROBLEMS WITH NUMBERS. NUMBER THEORY PROBLEMS INVOLVED MORE THEORIES, ESPECIALLY THE PROBLEM ABOUT PERFECT NUMBERS.

3. As you worked on the number theory problems, did you feel bored or discouraged? NO. If so, explain why you think you felt that way.

4. Did you enjoy working on the number theory problems? YES. Why or why not? THE NUMBER THEORY PROBLEMS WERE NEW AND INVOLVED THINGS I HAD NEVER SEEN BEFORE.

5. Do you usually discuss mathematics problems with your friends? SOMETIMES, BUT ONLY CERTAIN PROBLEMS FROM CLASS. Is there anything about the number theory problems that made you want to discuss them with your friends? YES. If so, explain. DISCUSSED THE PALINDROME PROBLEM WITH A FRIEND, BECAUSE IT WAS INTERESTING. If you discussed the number theory problems with your friends, about how much time did you spend discussing the problems? 5-10 MINUTES. Was this discussion longer than other discussions you may have previously had with your friends on mathematics topics? ABOUT THE SAME LENGTH OF TIME.

7. Before working on the number theory problems, what were your attitudes towards mathematics? I ENJOY MATHEMATICS VERY MUCH. Have your attitudes towards mathematics changed since you worked on the number theory problems? NO. Explain. MY ATTITUDES STAYED THE SAME. 218

8. As you worked on the number theory problems, were you aware of what you were thinking about? YES. WAS THINKING OF A DIFFERENT WAY TO REACH THE ANSWER, WITHOUT USING A FORMULA. As you worked on the problems, were you evaluating your thoughts? DO NOT REMEMBER. For any of the problems, did you decide to change your thought processes and strategies if you found that you were not getting closer to a solution? YES, FOR THE PROBLEM ABOUT MOD M. Did you go through any of these steps in any of your mathematics classes before you started working on the number theory problems? YES. Explain further. IN GEOMETRY CLASS.

9. As you were working on the number theory problems, did you prefer to think on your own about how to find the solutions? YES. Why or why not? I LIKE TO TRY PROBLEMS ON MY OWN, AND THEN ASK MY FRIENDS ABOUT THEM.

10. Were you able to solve the number theory problems more easily if you worked on them with your fellow students? YES. If so, how did the group work help you? I WORKED ON PROBLEMS 6 AND 7 WITH A FRIEND. THIS HELPED BECAUSE MY FRIEND DID THE PROBLEMS IN A DIFFERENT WAY.

11. Before starting your work on the number theory problems, to what extent, if any, did you believe that new mathematical concepts and problems continue to be discovered? BELIEVED THAT PROBABLY THERE ARE NEW CONCEPTS AND PROBLEMS. Do you have that same belief after having completed your work on the number theory problems? YES.

12. How do you feel about making mistakes in mathematics? I DO NOT TAKE IT TOO BADLY. MISTAKES HELP ME TO FIX THE PROBLEM LATER. Do you see mistakes as something to be ashamed of, or as a part of learning? IT IS PART OF LEARNING. Do you feel the same way about mistakes which you may have made on the number theory problems? YES. Explain. I FEEL THE SAME WAY.

13. As you were working on the number theory problems, did you quietly explain to yourself the procedures which you needed to follow in order to solve the problems? NOT CONSCIOUSLY. BUT THOUGHT ABOUT IT IN MY MIND. If not, what exactly were you thinking as you worked on the problems?

14. In your past mathematics courses, did you ever think about how you could improve your mathematics ability? YES. If so, what ideas did you have? USE EXTRA EXAMPLES TO IMPROVE SKILLS AND ASK MY FAMILY FOR HELP. Did you think of any new ideas for improving your mathematics ability after working on the number theory problems? YES. If so, what new ideas do you think of? BOUNCING IDEAS OFF ONE OF MY FRIENDS.

15. While working on the number theory problems, which of the following activities did you engage in: analyzing the problem YES, SEE WHAT IS GIVEN AND WHAT NEEDS TO BE FOUND, planning the method of the solution YES, FOR THE FIRST 4 PROBLEMS, FIGURING OUT AS I WENT ALONG FOR THE LAST 4 219

PROBLEMS, checking the work, YES, reflecting on the problem YES, FOR THE PERFECT NUMBER PROBLEM. Did you engage in those activities to a greater extent while working on the number theory problems, as compared to your work in your prior mathematics classes? YES, I DID THIS MORE FOR THE NUMBER THEORY PROBLEMS, BECAUSE THERE WERE SOME NEW QUESTIONS. MY NORMAL MATH INVOLVES THINGS I ALREADY KNOW.

16. Had you ever previously studied or read about the topics covered by the number theory problems? YES. I HEARD ABOUT THE PALINDROME PROBLEM IN ALGEBRA I, AND I KNEW ABOUT PROBLEM 8. If so, how did that prior learning affect your work on the number theory problems? IT HELPED ME UNDERSTAND. If the number theory problems presented topics which you had never heard of before, did the problems make you feel surprised A LITTLE, anxious NO, frustrated NO, or curious? YES, TO LEARN MORE ABOUT PERFECT NUMBERS AND OTHER TOPICS THAT I HAD NEVER SEEN BEFORE.

17. Having worked on the number theory problems, do you feel interested in doing Internet searches to learn more about number theory topics? DEPENDS. Why or why not? IT DEPENDS ON THE TOPICS. I WOULD WANT TO SEARCH MORE ABOUT PERFECT NUMBERS BECAUSE I FOUND THAT INTERESTING. THE PROBLEMS OPENED MY EYES TO NEW TOPICS THAT I WANT TO SEE MORE OF.

THE NUMBER THEORY PROBLEMS HAD MORE ABSTRACT CONCEPTS, AND FOR MORE ABSTRACT CONCEPTS I DREW PICTURES TO HELP ME WORK ON THEM. I DECIDED ON GOALS BEFORE WORKING ON THE PROBLEMS, AND I DID THAT IN OTHER MATH COURSES.

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Appendix L

Student Q7 Responses

221

Table Q7A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 5 2. My effort and persistence determine how well I perform on a mathematics test. 5 3. When I am studying mathematics, I attempt to use study methods that have been used 3 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about it. 4 5. If I have a real interest in a mathematics topic, I can learn more about it. 3 6. When I want to improve my understanding of a mathematics concept, I draw sketches, 2 pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 2 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 2 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 1 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 2 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 2 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 2 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 3 14. In order to help myself understand a mathematics concept, I think of my own 2 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 4 16. After I have read a mathematics problem, I know if I am able to solve it. 3 17. When I am asked to solve a mathematics problem, I focus on whatever data are 4 presented. 18. If I write the data presented by a problem, I have a better understanding of the 3 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 4 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 2 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 3 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 4 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 2 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 2 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 4 my teacher.

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Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 2 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 4 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 5 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 1 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 2 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 1 34. I enjoy finding mathematics problems which have never been solved. 2 35. When I am with my friends, I discuss new mathematics concepts which I have 2 recently learned.

Notes: i.) Student Q7’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 93/150 = 62.00% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 8/25 = 32.00%

223

Table Q7C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 4 concepts. 2. My effort and persistence determined how well I was able to do on the number theory 5 problems. 3. When I worked on the number theory problems, I attempted to use techniques that 4 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 4 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 3 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 4 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 2 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 2 9. After completing my work on the number theory problems, I asked myself if I learned 2 anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 4 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the most 2 important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 4 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 1 help. 14. In order to help myself understand the concepts relating to the number theory 4 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 2 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 3 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 4 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 5 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 5 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 4 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 2 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear up 4 my confusion. 24. When I was working on the number theory problems, I thought about whether or not 1 I was working on the main questions presented by the problems.

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Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 1 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 1 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 5 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 4 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 5 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 4 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 1 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 1 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 1 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 1 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 1 learned from working on the number theory problems.

Notes: i.) Student Q7’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 100/150 = 66.67% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 5/25 = 20.00%

225

Table Q7A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 3. When I am studying mathematics, I attempt 3 3. When I worked on the number 4 to use study methods that have been used by theory problems, I attempted to use successful mathematics students. techniques that have been used by successful mathematics students. 6. When I want to improve my understanding 2 6. When I wanted to improve my 4 of a mathematics concept, I draw sketches, understanding of the concepts pictures or diagrams relating to the concept. referenced by the number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 9. After completing my mathematics 1 9. After completing my work on the 2 homework, I ask myself if I have learned number theory problems, I asked anything new and valuable. myself if I learned anything new and valuable. 10. As soon as I solve a mathematics problem, 2 10. As soon as I solved one or more of 4 I wonder if I could have found an easier or the number theory problems, I quicker way to solve it. wondered if I could have found easier or quicker ways to arrive at the solution(s). 12. I try different ways of learning 2 12. I tried different ways of 4 mathematics, depending on the type of approaching the number theory mathematics involved. problems, depending on the type of concepts involved. 14. In order to help myself understand a 2 14. In order to help myself understand 4 mathematics concept, I think of my own the concepts relating to the number examples or illustrations of the concept. theory problems, I thought of my own examples or illustrations of the concepts. 18. If I write the data presented by a problem, 3 18. If I wrote the data presented by the 5 I have a better understanding of the problem. number theory problems, I had a better understanding of the problems. 19. When I am trying to solve a mathematics 4 19. In attempting to solve the number 5 problem, I try to use solutions of similar theory problems, I tried to use solutions problems which I encountered in the past. of similar problems which I had encountered in the past. 22. If I am having difficulty with a 3 22. If I had difficulty with the number 5 mathematics problem, I read the problem theory problems, I read the problems several times to make sure I understand it. several times. 28. When I cannot find the solution to a 2 28. When I could not find solutions to 4 mathematics problem, I know the nature of the the number theory problems, I knew difficulty I am experiencing. the nature of the difficulty I was experiencing. 29. I believe that mathematical problem 4 29. I believe that the number theory 5 solving techniques vary in their level of problems require problem solving difficulty. techniques which vary in their level of difficulty. Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q7’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 226

Student Q7 Interview Responses

(As used in the questions below, the “number theory problems” refer to the elementary number theory problems in Part B of the questionnaire used in this study.)

1. What types of strategies or research methods did you use as you tried to solve the number theory problems? NO RESEARCH METHODS. LOOKED AT PROBLEMS AND TRIED TO FIND OBVIOUS WAYS TO SOLVE THEM, JUST LIKE I TRY TO SOLVE ALGEBRA PROBLEMS. Were any of these strategies or methods different from the strategies you previously used in your mathematics courses? FOR THE LAST TWO NUMBER THEORY PROBLEMS, I NEVER SAW THEM BEFORE AND I DID NOT KNOW HOW TO DO THEM, SO I COULD NOT USE THE METHODS I USED ON ALGEBRA PROBLEMS. If so, explain how they were different. COULD NOT JUST LOOK AT THE NUMBER THEORY PROBLEMS AND SEE AN OBVIOUS WAY OF DOING THEM. Was there one number theory problem that you liked the most? I DON’T REMEMBER. If so, explain the thinking process and strategy you used in your attempt to solve that particular problem.

2. Would you say that the number theory problems were more interesting or less interesting than your other mathematics coursework? THE NUMBER THEORY PROBLEMS WERE MORE INTERESTING. Explain your answer. THE NUMBER THEORY PROBLEMS INVOLVED MORE PROBLEM-SOLVING , RATHER THAN BEING STRAIGHTFORWARD. I HAD TO THINK MORE ABOUT THE PROBLEMS AND SO THEY WERE MORE INTERESTING. I HAD A SENSE OF ACCOMPLISHMENT IF I COULD SOLVE THEM.

3. As you worked on the number theory problems, did you feel bored or discouraged? BORED, NOT REALLY. DISCOURAGED, YES. If so, explain why you think you felt that way. I FELT DISHEARTENED WHEN I DIDN’T KNOW HOW TO DO THE PROBLEMS.

4. Did you enjoy working on the number theory problems? YES. Why or why not? I LIKE MATH. I AM PRETTY GOOD AT IT. THERE IS ALWAYS AN ANSWER FOR A MATH PROBLEM. THERE IS ALWAYS A WAY TO SOLVE IT.

5. Do you usually discuss mathematics problems with your friends? NOT REALLY. ONLY RIGHT AFTER A MATH TEST. Is there anything about the number theory problems that made you want to discuss them with your friends? NO, BUT I DISCUSSED THEM WITH MY SISTER, WHO WAS ALSO WORKING ON THEM. If so, explain. WHEN I DON’T KNOW HOW TO DO A PROBLEM, I WANT TO ASK SOMEONE ELSE ABOUT IT. If you discussed the number theory problems with your friends, about how much time did you spend discussing the problems? FOR EACH PROBLEM I COULDN’T DO, I ASKED MY SISTER ABOUT IT FOR ABOUT 10-15 SECONDS. Was this discussion longer than other discussions you may have previously had with your friends on mathematics topics? NO, IT WAS A SHORTER TIME. WITH MY FRIENDS, I USUALLY JUST ASK IF THEY KNOW HOW TO DO A MATH PROBLEM. 227

6. After completing your work on the number theory problems, did you want to look through a number theory textbook to find information about other number theory concepts? YES. BUT I DIDN’T HAVE A NUMBER THEORY TEXTBOOK. If so, explain. I HAD NEVER HEARD OF NUMBER THEORY BEFORE. Did you think about changing any of the number theory problems? NO. If so, explain.

7. Before working on the number theory problems, what were your attitudes towards mathematics? I LIKED MATH. Have your attitudes towards mathematics changed since you worked on the number theory problems? NO. Explain. SAME ATTITUDES BEFORE AND AFTER DOING THE PROBLEMS.

8. As you worked on the number theory problems, were you aware of what you were thinking about? NO. As you worked on the problems, were you evaluating your thoughts? NO. For any of the problems, did you decide to change your thought processes and strategies if you found that you were not getting closer to a solution? ONCE IN A WHILE. Did you go through any of these steps in any of your mathematics classes before you started working on the number theory problems? YES. Explain further. I CHANGED MY STRATEGIES WHILE WORKING ON PRE-CALCULUS PROBLEMS.

9. As you were working on the number theory problems, did you prefer to think on your own about how to find the solutions? YES. Why or why not? IT IS MORE OF AN ACCOMPLISHMENT TO DO A PROBLEM ALONE.

10. Were you able to solve the number theory problems more easily if you worked on them with your fellow students? NO. If so, how did the group work help you? THERE WAS NO GROUP WORK.

12. How do you feel about making mistakes in mathematics? MISTAKES USUALLY RESULT FROM CARELESSNESS, OR THINKING ABOUT SOMETHING ELSE WHILE WORKING ON A PROBLEM. YOU CAN SOLVE EVERY PROBLEM IF YOU PUT YOUR MIND TO IT. Do you see mistakes as something to be ashamed of, or as a part of learning? PART OF LEARNING. Do you feel the same way about mistakes which you may have made on the number theory problems? Explain. YES. MISTAKES WERE PART OF LEARNING.

13. As you were working on the number theory problems, did you quietly explain to yourself the procedures which you needed to follow in order to solve the problems? NO. If not, what exactly were you thinking as you worked on the problems? HOW THESE WERE DIFFERENT KINDS OF PROBLEMS.

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14. In your past mathematics courses, did you ever think about how you could improve your mathematics ability? NO. If so, what ideas did you have? Did you think of any new ideas for improving your mathematics ability after working on the number theory problems? NO. If so, what new ideas do you think of?

15. While working on the number theory problems, which of the following activities did you engage in: analyzing the problem, planning the method of the solution, checking the work, reflecting on the problem. ANALYZING, YES. PLANNING, YES. CHECKING, NO. REFLECTING, NO. Did you engage in those activities to a greater extent while working on the number theory problems, as compared to your work in your prior mathematics classes? I ANALYZED THE PROBLEMS MORE WHEN I WAS WORKING ON THE NUMBER THEORY PROBLEMS.

16. Had you ever previously studied or read about the topics covered by the number theory problems? FOR SOME OF THEM, YES. If so, how did that prior learning affect your work on the number theory problems? IT HELPED ME SOLVE THEM. If the number theory problems presented topics which you had never heard of before, did the problems make you feel surprised, anxious, frustrated or curious? SURPRISED, NO. ANXIOUS, A LITTLE. FRUSTRATED, YES. CURIOUS, YES.

17. Having worked on the number theory problems, do you feel interested in doing Internet searches to learn more about number theory topics? YES. Why or why not? IT IS SOMETHING THAT I HAVEN’T LEARNED BEFORE. BEFORE SEEING THE NUMBER THEORY PROBLEMS, I THOUGHT I KNEW A LOT OF MATH. I AM CURIOUS TO LEARN MORE ABOUT NUMBER THEORY NOW.

MY ANSWERS ON THE QUESTIONNAIRE CHANGED AFTER DOING THE NUMBER THEORY PROBLEMS, BECAUSE OF THE DIFFICULTY OF THE PROBLEMS.

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Appendix M

Student Q8 Responses

230

Table Q8A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 5 3. When I am studying mathematics, I attempt to use study methods that have been used 3 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about it. 3 5. If I have a real interest in a mathematics topic, I can learn more about it. 2 6. When I want to improve my understanding of a mathematics concept, I draw sketches, 4 pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 1 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 3 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 2 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 3 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 2 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 3 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 4 14. In order to help myself understand a mathematics concept, I think of my own 2 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 2 16. After I have read a mathematics problem, I know if I am able to solve it. 3 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 4 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 3 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 3 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 4 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 4 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 3 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 3 my teacher.

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Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 5 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 3 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 4 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 1 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 3 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 1 34. I enjoy finding mathematics problems which have never been solved. 3 35. When I am with my friends, I discuss new mathematics concepts which I have 1 recently learned.

Notes: i.) Student Q8’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 101/150 = 67.33% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 9/25 = 36.00%

232

Table Q8C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 3 concepts. 2. My effort and persistence determined how well I was able to do on the number theory 4 problems. 3. When I worked on the number theory problems, I attempted to use techniques that 4 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 4 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 3 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 3 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 3 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 4 9. After completing my work on the number theory problems, I asked myself if I learned 3 anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 3 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the most 4 important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 2 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 4 help. 14. In order to help myself understand the concepts relating to the number theory 2 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 4 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 3 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 4 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 5 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 4 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 3 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 4 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear up 4 my confusion. 24. When I was working on the number theory problems, I thought about whether or not 4 I was working on the main questions presented by the problems.

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Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 2 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 4 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 4 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 3 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 3 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 4 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 2 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 2 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 2 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 2 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 1 learned from working on the number theory problems.

Notes: i.) Student Q8’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 106/150 = 70.67% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 9/25 = 36.00%

234

Table Q8A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part Response 3. When I am studying mathematics, I attempt to 3 3. When I worked on the number theory 4 use study methods that have been used by problems, I attempted to use techniques that successful mathematics students. have been used by successful mathematics students. 4. If I already know something about a mathematics 3 4. If I already know something about the 4 topic, I can learn even more about it. concepts referenced in the number theory problems, I can learn even more about them. 5. If I have a real interest in a mathematics topic, I 2 5. If I have a real interest in the number 3 can learn more about it. theory problems and the concepts to which they relate, I can learn more about it. 7. Before I start trying to learn a new mathematics 1 7. Before I started trying to solve one or 3 concept, I decide on clear goals that I must reach as more of the number theory problems, I part of the learning process. decided on clear goals that I needed to reach as part of the problem solving process. 8. As I am studying a new mathematics lesson, I 3 8. As I was working on the number theory 4 evaluate my progress in learning the material. problems, I evaluated my progress.

9. After completing my mathematics homework, I 2 9. After completing my work on the number 3 ask myself if I have learned anything new and theory problems, I asked myself if I learned valuable. anything new and valuable. 11. After I complete my work on a mathematics 2 11. After I completed my work on the 4 problem, I think about the most important elements number theory problems, I thought about of the problem to make sure I really understand the most important elements of the them. problems to make sure I really understood them. 15. I have my own ways of remembering what I 2 15. I have my own ways of remembering 4 have learned in mathematics what I have learned from the number theory problems. 18. If I write the data presented by a problem, I 4 18. If I wrote the data presented by the 5 have a better understanding of the problem. number theory problems, I had a better understanding of the problems. 21. When I am trying to solve a mathematics 3 21. When I was trying to solve the number 4 problem, I ask myself questions to help focus my theory problems, I asked myself questions attention on the problem and my attempts to solve to help focus my attention on the problems it. and my attempts to solve them. 26. I know how I performed on my mathematics 3 26. I knew how I performed on the number 4 test before I receive any feedback from my teacher. theory problems as soon as I finished my work on the problems. 31. When I am browsing in a bookstore or library, I 1 31. Having completed my work on the 2 flip through the pages of mathematics books. number theory problems, I flip through the pages of mathematics books while I am browsing in a bookstore or library. 33. I enjoy preparing my own mathematics tests, 1 33. Having completed my work on the 2 quizzes and puzzles. number theory problems, I enjoy preparing my own mathematics tests, quizzes and puzzles.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q8’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

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Student Q8 Interview Responses

(As used in the questions below, the “number theory problems” refer to the elementary number theory problems in Part B of the questionnaire used in this study.)

1. What types of strategies or research methods did you use as you tried to solve the number theory problems? JUST TRIED TO SOLVE THEM. Were any of these strategies or methods different from the strategies you previously used in your mathematics courses? NO. If so, explain how they were different. Was there one number theory problem that you liked the most? NO. If so, explain the thinking process and strategy you used in your attempt to solve that particular problem.

2. Would you say that the number theory problems were more interesting or less interesting than your other mathematics coursework? Explain your answer. THEY WERE LESS INTERESTING. THEY WERE QUITE DIFFERENT FROM OTHER COURSEWORK.

3. As you worked on the number theory problems, did you feel bored or discouraged? A LITTLE BORED. If so, explain why you think you felt that way. THE PROBLEMS WERE TOO DIFFICULT.

4. Did you enjoy working on the number theory problems? YES, FOR SOME. Why or why not? I FELT HAPPY IF I WAS ABLE TO SOLVE SOME OF THEM.

7. Before working on the number theory problems, what were your attitudes towards mathematics? I THOUGHT MATHEMATICS WAS SO-SO. Have your attitudes towards mathematics changed since you worked on the number theory problems? YES. Explain. SOMETIMES I FEEL CURIOUS ABOUT IT.

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9. As you were working on the number theory problems, did you prefer to think on your own about how to find the solutions? NO. Why or why not? I WORKED ON THEM ALONE AND ALSO WITH A TEACHER.

10. Were you able to solve the number theory problems more easily if you worked on them with your fellow students? YES. If so, how did the group work help you? WORKING WITH THE TEACHER GAVE ME DIFFERENT IDEAS ABOUT HOW TO SOLVE THE PROBLEMS.

12. How do you feel about making mistakes in mathematics? MISTAKES ARE AWFUL. Do you see mistakes as something to be ashamed of, or as a part of learning? NOT PART OF LEARNING. Do you feel the same way about mistakes which you may have made on the number theory problems? YES. Explain. MISTAKES ARE AWFUL.

14. In your past mathematics courses, did you ever think about how you could improve your mathematics ability? YES, ALWAYS. If so, what ideas did you have? TALK WITH MY TEACHER ABOUT IT. Did you think of any new ideas for improving your mathematics ability after working on the number theory problems? YES. If so, what new ideas do you think of? I THOUGHT OF SOME IDEAS WHICH WERE DIFFERENT FROM MY TEACHER’S.

15. While working on the number theory problems, which of the following activities did you engage in: analyzing the problem, planning the method of the solution, checking the work, reflecting on the problem. ANALYZING, YES. PLANNING, YES. CHECKING, OF COURSE. REFLECTING, SOMETIMES. Did you engage in those activities to a greater extent while working on the number theory problems, as compared to your work in your prior mathematics classes? I ANALYZED PROBLEMS MORE WHEN I WAS WORKING ON THE NUMBER THEORY PROBLEMS. I ANALYZED PROBLEMS SOMETIMES IN OTHER COURSES.

16. Had you ever previously studied or read about the topics covered by the number theory problems? YES, A LITTLE. If so, how did that prior learning affect your work on the number theory problems? IT HELPED ME A LITTLE. If the number theory problems presented topics which you had never heard of before, did the problems make you feel surprised, anxious, frustrated or curious? A LITTLE SURPRISED, NOT ANXIOUS, SOMETIMES FRUSTRATED AND SOMETIMES CURIOUS.

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17. Having worked on the number theory problems, do you feel interested in doing Internet searches to learn more about number theory topics? NO. Why or why not? I DO NOT LIKE NUMBER THEORY.

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Appendix N

Student Q9 Responses

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Table Q9A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 3 2. My effort and persistence determine how well I perform on a mathematics test. 5 3. When I am studying mathematics, I attempt to use study methods that have been used 3 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about it. 5 5. If I have a real interest in a mathematics topic, I can learn more about it. 5 6. When I want to improve my understanding of a mathematics concept, I draw sketches, 4 pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 2 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 4 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 4 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 3 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 2 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 2 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 3 14. In order to help myself understand a mathematics concept, I think of my own 4 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 5 16. After I have read a mathematics problem, I know if I am able to solve it. 2 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 4 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 3 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 5 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 4 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 5 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 3 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 2 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 1 my teacher.

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Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 2 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 5 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 4 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 1 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 4 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 5 34. I enjoy finding mathematics problems which have never been solved. 1 35. When I am with my friends, I discuss new mathematics concepts which I have 4 recently learned.

Notes: i.) Student Q9’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 108/150 = 72.00% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 15/25 = 60.00%

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Table Q9C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 3 concepts. 2. My effort and persistence determined how well I was able to do on the number theory 5 problems. 3. When I worked on the number theory problems, I attempted to use techniques that 4 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 2 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 3 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 4 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 3 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 2 9. After completing my work on the number theory problems, I asked myself if I learned 3 anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 4 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the most 3 important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 2 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 3 help. 14. In order to help myself understand the concepts relating to the number theory 4 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 3 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 1 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 2 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 5 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 3 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 2 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 4 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear up 4 my confusion.

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Question from Questionnaire Part C Score 24. When I was working on the number theory problems, I thought about whether or not 3 I was working on the main questions presented by the problems. 25. Once I found a solution to the number theory problems, I tried to find some other, 4 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 4 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 1 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 1 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 5 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 4 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 3 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 2 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 2 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 2 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 5 learned from working on the number theory problems.

Notes: i.) Student Q9’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 96/150 = 64.00% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 14/25 = 56.00%

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Table Q9A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 3. When I am studying mathematics, I attempt 3 3. When I worked on the number 4 to use study methods that have been used by theory problems, I attempted to use successful mathematics students. techniques that have been used by successful mathematics students. 7. Before I start trying to learn a new 2 7. Before I started trying to solve one 3 mathematics concept, I decide on clear goals or more of the number theory that I must reach as part of the learning problems, I decided on clear goals that process. I needed to reach as part of the problem solving process. 10. As soon as I solve a mathematics problem, 3 10. As soon as I solved one or more of 4 I wonder if I could have found an easier or the number theory problems, I quicker way to solve it. wondered if I could have found easier or quicker ways to arrive at the solution(s). 11. After I complete my work on a 2 11. After I completed my work on the 3 mathematics problem, I think about the most number theory problems, I thought important elements of the problem to make about the most important elements of sure I really understand them. the problems to make sure I really understood them. 18. If I write the data presented by a problem, 4 18. If I wrote the data presented by the 5 I have a better understanding of the problem. number theory problems, I had a better understanding of the problems. 22. If I am having difficulty with a 4 22. If I had difficulty with the number 5 mathematics problem, I read the problem theory problems, I read the problems several times to make sure I understand it. several times. 25. Once I find a solution to a mathematics 2 25. Once I found a solution to the 4 problem, I try to find some other, different number theory problems, I tried to find solutions to the same problem. some other, different solutions to the same problems. 26. I know how I performed on my 1 26. I knew how I performed on the 4 mathematics test before I receive any feedback number theory problems as soon as I from my teacher. finished my work on the problems. 31. When I am browsing in a bookstore or 1 31. Having completed my work on the 3 library, I flip through the pages of number theory problems, I flip through mathematics books. the pages of mathematics books while I am browsing in a bookstore or library. 34. I enjoy finding mathematics problems 1 34. Having completed my work on the 2 which have never been solved. number theory problems, I enjoy finding mathematics problems which have never been solved. 35. When I am with my friends, I discuss new 4 35. When I am with my friends, I 5 mathematics concepts which I have recently discuss new mathematics concepts learned. which I have learned from working on the number theory problems.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q9’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 244

Student Q9 Interview Responses

(As used in the questions below, the “number theory problems” refer to the elementary number theory problems in Part B of the questionnaire used in this study.)

1. What types of strategies or research methods did you use as you tried to solve the number theory problems? I JUST TRIED MY BEST AND LISTENED TO MY TEACHER. Were any of these strategies or methods different from the strategies you previously used in your mathematics courses? NO. If so, explain how they were different. Was there one number theory problem that you liked the most? NO. If so, explain the thinking process and strategy you used in your attempt to solve that particular problem.

3. As you worked on the number theory problems, did you feel bored or discouraged? A LITTLE BORED AND SOMETIMES DISCOURAGED. If so, explain why you think you felt that way. PROBLEMS WERE TOO DIFFICULT.

4. Did you enjoy working on the number theory problems? YES. Why or why not? I ENJOY SOLVING PROBLEMS.

5. Do you usually discuss mathematics problems with your friends? NO. Is there anything about the number theory problems that made you want to discuss them with your friends? YES. If so, explain. I ASKED MY FRIENDS IF I COULD NOT SOLVE THE PROBLEMS MYSELF. If you discussed the number theory problems with your friends, about how much time did you spend discussing the problems? NO MORE THAN ONE HOUR. Was this discussion longer than other discussions you may have previously had with your friends on mathematics topics? YES.

7. Before working on the number theory problems, what were your attitudes towards mathematics? MATHEMATICS WAS SO-SO. Have your attitudes towards mathematics changed since you worked on the number theory problems? YES. Explain. MATHEMATICS IS SOMETIMES INTERESTING. I LIKE IT MUCH MORE. IT CAN BE VERY FANTASTIC.

processes and strategies if you found that you were not getting closer to a solution? YES. Did you go through any of these steps in any of your mathematics classes before you started working on the number theory problems? YES. Explain further. I DID THIS IN OTHER CLASSES. THE NUMBER THEORY PROBLEMS CAN CHANGE MY THOUGHT PROCESS AND HELP ME SOLVE MATH PROBLEMS MORE AND MORE QUICKLY.

9. As you were working on the number theory problems, did you prefer to think on your own about how to find the solutions? YES. Why or why not? I LIKE TO DO MATH MYSELF.

10. Were you able to solve the number theory problems more easily if you worked on them with your fellow students? NO. If so, how did the group work help you?

12. How do you feel about making mistakes in mathematics? I FEEL REALLY BAD ABOUT IT. Do you see mistakes as something to be ashamed of, or as a part of learning? PART OF LEARNING, OF COURSE. Do you feel the same way about mistakes which you may have made on the number theory problems? YES. Explain. MISTAKES ARE PART OF LEARNING.

14. In your past mathematics courses, did you ever think about how you could improve your mathematics ability? YES. If so, what ideas did you have? TO ALWAYS TRY MY BEST. Did you think of any new ideas for improving your mathematics ability after working on the number theory problems? YES. If so, what new ideas do you think of? SOME IDEAS WHICH WERE DIFFERENT FROM MY TEACHER’S AND DIFFERENT FROM MY FRIENDS.

15. While working on the number theory problems, which of the following activities did you engage in: analyzing the problem, planning the method of the solution, checking the work, reflecting on the problem. ANALYZING, YES. PLANNING, YES. CHECKING, YES, ALWAYS. REFLECTING, YES. Did you engage in those activities to a greater extent while working on the number theory problems, as compared to your work in your prior mathematics classes? YES.

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16. Had you ever previously studied or read about the topics covered by the number theory problems? YES. If so, how did that prior learning affect your work on the number theory problems? IT MADE THE PROBLEMS A LITTLE EASIER. If the number theory problems presented topics which you had never heard of before, did the problems make you feel surprised, anxious, frustrated or curious? SURPRISED, YES. SOMETIMES ANXIOUS. FRUSTRATED, YES. AND SOMETIMES CURIOUS, BECAUSE OF THE DIFFICULTY OF THE PROBLEMS.

17. Having worked on the number theory problems, do you feel interested in doing Internet searches to learn more about number theory topics? NOT REALLY. Why or why not? I COULD NOT DO THE PROBLEMS.

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Appendix O

Student Q10 Responses

248

Table Q10A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 3 2. My effort and persistence determine how well I perform on a mathematics test. 5 3. When I am studying mathematics, I attempt to use study methods that have been used 4 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 3 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 4 6. When I want to improve my understanding of a mathematics concept, I draw 5 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 2 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 3 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 3 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 2 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 4 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 2 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 4 14. In order to help myself understand a mathematics concept, I think of my own 5 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 5 16. After I have read a mathematics problem, I know if I am able to solve it. 4 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 4 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 5 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 3 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 4 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 5 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 4 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 2 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 3 my teacher.

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Notes: i.) Student Q10’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 113/150 = 75.33% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 11/25 = 44.00%

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Table Q10C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 1 concepts. 2. My effort and persistence determined how well I was able to do on the number 3 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 4 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 2 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which 4 they relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 5 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I 4 decided on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 4 9. After completing my work on the number theory problems, I asked myself if I 2 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 5 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 2 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on 3 the type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else 5 for help. 14. In order to help myself understand the concepts relating to the number theory 5 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 3 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 4 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 4 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 4 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of 3 similar problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of 3 the problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions 4 to help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear 4 up my confusion. 24. When I was working on the number theory problems, I thought about whether or 4 not I was working on the main questions presented by the problems.

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Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 1 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 4 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 3 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature 3 of the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques 3 which vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 4 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the 3 pages of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of 1 new mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 1 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 1 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 4 learned from working on the number theory problems.

Notes: i.) Student Q10’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 105/150 = 70.00% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 10/25 = 40.00%

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Table Q10A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 7. Before I start trying to learn a new 2 7. Before I started trying to solve one 4 mathematics concept, I decide on clear goals or more of the number theory that I must reach as part of the learning problems, I decided on clear goals that process. I needed to reach as part of the problem solving process. 8. As I am studying a new mathematics lesson, 3 8. As I was working on the number 4 I evaluate my progress in learning the theory problems, I evaluated my material. progress. 10. As soon as I solve a mathematics problem, 2 10. As soon as I solved one or more of 5 I wonder if I could have found an easier or the number theory problems, I quicker way to solve it. wondered if I could have found easier or quicker ways to arrive at the solution(s). 12. I try different ways of learning 2 12. I tried different ways of 3 mathematics, depending on the type of approaching the number theory mathematics involved. problems, depending on the type of concepts involved. 13. Whenever I am confused in mathematics 4 13. Whenever I was confused by the 5 class, I ask someone else for help. number theory problems, I asked someone else for help. 26. I know how I performed on my 3 26. I knew how I performed on the 4 mathematics test before I receive any feedback number theory problems as soon as I from my teacher. finished my work on the problems. 31. When I am browsing in a bookstore or 2 31. Having completed my work on the 3 library, I flip through the pages of number theory problems, I flip through mathematics books. the pages of mathematics books while I am browsing in a bookstore or library. 35. When I am with my friends, I discuss new 3 35. When I am with my friends, I 4 mathematics concepts which I have recently discuss new mathematics concepts learned. which I have learned from working on the number theory problems..

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q10’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

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Student Q10 Interview Responses

(As used in the questions below, the “number theory problems” refer to the elementary number theory problems in Part B of the questionnaire used in this study.)

1. What types of strategies or research methods did you use as you tried to solve the number theory problems? I LISTENED TO MY TEACHER. Were any of these strategies or methods different from the strategies you previously used in your mathematics courses? NO. If so, explain how they were different. Was there one number theory problem that you liked the most? NO. If so, explain the thinking process and strategy you used in your attempt to solve that particular problem. I DID NOT LIKE MATH AND STILL DO NOT LIKE MATH.

2. Would you say that the number theory problems were more interesting or less interesting than your other mathematics coursework? Explain your answer. THEY WERE NOT MORE INTERESTING BECAUSE THEY WERE TOO DIFFICULT.

3. As you worked on the number theory problems, did you feel bored or discouraged? I WAS BORED AND SOMETIMES DISCOURAGED. If so, explain why you think you felt that way. I FELT THAT WAY WHEN I COULD NOT SOLVE THE PROBLEMS.

4. Did you enjoy working on the number theory problems? NO. Why or why not? TOO DIFFICULT.

5. Do you usually discuss mathematics problems with your friends? NEVER. Is there anything about the number theory problems that made you want to discuss them with your friends? NO. If so, explain. If you discussed the number theory problems with your friends, about how much time did you spend discussing the problems? Was this discussion longer than other discussions you may have previously had with your friends on mathematics topics?

7. Before working on the number theory problems, what were your attitudes towards mathematics? DID NOT LIKE MATHEMATICS. Have your attitudes towards mathematics changed since you worked on the number theory problems? NO. Explain.

8. As you worked on the number theory problems, were you aware of what you were thinking about? NOT ALWAYS. As you worked on the problems, were you evaluating your thoughts? USUALLY. For any of the problems, did you decide to change your thought processes and strategies if you found that you were not getting closer to a solution? NO. Did you go through any of these steps in any of your mathematics classes before you started working on the number theory problems? Explain further. YES,

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SOMETIMES EVALUATED MY THOUGHTS IN OTHER CLASSES. THE NUMBER THEORY PROBLEMS DID NOT CHANGE MY THOUGHT PROCESS.

9. As you were working on the number theory problems, did you prefer to think on your own about how to find the solutions? NO. Why or why not? I DID NOT LIKE THE PROBLEMS.

10. Were you able to solve the number theory problems more easily if you worked on them with your fellow students? NO. If so, how did the group work help you?

11. Before starting your work on the number theory problems, to what extent, if any, did you believe that new mathematical concepts and problems continue to be discovered? I DID NOT BELIEVE THIS. Do you have that same belief after having completed your work on the number theory problems? NO.

12. How do you feel about making mistakes in mathematics? ASHAMED. Do you see mistakes as something to be ashamed of, or as a part of learning? SOMETHING TO BE ASHAMED OF, BUT MAYBE PART OF LEARNING. Do you feel the same way about mistakes which you may have made on the number theory problems? YES. Explain. MISTAKES ARE PART OF LEARNING ABOUT NUMBER THEORY.

15. While working on the number theory problems, which of the following activities did you engage in: analyzing the problem, planning the method of the solution, checking the work, reflecting on the problem. ANALYZING, NO. PLANNING, NOT USUALLY. CHECKING, NEVER. REFLECTING, NO. Did you engage in those activities to a greater extent while working on the number theory problems, as compared to your work in your prior mathematics classes? NO.

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17. Having worked on the number theory problems, do you feel interested in doing Internet searches to learn more about number theory topics? NO. Why or why not? DO NOT LIKE MATH.

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Appendix P

Student Q11 Responses

257

Table Q11A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 4 3. When I am studying mathematics, I attempt to use study methods that have been used 5 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about it. 3 5. If I have a real interest in a mathematics topic, I can learn more about it. 2 6. When I want to improve my understanding of a mathematics concept, I draw sketches, 4 pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 4 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 3 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 4 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 3 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 4 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 4 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 5 14. In order to help myself understand a mathematics concept, I think of my own 4 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 3 16. After I have read a mathematics problem, I know if I am able to solve it. 2 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 3 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 5 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 4 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 2 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 4 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 3 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 2 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 2 my teacher.

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Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 4 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 3 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 5 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 2 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 2 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 2 34. I enjoy finding mathematics problems which have never been solved. 2 35. When I am with my friends, I discuss new mathematics concepts which I have 1 recently learned.

Notes: i.) Student Q11’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 110/150 = 73.33% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 9/25 = 36.00%

259

Table Q11C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 3 concepts. 2. My effort and persistence determined how well I was able to do on the number theory 3 problems. 3. When I worked on the number theory problems, I attempted to use techniques that 5 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 2 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 4 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 4 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 2 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I learned 3 anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 2 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the most 3 important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 4 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 5 help. 14. In order to help myself understand the concepts relating to the number theory 3 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 3 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 4 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 5 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 5 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 5 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 2 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 3 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear up 2 my confusion. 24. When I was working on the number theory problems, I thought about whether or not 5 I was working on the main questions presented by the problems.

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Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 4 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 4 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 2 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 3 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 4 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 5 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 2 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 3 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 2 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 2 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 5 learned from working on the number theory problems.

Notes: i.) Student Q11’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 107/150 = 71.33% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 14/25 = 56.00%

261

Table Q11A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 5. If I have a real interest in a mathematics 2 5. If I have a real interest in the number 4 topic, I can learn more about it. theory problems and the concepts to which they relate, I can learn more about it. 16. After I have read a mathematics problem, I 2 16. After I finished reading the number 4 know if I am able to solve it. theory problems, I knew if I was able to solve them. 18. If I write the data presented by a problem, 3 18. If I wrote the data presented by the 5 I have a better understanding of the problem. number theory problems, I had a better understanding of the problems. 21. When I am trying to solve a mathematics 2 21. When I was trying to solve the 3 problem, I ask myself questions to help focus number theory problems, I asked my attention on the problem and my attempts myself questions to help focus my to solve it. attention on the problems and my attempts to solve them. 24. When I am working on a mathematics 3 24. When I was working on the number 5 problem, I think about whether or not I am theory problems, I thought about working on what the problem is really asking. whether or not I was working on the main questions presented by the problems. 25. Once I find a solution to a mathematics 2 25. Once I found a solution to the 4 problem, I try to find some other, different number theory problems, I tried to find solutions to the same problem. some other, different solutions to the same problems. 26. I know how I performed on my 2 26. I knew how I performed on the 4 mathematics test before I receive any feedback number theory problems as soon as I from my teacher. finished my work on the problems. 29. I believe that mathematical problem 3 29. I believe that the number theory 4 solving techniques vary in their level of problems require problem solving difficulty. techniques which vary in their level of difficulty. 32. I enjoy thinking of new mathematics 2 32. Having completed my work on the 3 problems to work on. number theory problems, I enjoy thinking of new mathematics problems to work on. 35. When I am with my friends, I discuss new 1 35. When I am with my friends, I 5 mathematics concepts which I have recently discuss new mathematics concepts learned. which I have learned from working on the number theory problems.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q11’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

262

Student Q11 Interview Responses

(As used in the questions below, the “number theory problems” refer to the elementary number theory problems in Part B of the questionnaire used in this study.)

1. What types of strategies or research methods did you use as you tried to solve the number theory problems? TRIED TO LEARN BY MYSELF AND FRM MY TEACHER. Were any of these strategies or methods different from the strategies you previously used in your mathematics courses? YES. If so, explain how they were different. HAD TO UNDERSTAND THE NUMBER THEORY PROBLEMS FIRST. Was there one number theory problem that you liked the most? YES, THE ONE ABOUT THE FORMULA FOR PRIMES. If so, explain the thinking process and strategy you used in your attempt to solve that particular problem. USED MY OWN METHOD AND THE METHOD OF MY TEACHER.

3. As you worked on the number theory problems, did you feel bored or discouraged? NO. If so, explain why you think you felt that way.

4. Did you enjoy working on the number theory problems? SOMETIMES, BUT RARELY. Why or why not? I CAN’T UNDERSTAND NUMBER THEORY AND I DON’T LIKE MATH.

5. Do you usually discuss mathematics problems with your friends? NO. Is there anything about the number theory problems that made you want to discuss them with your friends? YES. If so, explain. CAN’T RECALL. If you discussed the number theory problems with your friends, about how much time did you spend discussing the problems? NOT MUCH TIME. Was this discussion longer than other discussions you may have previously had with your friends on mathematics topics? NO.

6. After completing your work on the number theory problems, did you want to look through a number theory textbook to find information about other number theory concepts? YES, IF I HAD THE TIME. If so, explain. I DID NOT LOOK THROUGH A NUMBER THEORY TEXTBOOK BECAUSE I DID NOT HAVE TIME. Did you think about changing any of the number theory problems? NO. If so, explain.

7. Before working on the number theory problems, what were your attitudes towards mathematics? IT WAS SO-SO. Have your attitudes towards mathematics changed since you worked on the number theory problems? NO. Explain. MATH IS STILL A LITTLE BORING.

processes and strategies if you found that you were not getting closer to a solution? YES. Did you go through any of these steps in any of your mathematics classes before you started working on the number theory problems? YES. Explain further. I THINK ABOUT HOW TO DO MY BEST TO SOLVE MATH PROBLEMS.

9. As you were working on the number theory problems, did you prefer to think on your own about how to find the solutions? YES. Why or why not? I FEEL BETTER IF I SOLVE PROBLEMS ALONE.

10. Were you able to solve the number theory problems more easily if you worked on them with your fellow students? YES. If so, how did the group work help you? IT MADE IT EASIER.

12. How do you feel about making mistakes in mathematics? MISTAKES ARE NORMAL. Do you see mistakes as something to be ashamed of, or as a part of learning? PART OF LEARNING, NOT SOMETHING TO BE ASHAMED OF. Do you feel the same way about mistakes which you may have made on the number theory problems? YES. Explain. MISTAKES ARE PART OF LEARNING.

14. In your past mathematics courses, did you ever think about how you could improve your mathematics ability? YES. If so, what ideas did you have? DO MORE EXERCISES. Did you think of any new ideas for improving your mathematics ability after working on the number theory problems? NO. If so, what new ideas do you think of?

15. While working on the number theory problems, which of the following activities did you engage in: analyzing the problem, planning the method of the solution, checking the work, reflecting on the problem. ANALYZING, YES. PLANNING, YES. CHECKING, YES. REFLECTING, MAYBE. Did you engage in those activities to a greater extent while working on the number theory problems, as compared to your work in your prior mathematics classes? YES.

16. Had you ever previously studied or read about the topics covered by the number theory problems? YES, A LITTLE. If so, how did that prior learning affect your work on the number theory problems? IT HELPED A LITTLE. If the number theory problems presented topics which you had never heard of before, did the problems make you feel surprised, anxious, frustrated or curious? SURPRISED, MAYBE. ANXIOUS, NO.

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FRUSTRATED, SOMETIMES. AND SOMETIMES CURIOUS, BECAUSE SOME OF THE PROBLEMS WERE SO DIFFICULT.

17. Having worked on the number theory problems, do you feel interested in doing Internet searches to learn more about number theory topics? YES. Why or why not? I LOVE TO LEARN MORE. THE NUMBER THEORY PROBLEMS SOMETIMES CHANGED MY THOUGHT PROCESS ABOUT MATH. THE PROBLEMS SHOWED THAT I CAN LEARN MATH IN MORE WAYS.

265

Appendix Q

Student Q12 Responses

266

Table Q12A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 3 2. My effort and persistence determine how well I perform on a mathematics test. 3 3. When I am studying mathematics, I attempt to use study methods that have been used 2 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about it. 5 5. If I have a real interest in a mathematics topic, I can learn more about it. 4 6. When I want to improve my understanding of a mathematics concept, I draw sketches, 5 pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 3 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 4 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 2 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 2 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 3 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 3 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 5 14. In order to help myself understand a mathematics concept, I think of my own 5 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 5 16. After I have read a mathematics problem, I know if I am able to solve it. 2 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 5 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 5 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 2 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 1 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 3 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 3 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 2 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 5 my teacher.

267

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 3 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 1 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 5 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 1 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 2 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 1 34. I enjoy finding mathematics problems which have never been solved. 1 35. When I am with my friends, I discuss new mathematics concepts which I have 3 recently learned.

Notes: i.) Student Q12’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 106/150 = 70.67% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 8/25 = 32.00%

268

Table Q12C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 4 concepts. 2. My effort and persistence determined how well I was able to do on the number theory 3 problems. 3. When I worked on the number theory problems, I attempted to use techniques that 2 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 3 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 5 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 5 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 2 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 4 9. After completing my work on the number theory problems, I asked myself if I learned 4 anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 2 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the most 2 important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 1 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 5 help. 14. In order to help myself understand the concepts relating to the number theory 5 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 5 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 3 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 5 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 3 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 5 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 2 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 2 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear up 5 my confusion. 24. When I was working on the number theory problems, I thought about whether or not 5 I was working on the main questions presented by the problems.

269

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 2 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 3 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 5 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 2 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 5 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 5 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 1 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 1 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 1 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 1 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 1 learned from working on the number theory problems.

Notes: i.) Student Q12’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 109/150 = 72.67% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 5/25 = 20.00%

270

Table Q12A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 1. After I have studied a mathematics lesson, I 3 1. After I worked on the number theory 4 know how well I understand it. problems, I knew how well I understood the concepts. 5. If I have a real interest in a mathematics 4 5. If I have a real interest in the number 5 topic, I can learn more about it. theory problems and the concepts to which they relate, I can learn more about it. 9. After completing my mathematics 2 9. After completing my work on the 4 homework, I ask myself if I have learned number theory problems, I asked anything new and valuable. myself if I learned anything new and valuable. 16. After I have read a mathematics problem, I 2 16. After I finished reading the number 3 know if I am able to solve it. theory problems, I knew if I was able to solve them. 21. When I am trying to solve a mathematics 1 21. When I was trying to solve the 2 problem, I ask myself questions to help focus number theory problems, I asked my attention on the problem and my attempts myself questions to help focus my to solve it. attention on the problems and my attempts to solve them. 23. If I become confused in trying to solve a 3 23. If I became confused in trying to 5 mathematics problem, I try to clear up my solve the number theory problems, I confusion. tried to clear up my confusion. 24. When I am working on a mathematics 3 24. When I was working on the number 5 problem, I think about whether or not I am theory problems, I thought about working on what the problem is really asking. whether or not I was working on the main questions presented by the problems. 29. I believe that mathematical problem 1 29. I believe that the number theory 5 solving techniques vary in their level of problems require problem solving difficulty. techniques which vary in their level of difficulty.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q12’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

271

Student Q12 Interview Responses

(As used in the questions below, the “number theory problems” refer to the elementary number theory problems in Part B of the questionnaire used in this study.)

1. What types of strategies or research methods did you use as you tried to solve the number theory problems? I DID A LOT OF WRITING AND THINKING ABOUT THEM. Were any of these strategies or methods different from the strategies you previously used in your mathematics courses? NO. If so, explain how they were different. Was there one number theory problem that you liked the most? I DO NOT KNOW. If so, explain the thinking process and strategy you used in your attempt to solve that particular problem.

2. Would you say that the number theory problems were more interesting or less interesting than your other mathematics coursework? Explain your answer. THEY WERE MORE INTERESTING. I LOVE NUMBERS.

3. As you worked on the number theory problems, did you feel bored or discouraged? SOMETIMES BORED AND DISCOURAGED. If so, explain why you think you felt that way. IF THE PROBLEMS WERE DIFFICULT, I FELT THAT WAY.

4. Did you enjoy working on the number theory problems? SOMETIMES. Why or why not? I FELT GOOD IF I SOLVED THEM.

5. Do you usually discuss mathematics problems with your friends? ONLY IN CLASS. Is there anything about the number theory problems that made you want to discuss them with your friends? NO. If so, explain. If you discussed the number theory problems with your friends, about how much time did you spend discussing the problems? DO NOT KNOW. Was this discussion longer than other discussions you may have previously had with your friends on mathematics topics? NO.

6. After completing your work on the number theory problems, did you want to look through a number theory textbook to find information about other number theory concepts? SOMETIMES. If so, explain. TO FIND A MTHOD TO SOLVE THEM. Did you think about changing any of the number theory problems? YES. If so, explain. I WANTED TO MAKE THEM EASIER.

7. Before working on the number theory problems, what were your attitudes towards mathematics? I THOUGHT MATHEMATICS WAS BORING. Have your attitudes towards mathematics changed since you worked on the number theory problems? YES. Explain. I THINK ABOUT MATH MORE AFTER WORKING ON THE NUMBER THEORY PROBLEMS. MATH IS BORING IN SCHOOL, BUT THE NUMBER THEORY PROBLEMS WERE DIFFERENT.

Did you go through any of these steps in any of your mathematics classes before you started working on the number theory problems? YES. Explain further. I ANALYZED MY THOUGHTS IN MATH CLASSES. THE NUMBER THEORY PROBLEMS CHANGED MY THOUGHT PROCESS. THEY MADE ME TRY TO FIND EASIER WAYS TO SOLVE THE PROBLEMS.

9. As you were working on the number theory problems, did you prefer to think on your own about how to find the solutions? NO. Why or why not? I WANTED TO WORK WITH OTHER STUDENTS, TO MAKE THE WORK QUICKER AND EASIER.

10. Were you able to solve the number theory problems more easily if you worked on them with your fellow students? YES. If so, how did the group work help you? MADE IT QUICKER AND EASIER.

12. How do you feel about making mistakes in mathematics? MISTAKES ARE PART OF LEARNING AND NOT TO BE ASHAMED OF. Do you see mistakes as something to be ashamed of, or as a part of learning? PART OF LEARNING. Do you feel the same way about mistakes which you may have made on the number theory problems? YES. Explain. MISTAKES ARE PART OF LEARNING.

14. In your past mathematics courses, did you ever think about how you could improve your mathematics ability? YES. If so, what ideas did you have? REVIEW. Did you think of any new ideas for improving your mathematics ability after working on the number theory problems? RARELY. If so, what new ideas do you think of?

15. While working on the number theory problems, which of the following activities did you engage in: analyzing the problem, planning the method of the solution, checking the work, reflecting on the problem. ANALYZING, YES. PLANNING, YES. CHECKING, YES. REFLECTING, SOMETIMES. Did you engage in those activities to a greater extent while working on the number theory problems, as compared to your work in your prior mathematics classes? YES. I DID NOT DO IT MORE IN SCHOOL.

16. Had you ever previously studied or read about the topics covered by the number theory problems? STUDIED IT A LITTLE AND READ ABOUT IT. If so, how did that prior learning affect your work on the number theory problems? IT HELPED. If the number theory problems presented topics which you had never heard of before, did the problems make you feel surprised, anxious, frustrated or curious? SURPRISED, YES. NOT 273

ANXIOUS. SOMETIMES FRUSTRATED. CURIOUS, YES, BECAUSE I COULD NOT SOLVE THEM ALL.

17. Having worked on the number theory problems, do you feel interested in doing Internet searches to learn more about number theory topics? YES. Why or why not? I NEED MORE KNOWLEDGE ABOUT NUMBER THEORY.

274

Appendix R

Student Q13 Responses

275

Table Q13A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 5 3. When I am studying mathematics, I attempt to use study methods that have been 3 used by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 3 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 5 6. When I want to improve my understanding of a mathematics concept, I draw 3 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that 3 I must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 3 material. 9. After completing my mathematics homework, I ask myself if I have learned 3 anything new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 3 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 4 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 4 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 3 14. In order to help myself understand a mathematics concept, I think of my own 3 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 5 16. After I have read a mathematics problem, I know if I am able to solve it. 3 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 5 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 3 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 3 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 5 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 5 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 3 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback 1 from my teacher.

276

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 4 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 5 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 5 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 4 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 3 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 3 34. I enjoy finding mathematics problems which have never been solved. 1 35. When I am with my friends, I discuss new mathematics concepts which I have 3 recently learned.

Notes: i.) Student Q13’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 115/150 = 76.67% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 14/25 = 56.00%

277

Table Q13C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 5 concepts. 2. My effort and persistence determined how well I was able to do on the number 5 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 3 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 2 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which 5 they relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 3 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I 4 decided on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I 3 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 3 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 3 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on 4 the type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else 3 for help. 14. In order to help myself understand the concepts relating to the number theory 3 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 5 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 3 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 5 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 5 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of 4 similar problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of 3 the problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions 3 to help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear 5 up my confusion.

278

Question from Questionnaire Part C Score 24. When I was working on the number theory problems, I thought about whether or 5 not I was working on the main questions presented by the problems. 25. Once I found a solution to the number theory problems, I tried to find some other, 4 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 5 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 5 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature 4 of the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques 5 which vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 5 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the 3 pages of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of 3 new mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 2 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 2 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 3 learned from working on the number theory problems.

Notes: i.) Student Q13’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 120/150 = 80.00% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 13/25 = 52.00%

279

Table Q13A/C

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q13’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

280

Student Q13 Interview Responses

(As used in the questions below, the “number theory problems” refer to the elementary number theory problems in Part B of the questionnaire used in this study.)

1. What types of strategies or research methods did you use as you tried to solve the number theory problems? MY TEACHER TAUGHT ME. Were any of these strategies or methods different from the strategies you previously used in your mathematics courses? NO. If so, explain how they were different. Was there one number theory problem that you liked the most? NO. If so, explain the thinking process and strategy you used in your attempt to solve that particular problem.

3. As you worked on the number theory problems, did you feel bored or discouraged? YES, USUALLY. If so, explain why you think you felt that way. I COULD NOT SOLVE THE PROBLEMS BY MYSELF.

4. Did you enjoy working on the number theory problems? NOT REALLY. Why or why not? TOO DIFFICULT.

7. Before working on the number theory problems, what were your attitudes towards mathematics? I JUST WANT TO FINISH THE MATH PROBLEMS THAT I HAVE TO DO. Have your attitudes towards mathematics changed since you worked on the number theory problems? NO. Explain. I STILL DON’T LIKE MATH.

8. As you worked on the number theory problems, were you aware of what you were thinking about? SOMETIMES. As you worked on the problems, were you evaluating your thoughts? YES. For any of the problems, did you decide to change your thought processes and strategies if you found that you were not getting closer to a solution? SOMETIMES. Did you go through any of these steps in any of your mathematics classes before you started working on the number theory problems? NO. Explain further. DID NOT DO THIS IN MATH CLASS. 281

9. As you were working on the number theory problems, did you prefer to think on your own about how to find the solutions? YES. Why or why not? I FEEL BETTER WHEN I DO PROBLEMS MYSELF.

10. Were you able to solve the number theory problems more easily if you worked on them with your fellow students? NO. If so, how did the group work help you? I NEVER DO THAT.

11. Before starting your work on the number theory problems, to what extent, if any, did you believe that new mathematical concepts and problems continue to be discovered? MAYBE, NOT SURE. Do you have that same belief after having completed your work on the number theory problems? YES.

12. How do you feel about making mistakes in mathematics? A LITTLE BAD. Do you see mistakes as something to be ashamed of, or as a part of learning? NOT SOMETHING TO BE ASHAMED OF, NEVER. IT IS WRONG TO FEEL THAT WAY. IT IS PART OF LEARNING. MISTAKES HAPPEN. Do you feel the same way about mistakes which you may have made on the number theory problems? YES. Explain. MISTAKES HAPPEN.

13. As you were working on the number theory problems, did you quietly explain to yourself the procedures which you needed to follow in order to solve the problems? YES, I DID. If not, what exactly were you thinking as you worked on the problems?

14. In your past mathematics courses, did you ever think about how you could improve your mathematics ability? NOT USUALLY. If so, what ideas did you have? Did you think of any new ideas for improving your mathematics ability after working on the number theory problems? NO. If so, what new ideas do you think of?

15. While working on the number theory problems, which of the following activities did you engage in: analyzing the problem, planning the method of the solution, checking the work, reflecting on the problem. ANALYZING, NO. PLANNING, NOT USUALLY. CHECKING, NO. REFLECTING, MAYBE SOMETIMES. Did you engage in those activities to a greater extent while working on the number theory problems, as compared to your work in your prior mathematics classes? NO.

17. Having worked on the number theory problems, do you feel interested in doing Internet searches to learn more about number theory topics? NO. Why or why not? I DON’T LIKE MATH.

282

Appendix S

Student Q14 Responses

283

Table Q14A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 5 2. My effort and persistence determine how well I perform on a mathematics test. 5 3. When I am studying mathematics, I attempt to use study methods that have been used 3 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about it. 3 5. If I have a real interest in a mathematics topic, I can learn more about it. 4 6. When I want to improve my understanding of a mathematics concept, I draw sketches, 4 pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 2 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 4 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 2 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 4 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 4 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 4 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 5 14. In order to help myself understand a mathematics concept, I think of my own 4 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 4 16. After I have read a mathematics problem, I know if I am able to solve it. 3 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 3 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 2 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 3 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 4 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 2 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 5 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 3 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 4 my teacher.

284

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 3 28. When I cannot find the solution to a mathematics problem, I know the nature of the 3 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 1 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 4 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 2 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 4 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 2 34. I enjoy finding mathematics problems which have never been solved. 1 35. When I am with my friends, I discuss new mathematics concepts which I have 3 recently learned.

Notes: i.) Student Q14’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 106/150 = 70.67% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 12/25 = 48.00%

285

Table Q14C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 3 concepts. 2. My effort and persistence determined how well I was able to do on the number theory 2 problems. 3. When I worked on the number theory problems, I attempted to use techniques that 1 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 2 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 4 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 1 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 2 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I learned 2 anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 2 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the most 2 important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 3 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 2 help. 14. In order to help myself understand the concepts relating to the number theory 3 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 4 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 2 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 5 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 2 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 4 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 3 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 3 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 4 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear up 4 my confusion. 24. When I was working on the number theory problems, I thought about whether or not 4 I was working on the main questions presented by the problems.

286

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 2 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 5 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 5 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 2 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 3 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 4 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 4 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 4 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 4 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 2 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 3 learned from working on the number theory problems.

Notes: i.) Student Q14’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 88/150 = 58.67% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 17/25 = 68.00%

287

Table Q14A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 20. When I am trying to solve a mathematics 2 20. In attempting to solve the number 3 problem, I try to identify the parts of the theory problems, I tried to identify the problem which I do not understand. parts of the problems which I did not understand. 23. If I become confused in trying to solve a 2 23. If I became confused in trying to 4 mathematics problem, I try to clear up my solve the number theory problems, I confusion. tried to clear up my confusion. 26. I know how I performed on my 4 26. I knew how I performed on the 5 mathematics test before I receive any feedback number theory problems as soon as I from my teacher. finished my work on the problems. 27. I believe that mathematics topics vary in 3 27. I believe that the concepts 5 their level of difficulty. referenced by the number theory problems vary in their level of difficulty. 29. I believe that mathematical problem 1 29. I believe that the number theory 3 solving techniques vary in their level of problems require problem solving difficulty. techniques which vary in their level of difficulty. 31. When I am browsing in a bookstore or 2 31. Having completed my work on the 4 library, I flip through the pages of number theory problems, I flip through mathematics books. the pages of mathematics books while I am browsing in a bookstore or library. 33. I enjoy preparing my own mathematics 2 33. Having completed my work on the 4 tests, quizzes and puzzles. number theory problems, I enjoy preparing my own mathematics tests, quizzes and puzzles. 34. I enjoy finding mathematics problems 1 34. Having completed my work on the 2 which have never been solved. number theory problems, I enjoy finding mathematics problems which have never been solved.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q14’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

288

Appendix T

Student Q15 Responses

289

Table Q15A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 5 2. My effort and persistence determine how well I perform on a mathematics test. 5 3. When I am studying mathematics, I attempt to use study methods that have been used 3 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about it. 3 5. If I have a real interest in a mathematics topic, I can learn more about it. 5 6. When I want to improve my understanding of a mathematics concept, I draw sketches, 2 pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 2 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 2 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 3 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 5 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 4 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 5 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 3 14. In order to help myself understand a mathematics concept, I think of my own 4 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 5 16. After I have read a mathematics problem, I know if I am able to solve it. 4 17. When I am asked to solve a mathematics problem, I focus on whatever data are 3 presented. 18. If I write the data presented by a problem, I have a better understanding of the 5 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 3 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 3 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 2 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 5 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 3 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 4 my teacher.

290

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 5 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 2 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 3 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 2 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 5 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 2 34. I enjoy finding mathematics problems which have never been solved. 4 35. When I am with my friends, I discuss new mathematics concepts which I have 3 recently learned.

Notes: i.) Student Q15’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 112/150 = 74.67% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 16/25 = 64.00%

291

Table Q15C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 3 concepts. 2. My effort and persistence determined how well I was able to do on the number 2 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 1 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 2 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 4 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 1 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 2 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I 2 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 2 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 2 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 3 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 2 help. 14. In order to help myself understand the concepts relating to the number theory 3 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 4 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 2 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 5 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 2 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 4 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 3 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 3 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 4 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear 4 up my confusion. 24. When I was working on the number theory problems, I thought about whether or not 4 I was working on the main questions presented by the problems.

292

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 2 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 5 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 5 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 2 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 3 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 4 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 3 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 4 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 4 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 5 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 5 learned from working on the number theory problems.

Notes: i.) Student Q15’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 88/150 = 58.67% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 21/25 = 84.00%

293

Table Q15A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 8. As I am studying a new mathematics lesson, 2 8. As I was working on the number 3 I evaluate my progress in learning the theory problems, I evaluated my material. progress. 17. When I am asked to solve a mathematics 3 17. In attempting to solve the number 5 problem, I focus on whatever data are theory problems, I focused on whatever presented. data had been presented. 23. If I become confused in trying to solve a 2 23. If I became confused in trying to 4 mathematics problem, I try to clear up my solve the number theory problems, I confusion. tried to clear up my confusion. 26. I know how I performed on my 4 26. I knew how I performed on the 5 mathematics test before I receive any feedback number theory problems as soon as I from my teacher. finished my work on the problems. 29. I believe that mathematical problem 2 29. I believe that the number theory 3 solving techniques vary in their level of problems require problem solving difficulty. techniques which vary in their level of difficulty. 30. If I encounter difficulty in solving a 3 30. If I encountered difficulty in 4 mathematics problem, I am not embarrassed to solving the number theory problems, I ask my teacher for help. was not embarrassed to ask my teacher for help. 31. When I am browsing in a bookstore or 2 31. Having completed my work on the 3 library, I flip through the pages of number theory problems, I flip through mathematics books. the pages of mathematics books while I am browsing in a bookstore or library. 33. I enjoy preparing my own mathematics 2 33. Having completed my work on the 4 tests, quizzes and puzzles. number theory problems, I enjoy preparing my own mathematics tests, quizzes and puzzles. 34. I enjoy finding mathematics problems 4 34. Having completed my work on the 5 which have never been solved. number theory problems, I enjoy finding mathematics problems which have never been solved. 35. When I am with my friends, I discuss new 3 35. When I am with my friends, I 5 mathematics concepts which I have recently discuss new mathematics concepts learned. which I have learned from working on the number theory problems.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q15’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

294

Appendix U

Student Q16 Responses

295

Table Q16A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 5 3. When I am studying mathematics, I attempt to use study methods that have been used 4 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 4 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 5 6. When I want to improve my understanding of a mathematics concept, I draw 5 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 3 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 3 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 3 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 4 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 3 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 4 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 3 14. In order to help myself understand a mathematics concept, I think of my own 3 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 4 16. After I have read a mathematics problem, I know if I am able to solve it. 3 17. When I am asked to solve a mathematics problem, I focus on whatever data are 4 presented. 18. If I write the data presented by a problem, I have a better understanding of the 5 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 3 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 3 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 4 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 4 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 4 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 4 my teacher.

296

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 5 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 4 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 3 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 3 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 3 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 1 34. I enjoy finding mathematics problems which have never been solved. 2 35. When I am with my friends, I discuss new mathematics concepts which I have 3 recently learned.

Notes: i.) Student Q16’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 117/150 = 78.00% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 12/25 = 48.00%

297

Table Q16C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 2 concepts. 2. My effort and persistence determined how well I was able to do on the number 4 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 2 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 3 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 4 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 4 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 3 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I 4 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 3 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 3 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 4 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 4 help. 14. In order to help myself understand the concepts relating to the number theory 4 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 4 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 3 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 4 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 4 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 5 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 5 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 4 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear 4 up my confusion. 24. When I was working on the number theory problems, I thought about whether or not 4 I was working on the main questions presented by the problems.

298

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 3 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 3 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 5 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 3 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 4 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 4 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 4 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 4 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 3 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 4 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 3 learned from working on the number theory problems.

Notes: i.) Student Q16’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 111/150 = 74.00% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 18/25 = 72.00%

299

Table Q16A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 9. After completing my mathematics 3 9. After completing my work on the 4 homework, I ask myself if I have learned number theory problems, I asked myself if anything new and valuable. I learned anything new and valuable. 13. Whenever I am confused in 3 13. Whenever I was confused by the 4 mathematics class, I ask someone else for number theory problems, I asked someone help. else for help. 14. In order to help myself understand a 3 14. In order to help myself understand the 4 mathematics concept, I think of my own concepts relating to the number theory examples or illustrations of the concept. problems, I thought of my own examples or illustrations of the concepts. 19. When I am trying to solve a 4 19. In attempting to solve the number 5 mathematics problem, I try to use solutions theory problems, I tried to use solutions of of similar problems which I encountered in similar problems which I had encountered the past. in the past. 20. When I am trying to solve a 3 20. In attempting to solve the number 5 mathematics problem, I try to identify the theory problems, I tried to identify the parts of the problem which I do not parts of the problems which I did not understand. understand. 21. When I am trying to solve a 3 21. When I was trying to solve the number 4 mathematics problem, I ask myself theory problems, I asked myself questions questions to help focus my attention on the to help focus my attention on the problems problem and my attempts to solve it. and my attempts to solve them.

30. If I encounter difficulty in solving a 3 30. If I encountered difficulty in solving 4 mathematics problem, I am not the number theory problems, I was not embarrassed to ask my teacher for help. embarrassed to ask my teacher for help. 31. When I am browsing in a bookstore or 3 31. Having completed my work on the 4 library, I flip through the pages of number theory problems, I flip through the mathematics books. pages of mathematics books while I am browsing in a bookstore or library. 32. I enjoy thinking of new mathematics 3 32. Having completed my work on the 4 problems to work on. number theory problems, I enjoy thinking of new mathematics problems to work on. 33. I enjoy preparing my own mathematics 1 33. Having completed my work on the 3 tests, quizzes and puzzles. number theory problems, I enjoy preparing my own mathematics tests, quizzes and puzzles. 34. I enjoy finding mathematics problems 2 34. Having completed my work on the 4 which have never been solved. number theory problems, I enjoy finding mathematics problems which have never been solved.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q16’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

300

Appendix V

Student Q17 Responses

301

Table Q17A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 5 2. My effort and persistence determine how well I perform on a mathematics test. 3 3. When I am studying mathematics, I attempt to use study methods that have been 2 used by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 4 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 5 6. When I want to improve my understanding of a mathematics concept, I draw 4 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that 5 I must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 3 material. 9. After completing my mathematics homework, I ask myself if I have learned 3 anything new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 5 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 4 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 4 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 2 14. In order to help myself understand a mathematics concept, I think of my own 4 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 5 16. After I have read a mathematics problem, I know if I am able to solve it. 3 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 4 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 5 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 5 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 4 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 3 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 4 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 3 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback 3 from my teacher.

302

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 4 28. When I cannot find the solution to a mathematics problem, I know the nature of the 5 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 3 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 5 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 2 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 4 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 3 34. I enjoy finding mathematics problems which have never been solved. 3 35. When I am with my friends, I discuss new mathematics concepts which I have 3 recently learned.

Notes: i.) Student Q17’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 118/150 = 78.67% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 15/25 = 60.00%

303

Table Q17C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 5 concepts. 2. My effort and persistence determined how well I was able to do on the number 3 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 2 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 4 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 4 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 4 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 3 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I 4 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 5 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 3 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 5 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 3 help. 14. In order to help myself understand the concepts relating to the number theory 4 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 5 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 4 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 5 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 4 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 4 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 5 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 5 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear 5 up my confusion. 24. When I was working on the number theory problems, I thought about whether or not 4 I was working on the main questions presented by the problems.

304

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 4 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 3 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 4 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 4 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 4 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 4 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 3 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 3 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 3 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 3 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 3 learned from working on the number theory problems.

Notes: i.) Student Q17’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 121/150 = 80.67% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 15/25 = 60.00%

305

Table Q17A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 9. After completing my mathematics 3 9. After completing my work on the 4 homework, I ask myself if I have learned number theory problems, I asked anything new and valuable. myself if I learned anything new and valuable. 12. I try different ways of learning 4 12. I tried different ways of 5 mathematics, depending on the type of approaching the number theory mathematics involved. problems, depending on the type of concepts involved. 13. Whenever I am confused in mathematics 2 13. Whenever I was confused by the 3 class, I ask someone else for help. number theory problems, I asked someone else for help. 16. After I have read a mathematics problem, I 3 16. After I finished reading the number 4 know if I am able to solve it. theory problems, I knew if I was able to solve them. 22. If I am having difficulty with a 4 22. If I had difficulty with the number 5 mathematics problem, I read the problem theory problems, I read the problems several times to make sure I understand it. several times. 23. If I become confused in trying to solve a 3 23. If I became confused in trying to 5 mathematics problem, I try to clear up my solve the number theory problems, I confusion. tried to clear up my confusion. 25. Once I find a solution to a mathematics 3 25. Once I found a solution to the 4 problem, I try to find some other, different number theory problems, I tried to find solutions to the same problem. some other, different solutions to the same problems. 29. I believe that mathematical problem 3 29. I believe that the number theory 4 solving techniques vary in their level of problems require problem solving difficulty. techniques which vary in their level of difficulty. 31. When I am browsing in a bookstore or 2 31. Having completed my work on the 3 library, I flip through the pages of number theory problems, I flip through mathematics books. the pages of mathematics books while I am browsing in a bookstore or library.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q17’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

306

Appendix W

Student Q18 Responses

307

Table Q18A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 5 2. My effort and persistence determine how well I perform on a mathematics test. 5 3. When I am studying mathematics, I attempt to use study methods that have been used 3 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 4 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 4 6. When I want to improve my understanding of a mathematics concept, I draw 5 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 5 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 5 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 4 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 5 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 5 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 5 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 5 14. In order to help myself understand a mathematics concept, I think of my own 5 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 5 16. After I have read a mathematics problem, I know if I am able to solve it. 4 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 4 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 4 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 5 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 5 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 5 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 4 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 5 my teacher.

308

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 3 28. When I cannot find the solution to a mathematics problem, I know the nature of the 4 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 5 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 5 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 3 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 4 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 3 34. I enjoy finding mathematics problems which have never been solved. 3 35. When I am with my friends, I discuss new mathematics concepts which I have 3 recently learned.

Notes: i.) Student Q18’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 137/150 = 91.33% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 16/25 = 64.00%

309

Table Q18C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 4 concepts. 2. My effort and persistence determined how well I was able to do on the number 4 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 4 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 3 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 3 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 4 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 4 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 4 9. After completing my work on the number theory problems, I asked myself if I 4 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 4 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 4 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 4 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 5 help. 14. In order to help myself understand the concepts relating to the number theory 4 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 5 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 4 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 4 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 5 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 5 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 4 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 4 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear 5 up my confusion. 24. When I was working on the number theory problems, I thought about whether or not 5 I was working on the main questions presented by the problems.

310

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 3 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 4 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 3 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 3 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 4 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 5 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 3 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 3 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 3 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 3 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 3 learned from working on the number theory problems.

Notes: i.) Student Q18’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 123/150 = 82.00% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 15/25 = 60.00%

311

Table Q18A/C

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q18’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

312

Appendix X

Student Q19 Responses

313

Table Q19A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 5 2. My effort and persistence determine how well I perform on a mathematics test. 4 3. When I am studying mathematics, I attempt to use study methods that have been used 4 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 3 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 3 6. When I want to improve my understanding of a mathematics concept, I draw 3 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 2 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 4 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 3 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 3 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 3 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 4 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 3 14. In order to help myself understand a mathematics concept, I think of my own 2 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 4 16. After I have read a mathematics problem, I know if I am able to solve it. 3 17. When I am asked to solve a mathematics problem, I focus on whatever data are 4 presented. 18. If I write the data presented by a problem, I have a better understanding of the 3 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 5 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 2 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 2 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 4 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 4 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 3 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 5 my teacher.

314

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 4 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 3 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 3 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 3 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 2 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 3 34. I enjoy finding mathematics problems which have never been solved. 1 35. When I am with my friends, I discuss new mathematics concepts which I have 3 recently learned.

Notes: i.) Student Q19’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 105/150 = 70.00% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 12/25 = 48.00%

315

Table Q19C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 3 concepts. 2. My effort and persistence determined how well I was able to do on the number 3 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 3 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 2 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 3 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 2 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 1 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I 3 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 1 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 2 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 3 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 2 help. 14. In order to help myself understand the concepts relating to the number theory 1 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 3 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 3 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 4 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 3 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 4 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 4 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 1 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 4 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear 3 up my confusion. 24. When I was working on the number theory problems, I thought about whether or not 2 I was working on the main questions presented by the problems.

316

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 3 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 4 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 3 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 3 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 4 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 1 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 2 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 2 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 2 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 3 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 3 learned from working on the number theory problems.

Notes: i.) Student Q19’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 81/150 = 54.00% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 12/25 = 48.00%

317

Table Q19A/C

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q19’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

318

Appendix Y

Student Q20 Responses

319

Table Q20A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 5 2. My effort and persistence determine how well I perform on a mathematics test. 4 3. When I am studying mathematics, I attempt to use study methods that have been used 2 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 1 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 4 6. When I want to improve my understanding of a mathematics concept, I draw 4 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 3 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 3 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 3 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 4 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 4 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 4 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 5 14. In order to help myself understand a mathematics concept, I think of my own 4 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 4 16. After I have read a mathematics problem, I know if I am able to solve it. 3 17. When I am asked to solve a mathematics problem, I focus on whatever data are 3 presented. 18. If I write the data presented by a problem, I have a better understanding of the 4 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 4 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 3 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 4 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 3 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 4 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 3 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 4 my teacher.

320

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 4 28. When I cannot find the solution to a mathematics problem, I know the nature of the 4 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 1 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 2 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 1 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 3 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 2 34. I enjoy finding mathematics problems which have never been solved. 2 35. When I am with my friends, I discuss new mathematics concepts which I have 1 recently learned.

Notes: i.) Student Q20’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 104/150 = 69.33% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 9/25 = 36.00%

321

Table Q20C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 4 concepts. 2. My effort and persistence determined how well I was able to do on the number theory 5 problems. 3. When I worked on the number theory problems, I attempted to use techniques that 3 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 5 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 4 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 5 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 4 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 4 9. After completing my work on the number theory problems, I asked myself if I learned 3 anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 4 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the most 5 important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 5 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 3 help. 14. In order to help myself understand the concepts relating to the number theory 5 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 5 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 4 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 5 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 4 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 4 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 4 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 3 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 4 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear up 4 my confusion. 24. When I was working on the number theory problems, I thought about whether or not 3 I was working on the main questions presented by the problems.

322

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 3 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 5 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 5 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 5 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 5 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 4 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 2 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 3 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 2 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 3 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 1 learned from working on the number theory problems.

Notes: i.) Student Q20’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 126/150 = 84.00% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 11/25 = 44.00%

323

Table Q20A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 2. My effort and persistence determine how well I perform 4 2. My effort and persistence determined how well 5 on a mathematics test. I was able to do on the number theory problems. 3. When I am studying mathematics, I attempt to use 2 3. When I worked on the number theory 3 study methods that have been used by successful problems, I attempted to use techniques that have mathematics students. been used by successful mathematics students. 4. If I already know something about a mathematics topic, 1 4. If I already know something about the concepts 5 I can learn even more about it. referenced in the number theory problems, I can learn even more about them. 6. When I want to improve my understanding of a 4 6. When I wanted to improve my understanding 5 mathematics concept, I draw sketches, pictures or of the concepts referenced by the number theory diagrams relating to the concept. problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I start trying to learn a new mathematics 3 7. Before I started trying to solve one or more of 4 concept, I decide on clear goals that I must reach as part the number theory problems, I decided on clear of the learning process. goals that I needed to reach as part of the problem solving process. 8. As I am studying a new mathematics lesson, I evaluate 3 8. As I was working on the number theory 4 my progress in learning the material. problems, I evaluated my progress. 11. After I complete my work on a mathematics problem, 4 11. After I completed my work on the number 5 I think about the most important elements of the problem theory problems, I thought about the most to make sure I really understand them. important elements of the problems to make sure I really understood them. 12. I try different ways of learning mathematics, 4 12. I tried different ways of approaching the 5 depending on the type of mathematics involved. number theory problems, depending on the type of concepts involved. 14. In order to help myself understand a mathematics 4 14. In order to help myself understand the 5 concept, I think of my own examples or illustrations of the concepts relating to the number theory problems, concept. I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have 4 15. I have my own ways of remembering what I 5 learned in mathematics have learned from the number theory problems. 16. After I have read a mathematics problem, I know if I 3 16. After I finished reading the number theory 4 am able to solve it. problems, I knew if I was able to solve them.

17. When I am asked to solve a mathematics problem, I 3 17. In attempting to solve the number theory 5 focus on whatever data are presented. problems, I focused on whatever data had been presented. 23. If I become confused in trying to solve a mathematics 3 23. If I became confused in trying to solve the 4 problem, I try to clear up my confusion. number theory problems, I tried to clear up my confusion. 26. I know how I performed on my mathematics test 4 26. I knew how I performed on the number 5 before I receive any feedback from my teacher. theory problems as soon as I finished my work on the problems. 27. I believe that mathematics topics vary in their level of 4 27. I believe that the concepts referenced by the 5 difficulty. number theory problems vary in their level of difficulty. 28. When I cannot find the solution to a mathematics 4 28. When I could not find solutions to the number 5 problem, I know the nature of the difficulty I am theory problems, I knew the nature of the experiencing. difficulty I was experiencing.

29. I believe that mathematical problem solving 1 29. I believe that the number theory problems 5 techniques vary in their level of difficulty. require problem solving techniques which vary in their level of difficulty. 30. If I encounter difficulty in solving a mathematics 2 30. If I encountered difficulty in solving the 4 problem, I am not embarrassed to ask my teacher for help. number theory problems, I was not embarrassed to ask my teacher for help.

324

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 31. When I am browsing in a bookstore or library, I flip 1 31. Having completed my work on the number 2 through the pages of mathematics books. theory problems, I flip through the pages of mathematics books while I am browsing in a bookstore or library. 34. I enjoy finding mathematics problems which have 2 34. Having completed my work on the number 3 never been solved. theory problems, I enjoy finding mathematics problems which have never been solved.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q20’s responses evidenced an increase in scores.

ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

325

Appendix Z

Student Q21 Responses

326

Table Q21A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 5 3. When I am studying mathematics, I attempt to use study methods that have been used 4 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about it. 5 5. If I have a real interest in a mathematics topic, I can learn more about it. 5 6. When I want to improve my understanding of a mathematics concept, I draw sketches, 3 pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 2 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 3 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 4 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 5 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 5 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 4 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 3 14. In order to help myself understand a mathematics concept, I think of my own 3 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 4 16. After I have read a mathematics problem, I know if I am able to solve it. 3 17. When I am asked to solve a mathematics problem, I focus on whatever data are 3 presented. 18. If I write the data presented by a problem, I have a better understanding of the 4 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 3 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 5 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 3 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 4 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 5 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 5 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 3 my teacher.

327

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 4 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 5 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 3 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 1 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 2 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 1 34. I enjoy finding mathematics problems which have never been solved. 1 35. When I am with my friends, I discuss new mathematics concepts which I have 2 recently learned.

Notes: i.) Student Q21’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 119/150 = 79.33% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 7/25 = 28.00%

328

Table Q21C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 4 concepts. 2. My effort and persistence determined how well I was able to do on the number 4 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 4 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 4 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which 5 they relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 4 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I 3 decided on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 4 9. After completing my work on the number theory problems, I asked myself if I 5 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 5 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 5 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on 5 the type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else 3 for help. 14. In order to help myself understand the concepts relating to the number theory 3 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 3 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 3 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 3 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 5 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of 4 similar problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of 5 the problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions 4 to help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear 5 up my confusion.

329

Question from Questionnaire Part C Score 24. When I was working on the number theory problems, I thought about whether or 5 not I was working on the main questions presented by the problems. 25. Once I found a solution to the number theory problems, I tried to find some other, 5 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 3 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 4 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature 4 of the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques 4 which vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 2 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the 1 pages of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of 3 new mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 1 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 1 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 2 learned from working on the number theory problems.

Notes: i.) Student Q21’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 122/150 = 81.33% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 8/25 = 32.00%

330

Table Q21A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 6. When I want to improve my understanding 3 6. When I wanted to improve my 4 of a mathematics concept, I draw sketches, understanding of the concepts pictures or diagrams relating to the concept. referenced by the number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I start trying to learn a new 2 7. Before I started trying to solve one 3 mathematics concept, I decide on clear goals or more of the number theory that I must reach as part of the learning problems, I decided on clear goals that process. I needed to reach as part of the problem solving process. 8. As I am studying a new mathematics lesson, 3 8. As I was working on the number 4 I evaluate my progress in learning the theory problems, I evaluated my material. progress. 9. After completing my mathematics 4 9. After completing my work on the 5 homework, I ask myself if I have learned number theory problems, I asked anything new and valuable. myself if I learned anything new and valuable. 12. I try different ways of learning 4 12. I tried different ways of 5 mathematics, depending on the type of approaching the number theory mathematics involved. problems, depending on the type of concepts involved. 18. If I write the data presented by a problem, 4 18. If I wrote the data presented by the 5 I have a better understanding of the problem. number theory problems, I had a better understanding of the problems. 19. When I am trying to solve a mathematics 3 19. In attempting to solve the number 4 problem, I try to use solutions of similar theory problems, I tried to use solutions problems which I encountered in the past. of similar problems which I had encountered in the past. 21. When I am trying to solve a mathematics 3 21. When I was trying to solve the 4 problem, I ask myself questions to help focus number theory problems, I asked my attention on the problem and my attempts myself questions to help focus my to solve it. attention on the problems and my attempts to solve them. 23. If I become confused in trying to solve a 4 23. If I became confused in trying to 5 mathematics problem, I try to clear up my solve the number theory problems, I confusion. tried to clear up my confusion. 32. I enjoy thinking of new mathematics 2 32. Having completed my work on the 3 problems to work on. number theory problems, I enjoy thinking of new mathematics problems to work on.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q21’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

331

Appendix AA

Student Q22 Responses

332

Table Q22A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 5 3. When I am studying mathematics, I attempt to use study methods that have been 2 used by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 2 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 2 6. When I want to improve my understanding of a mathematics concept, I draw 3 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that 2 I must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 4 material. 9. After completing my mathematics homework, I ask myself if I have learned 2 anything new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 3 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 4 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 2 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 5 14. In order to help myself understand a mathematics concept, I think of my own 2 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 5 16. After I have read a mathematics problem, I know if I am able to solve it. 4 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 1 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 5 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 3 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 1 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 5 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 2 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 1 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback 5 from my teacher.

333

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 4 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 4 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 3 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 1 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 2 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 2 34. I enjoy finding mathematics problems which have never been solved. 1 35. When I am with my friends, I discuss new mathematics concepts which I have 4 recently learned.

Notes: i.) Student Q22’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 100/150 = 66.67% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 10/25 = 40.00%

334

Table Q22C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 2 concepts. 2. My effort and persistence determined how well I was able to do on the number 3 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 4 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 4 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 4 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 2 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 3 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 5 9. After completing my work on the number theory problems, I asked myself if I 3 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 2 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 5 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 2 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 5 help. 14. In order to help myself understand the concepts relating to the number theory 1 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 5 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 5 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 5 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 1 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 5 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 3 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 2 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear 5 up my confusion. 24. When I was working on the number theory problems, I thought about whether or not 2 I was working on the main questions presented by the problems.

335

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 2 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 5 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 5 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 3 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 5 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 4 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 3 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 4 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 2 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 3 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 4 learned from working on the number theory problems.

Notes: i.) Student Q22’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 107/150 = 71.33% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 16/25 = 64.00%

336

Table Q22A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 3. When I am studying mathematics, I attempt 2 3. When I worked on the number theory 4 to use study methods that have been used by problems, I attempted to use techniques that successful mathematics students. have been used by successful mathematics students. 4. If I already know something about a 2 4. If I already know something about the 4 mathematics topic, I can learn even more concepts referenced in the number theory about it. problems, I can learn even more about them. 5. If I have a real interest in a mathematics 2 5. If I have a real interest in the number 4 topic, I can learn more about it. theory problems and the concepts to which they relate, I can learn more about it. 7. Before I start trying to learn a new 2 7. Before I started trying to solve one or 3 mathematics concept, I decide on clear goals more of the number theory problems, I that I must reach as part of the learning decided on clear goals that I needed to reach process. as part of the problem solving process. 8. As I am studying a new mathematics lesson, 4 8. As I was working on the number theory 5 I evaluate my progress in learning the problems, I evaluated my progress. material. 9. After completing my mathematics 2 9. After completing my work on the number 3 homework, I ask myself if I have learned theory problems, I asked myself if I learned anything new and valuable. anything new and valuable. 11. After I complete my work on a 4 11. After I completed my work on the 5 mathematics problem, I think about the most number theory problems, I thought about important elements of the problem to make the most important elements of the sure I really understand them. problems to make sure I really understood them. 16. After I have read a mathematics problem, I 4 16. After I finished reading the number 5 know if I am able to solve it. theory problems, I knew if I was able to solve them. 21. When I am trying to solve a mathematics 1 21. When I was trying to solve the number 2 problem, I ask myself questions to help focus theory problems, I asked myself questions my attention on the problem and my attempts to help focus my attention on the problems to solve it. and my attempts to solve them.

25. Once I find a solution to a mathematics 1 25. Once I found a solution to the number 2 problem, I try to find some other, different theory problems, I tried to find some other, solutions to the same problem. different solutions to the same problems. 29. I believe that mathematical problem 4 29. I believe that the number theory 5 solving techniques vary in their level of problems require problem solving difficulty. techniques which vary in their level of difficulty. 30. If I encounter difficulty in solving a 3 30. If I encountered difficulty in solving the 4 mathematics problem, I am not embarrassed to number theory problems, I was not ask my teacher for help. embarrassed to ask my teacher for help. 31. When I am browsing in a bookstore or library, I 1 31. Having completed my work on the 3 flip through the pages of mathematics books. number theory problems, I flip through the pages of mathematics books while I am browsing in a bookstore or library.

337

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 32. I enjoy thinking of new mathematics problems 2 32. Having completed my work on the 4 to work on. number theory problems, I enjoy thinking of new mathematics problems to work on. 34. I enjoy finding mathematics problems which 1 34. Having completed my work on the 3 have never been solved. number theory problems, I enjoy finding mathematics problems which have never been solved.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q22’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

338

Appendix BB

Student Q23 Responses

339

Table Q23A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 5 2. My effort and persistence determine how well I perform on a mathematics test. 5 3. When I am studying mathematics, I attempt to use study methods that have been 3 used by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 5 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 5 6. When I want to improve my understanding of a mathematics concept, I draw 4 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that 5 I must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 5 material. 9. After completing my mathematics homework, I ask myself if I have learned 5 anything new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 5 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 5 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 4 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 5 14. In order to help myself understand a mathematics concept, I think of my own 5 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 3 16. After I have read a mathematics problem, I know if I am able to solve it. 5 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 3 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 5 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 4 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 5 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 5 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 4 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback 5 from my teacher.

340

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 2 28. When I cannot find the solution to a mathematics problem, I know the nature of the 5 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 4 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 5 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 5 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 5 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 4 34. I enjoy finding mathematics problems which have never been solved. 4 35. When I am with my friends, I discuss new mathematics concepts which I have 5 recently learned.

Notes: i.) Student Q23’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 135/150 = 90.00% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 23/25 = 92.00%

341

Table Q23C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 5 concepts. 2. My effort and persistence determined how well I was able to do on the number theory 5 problems. 3. When I worked on the number theory problems, I attempted to use techniques that 5 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 5 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 5 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 5 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 5 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 5 9. After completing my work on the number theory problems, I asked myself if I learned 5 anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 5 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the most 5 important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 5 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 5 help. 14. In order to help myself understand the concepts relating to the number theory 5 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 5 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 5 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 5 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 5 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 5 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 5 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 5 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear up 5 my confusion. 24. When I was working on the number theory problems, I thought about whether or not 5 I was working on the main questions presented by the problems.

342

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 5 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 5 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 5 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 5 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 5 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 5 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 5 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 5 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 5 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 5 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 5 learned from working on the number theory problems.

Notes: i.) Student Q23’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 150/150 = 100.00% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 25/25 = 100.00%

343

Table Q23A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 3. When I am studying mathematics, I attempt to 3 3. When I worked on the number theory 5 use study methods that have been used by problems, I attempted to use techniques that successful mathematics students. have been used by successful mathematics students. 6. When I want to improve my understanding of a 4 6. When I wanted to improve my 5 mathematics concept, I draw sketches, pictures or understanding of the concepts referenced by diagrams relating to the concept. the number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 12. I try different ways of learning mathematics, 4 12. I tried different ways of approaching the 5 depending on the type of mathematics involved. number theory problems, depending on the type of concepts involved. 15. I have my own ways of remembering what I 3 15. I have my own ways of remembering 5 have learned in mathematics what I have learned from the number theory problems. 18. If I write the data presented by a problem, I 3 18. If I wrote the data presented by the 5 have a better understanding of the problem. number theory problems, I had a better understanding of the problems. 19. When I am trying to solve a mathematics 4 19. In attempting to solve the number 5 problem, I try to use solutions of similar problems theory problems, I tried to use solutions of which I encountered in the past. similar problems which I had encountered in the past.

21. When I am trying to solve a mathematics 4 21. When I was trying to solve the number 5 problem, I ask myself questions to help focus my theory problems, I asked myself questions attention on the problem and my attempts to solve to help focus my attention on the problems it. and my attempts to solve them.

25. Once I find a solution to a mathematics 4 25. Once I found a solution to the number 5 problem, I try to find some other, different solutions theory problems, I tried to find some other, to the same problem. different solutions to the same problems. 27. I believe that mathematics topics vary in their 2 27. I believe that the concepts referenced by 5 level of difficulty. the number theory problems vary in their level of difficulty. 29. I believe that mathematical problem solving 4 29. I believe that the number theory 5 techniques vary in their level of difficulty. problems require problem solving techniques which vary in their level of difficulty. 33. I enjoy preparing my own mathematics tests, 4 33. Having completed my work on the 5 quizzes and puzzles. number theory problems, I enjoy preparing my own mathematics tests, quizzes and puzzles. 34. I enjoy finding mathematics problems which 4 34. Having completed my work on the 5 have never been solved. number theory problems, I enjoy finding mathematics problems which have never been solved.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q23’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

344

Appendix CC

Student Q24 Responses

345

Table Q24A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 3 3. When I am studying mathematics, I attempt to use study methods that have been used 2 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about it. 3 5. If I have a real interest in a mathematics topic, I can learn more about it. 4 6. When I want to improve my understanding of a mathematics concept, I draw sketches, 5 pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 2 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 3 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 3 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 4 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 4 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 3 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 2 14. In order to help myself understand a mathematics concept, I think of my own 2 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 5 16. After I have read a mathematics problem, I know if I am able to solve it. 4 17. When I am asked to solve a mathematics problem, I focus on whatever data are 4 presented. 18. If I write the data presented by a problem, I have a better understanding of the 4 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 5 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 4 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 5 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 3 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 3 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 1 my teacher.

346

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 4 28. When I cannot find the solution to a mathematics problem, I know the nature of the 4 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 3 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 5 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 2 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 3 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 2 34. I enjoy finding mathematics problems which have never been solved. 3 35. When I am with my friends, I discuss new mathematics concepts which I have 2 recently learned.

Notes: i.) Student Q24’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 107/150 = 71.33% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 12/25 = 48.00%

347

Table Q24C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 4 concepts. 2. My effort and persistence determined how well I was able to do on the number theory 3 problems. 3. When I worked on the number theory problems, I attempted to use techniques that 2 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 3 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 4 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 5 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 2 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I learned 3 anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 4 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the most 4 important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 3 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 3 help. 14. In order to help myself understand the concepts relating to the number theory 2 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 5 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 5 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 4 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 4 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 3 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 5 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 4 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear up 5 my confusion.

348

Question from Questionnaire Part C Score 24. When I was working on the number theory problems, I thought about whether or not 4 I was working on the main questions presented by the problems. 25. Once I found a solution to the number theory problems, I tried to find some other, 3 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 1 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 4 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 4 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 4 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 5 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 2 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 3 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 2 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 3 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 2 learned from working on the number theory problems.

Notes: i.) Student Q24’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 110/150 = 73.33% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 12/25 = 48.00%

349

Table Q24A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 13. Whenever I am confused in mathematics 2 13. Whenever I was confused by the 3 class, I ask someone else for help. number theory problems, I asked someone else for help. 16. After I have read a mathematics problem, I 4 16. After I finished reading the number 5 know if I am able to solve it. theory problems, I knew if I was able to solve them. 24. When I am working on a mathematics 3 24. When I was working on the number 4 problem, I think about whether or not I am theory problems, I thought about working on what the problem is really asking. whether or not I was working on the main questions presented by the problems. 29. I believe that mathematical problem 3 29. I believe that the number theory 4 solving techniques vary in their level of problems require problem solving difficulty. techniques which vary in their level of difficulty.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q24’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

350

Appendix DD

Student Q25 Responses

351

Table Q25A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 4 3. When I am studying mathematics, I attempt to use study methods that have been used 2 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about it. 3 5. If I have a real interest in a mathematics topic, I can learn more about it. 3 6. When I want to improve my understanding of a mathematics concept, I draw sketches, 5 pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 1 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 4 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 2 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 1 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 4 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 5 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 3 14. In order to help myself understand a mathematics concept, I think of my own 5 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 5 16. After I have read a mathematics problem, I know if I am able to solve it. 3 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 5 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 5 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 2 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 5 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 4 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 1 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 4 my teacher.

352

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 3 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 5 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 2 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 1 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 2 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 1 34. I enjoy finding mathematics problems which have never been solved. 1 35. When I am with my friends, I discuss new mathematics concepts which I have 3 recently learned.

Notes: i.) Student Q25’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 109/150 = 72.67% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 8/25 = 32.00%

353

Table Q25C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 4 concepts. 2. My effort and persistence determined how well I was able to do on the number theory 4 problems. 3. When I worked on the number theory problems, I attempted to use techniques that 4 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 4 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 5 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 4 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 3 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 4 9. After completing my work on the number theory problems, I asked myself if I learned 5 anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 5 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the most 5 important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 5 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 3 help. 14. In order to help myself understand the concepts relating to the number theory 3 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 3 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 3 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 3 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 5 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 4 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 5 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 4 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear up 5 my confusion. 24. When I was working on the number theory problems, I thought about whether or not 5 I was working on the main questions presented by the problems.

354

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 5 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 3 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 4 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 4 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 4 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 2 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 1 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 3 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 1 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 1 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 2 learned from working on the number theory problems.

Notes: i.) Student Q25’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 122/150 = 81.33% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 8/25 = 32.00%

355

Table Q25A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response

3. When I am studying mathematics, I attempt to 2 3. When I worked on the number theory 4 use study methods that have been used by problems, I attempted to use techniques that successful mathematics students. have been used by successful mathematics students. 4. If I already know something about a mathematics 3 4. If I already know something about the 4 topic, I can learn even more about it. concepts referenced in the number theory problems, I can learn even more about them. 5. If I have a real interest in a mathematics topic, I 3 5. If I have a real interest in the number 5 can learn more about it. theory problems and the concepts to which they relate, I can learn more about it. 7. Before I start trying to learn a new mathematics 1 7. Before I started trying to solve one or 3 concept, I decide on clear goals that I must reach as more of the number theory problems, I part of the learning process. decided on clear goals that I needed to reach as part of the problem solving process. 9. After completing my mathematics homework, I 2 9. After completing my work on the number 5 ask myself if I have learned anything new and theory problems, I asked myself if I learned valuable. anything new and valuable. 10. As soon as I solve a mathematics problem, I 1 10. As soon as I solved one or more of the 5 wonder if I could have found an easier or quicker number theory problems, I wondered if I way to solve it. could have found easier or quicker ways to arrive at the solution(s). 11. After I complete my work on a mathematics 4 11. After I completed my work on the 5 problem, I think about the most important elements number theory problems, I thought about of the problem to make sure I really understand the most important elements of the them. problems to make sure I really understood them. 21. When I am trying to solve a mathematics 2 21. When I was trying to solve the number 4 problem, I ask myself questions to help focus my theory problems, I asked myself questions attention on the problem and my attempts to solve to help focus my attention on the problems it. and my attempts to solve them. 24. When I am working on a mathematics problem, 4 24. When I was working on the number 5 I think about whether or not I am working on what theory problems, I thought about whether or the problem is really asking. not I was working on the main questions presented by the problems. 25. Once I find a solution to a mathematics 1 25. Once I found a solution to the number 5 problem, I try to find some other, different solutions theory problems, I tried to find some other, to the same problem. different solutions to the same problems. 28. When I cannot find the solution to a 3 28. When I could not find solutions to the 4 mathematics problem, I know the nature of the number theory problems, I knew the nature difficulty I am experiencing. of the difficulty I was experiencing. 32. I enjoy thinking of new mathematics problems 2 32. Having completed my work on the 3 to work on. number theory problems, I enjoy thinking of new mathematics problems to work on.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q25’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

356

Appendix EE

Student Q26 Responses

357

Table Q26A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 3 3. When I am studying mathematics, I attempt to use study methods that have been used 2 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about it. 4 5. If I have a real interest in a mathematics topic, I can learn more about it. 5 6. When I want to improve my understanding of a mathematics concept, I draw sketches, 3 pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 1 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 2 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 3 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 5 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 2 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 5 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 5 14. In order to help myself understand a mathematics concept, I think of my own 2 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 3 16. After I have read a mathematics problem, I know if I am able to solve it. 5 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 2 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 4 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 2 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 2 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 5 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 3 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 4 my teacher.

358

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 3 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 3 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 4 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 1 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 2 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 2 34. I enjoy finding mathematics problems which have never been solved. 2 35. When I am with my friends, I discuss new mathematics concepts which I have 2 recently learned.

Notes: i.) Student Q26’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 105/150 = 70.00% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 9/25 = 36.00%

359

Table Q26C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 3 concepts. 2. My effort and persistence determined how well I was able to do on the number 1 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 1 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 3 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 2 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 1 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 1 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 1 9. After completing my work on the number theory problems, I asked myself if I 3 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 1 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 1 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 3 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 1 help. 14. In order to help myself understand the concepts relating to the number theory 1 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 3 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 5 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 5 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 1 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 3 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 2 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 2 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear 3 up my confusion. 24. When I was working on the number theory problems, I thought about whether or not 2 I was working on the main questions presented by the problems.

360

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 1 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 4 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 5 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 2 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 5 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 1 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 3 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 3 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 1 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 3 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 1 learned from working on the number theory problems.

Notes: i.) Student Q26’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 72/150 = 48.00% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 11/25 = 44.00%

361

Table Q26A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 23. If I become confused in trying to solve a 2 23. If I became confused in trying to 3 mathematics problem, I try to clear up my solve the number theory problems, I confusion. tried to clear up my confusion. 29. I believe that mathematical problem 3 29. I believe that the number theory 5 solving techniques vary in their level of problems require problem solving difficulty. techniques which vary in their level of difficulty. 31. When I am browsing in a bookstore or 1 31. Having completed my work on the 3 library, I flip through the pages of number theory problems, I flip through mathematics books. the pages of mathematics books while I am browsing in a bookstore or library. 32. I enjoy thinking of new mathematics 2 32. Having completed my work on the 3 problems to work on. number theory problems, I enjoy thinking of new mathematics problems to work on. 34. I enjoy finding mathematics problems 2 34. Having completed my work on the 3 which have never been solved. number theory problems, I enjoy finding mathematics problems which have never been solved.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q26’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

362

Appendix FF

Student Q27 Responses

363

Table Q27A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 4 3. When I am studying mathematics, I attempt to use study methods that have been used 2 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 4 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 4 6. When I want to improve my understanding of a mathematics concept, I draw 3 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 3 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 4 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 3 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 3 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 3 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 4 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 3 14. In order to help myself understand a mathematics concept, I think of my own 3 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 4 16. After I have read a mathematics problem, I know if I am able to solve it. 3 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 3 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 3 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 2 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 2 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 3 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 5 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 4 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 2 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 4 my teacher.

364

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 3 28. When I cannot find the solution to a mathematics problem, I know the nature of the 4 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 3 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 5 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 2 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 4 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 2 34. I enjoy finding mathematics problems which have never been solved. 1 35. When I am with my friends, I discuss new mathematics concepts which I have 3 recently learned.

Notes: i.) Student Q27’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 102/150 = 68.00% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 12/25 = 48.00%

365

Table Q27C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 4 concepts. 2. My effort and persistence determined how well I was able to do on the number theory 3 problems. 3. When I worked on the number theory problems, I attempted to use techniques that have 4 been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 2 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 3 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the number 3 theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 2 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I learned 5 anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I could 3 have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the most 3 important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 4 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 2 help. 14. In order to help myself understand the concepts relating to the number theory 3 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 3 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 1 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 4 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 3 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 4 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 5 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 3 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several times. 5 23. If I became confused in trying to solve the number theory problems, I tried to clear up 3 my confusion. 24. When I was working on the number theory problems, I thought about whether or not I 4 was working on the main questions presented by the problems.

366

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 2 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my work 3 on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 3 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 3 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 1 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 5 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 2 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 3 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 2 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 1 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have learned 2 from working on the number theory problems.

Notes: i.) Student Q27’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 96/150 = 64.00% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 10/25 = 40.00%

367

Table Q27A/C

20. When I am trying to solve a mathematics 2 20. In attempting to solve the number 5 problem, I try to identify the parts of the theory problems, I tried to identify the problem which I do not understand. parts of the problems which I did not understand. 21. When I am trying to solve a mathematics 2 21. When I was trying to solve the 3 problem, I ask myself questions to help focus number theory problems, I asked my attention on the problem and my attempts myself questions to help focus my to solve it. attention on the problems and my attempts to solve them. 22. If I am having difficulty with a 3 22. If I had difficulty with the number 5 mathematics problem, I read the problem theory problems, I read the problems several times to make sure I understand it. several times.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q27’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

368

Appendix GG

Student Q28 Responses

369

Table Q28A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 3 3. When I am studying mathematics, I attempt to use study methods that have been used 5 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 3 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 2 6. When I want to improve my understanding of a mathematics concept, I draw 1 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 2 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 5 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 5 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 3 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 4 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 3 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 5 14. In order to help myself understand a mathematics concept, I think of my own 3 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 3 16. After I have read a mathematics problem, I know if I am able to solve it. 3 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 5 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 5 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 5 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 5 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 4 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 4 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 4 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 3 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 4 my teacher.

370

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 4 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 4 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 5 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 1 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 2 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 1 34. I enjoy finding mathematics problems which have never been solved. 1 35. When I am with my friends, I discuss new mathematics concepts which I have 3 recently learned.

Notes: i.) Student Q28’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 116/150 = 77.33% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 8/25 = 32.00%

371

Table Q28C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 4 concepts. 2. My effort and persistence determined how well I was able to do on the number 4 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 5 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 5 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 3 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 5 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 2 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I 3 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 3 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 5 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 5 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 5 help. 14. In order to help myself understand the concepts relating to the number theory 5 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 3 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 3 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 5 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 5 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 5 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 5 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 3 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 4 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear 4 up my confusion. 24. When I was working on the number theory problems, I thought about whether or not 4 I was working on the main questions presented by the problems.

372

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 4 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 4 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 5 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 3 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 3 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 5 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 1 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 3 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 1 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 1 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 3 learned from working on the number theory problems.

Notes: i.) Student Q28’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 122/150 = 81.33% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 9/25 = 36.00%

373

Table Q28A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 2. My effort and persistence determine how 3 2. My effort and persistence 4 well I perform on a mathematics test. determined how well I was able to do on the number theory problems. 4. If I already know something about a 3 4. If I already know something about 5 mathematics topic, I can learn even more the concepts referenced in the number about it. theory problems, I can learn even more about them. 5. If I have a real interest in a mathematics 2 5. If I have a real interest in the number 3 topic, I can learn more about it. theory problems and the concepts to which they relate, I can learn more about it. 6. When I want to improve my understanding 1 6. When I wanted to improve my 5 of a mathematics concept, I draw sketches, understanding of the concepts pictures or diagrams relating to the concept. referenced by the number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 11. After I complete my work on a 4 11. After I completed my work on the 5 mathematics problem, I think about the most number theory problems, I thought important elements of the problem to make about the most important elements of sure I really understand them. the problems to make sure I really understood them. 12. I try different ways of learning 3 12. I tried different ways of 5 mathematics, depending on the type of approaching the number theory mathematics involved. problems, depending on the type of concepts involved. 14. In order to help myself understand a 3 14. In order to help myself understand 5 mathematics concept, I think of my own the concepts relating to the number examples or illustrations of the concept. theory problems, I thought of my own examples or illustrations of the concepts. 25. Once I find a solution to a mathematics 3 25. Once I found a solution to the 4 problem, I try to find some other, different number theory problems, I tried to find solutions to the same problem. some other, different solutions to the same problems. 32. I enjoy thinking of new mathematics 2 32. Having completed my work on the 3 problems to work on. number theory problems, I enjoy thinking of new mathematics problems to work on.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q28’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

374

Appendix HH

Student Q29 Responses

375

Table Q29A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 3 3. When I am studying mathematics, I attempt to use study methods that have been used 3 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 3 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 3 6. When I want to improve my understanding of a mathematics concept, I draw 5 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 4 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 3 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 4 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 3 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 4 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 5 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 4 14. In order to help myself understand a mathematics concept, I think of my own 4 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 5 16. After I have read a mathematics problem, I know if I am able to solve it. 4 17. When I am asked to solve a mathematics problem, I focus on whatever data are 4 presented. 18. If I write the data presented by a problem, I have a better understanding of the 4 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 5 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 3 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 5 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 5 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 3 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 4 my teacher.

376

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 4 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 5 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 4 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 1 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 2 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 2 34. I enjoy finding mathematics problems which have never been solved. 3 35. When I am with my friends, I discuss new mathematics concepts which I have 2 recently learned.

Notes: i.) Student Q29’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 121/150 = 80.67% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 10/25 = 40.00%

377

Table Q29C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 3 concepts. 2. My effort and persistence determined how well I was able to do on the number 4 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 4 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 4 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 4 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 3 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 4 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I 4 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 4 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 4 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 4 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 4 help. 14. In order to help myself understand the concepts relating to the number theory 4 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 4 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 4 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 4 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 4 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 4 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 5 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 4 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 4 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear 3 up my confusion. 24. When I was working on the number theory problems, I thought about whether or not 4 I was working on the main questions presented by the problems.

378

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 5 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 3 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 4 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 5 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 5 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 4 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 3 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 4 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 5 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 4 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 4 learned from working on the number theory problems.

Notes: i.) Student Q29’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 119/150 = 79.33% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 20/25 = 80.00%

379

Table Q29A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 2. My effort and persistence determine how well I 3 2. My effort and persistence determined 4 perform on a mathematics test. how well I was able to do on the number theory problems. 3. When I am studying mathematics, I attempt to 3 3. When I worked on the number theory 4 use study methods that have been used by problems, I attempted to use techniques that successful mathematics students. have been used by successful mathematics students. 4. If I already know something about a mathematics 3 4. If I already know something about the 4 topic, I can learn even more about it. concepts referenced in the number theory problems, I can learn even more about them. 5. If I have a real interest in a mathematics topic, I 3 5. If I have a real interest in the number 4 can learn more about it. theory problems and the concepts to which they relate, I can learn more about it. 10. As soon as I solve a mathematics problem, I 3 10. As soon as I solved one or more of the 4 wonder if I could have found an easier or quicker number theory problems, I wondered if I way to solve it. could have found easier or quicker ways to arrive at the solution(s). 21. When I am trying to solve a mathematics 3 21. When I was trying to solve the number 4 problem, I ask myself questions to help focus my theory problems, I asked myself questions attention on the problem and my attempts to solve to help focus my attention on the problems it. and my attempts to solve them.

25. Once I find a solution to a mathematics 3 25. Once I found a solution to the number 5 problem, I try to find some other, different solutions theory problems, I tried to find some other, to the same problem. different solutions to the same problems. 28. When I cannot find the solution to a 4 28. When I could not find solutions to the 5 mathematics problem, I know the nature of the number theory problems, I knew the nature difficulty I am experiencing. of the difficulty I was experiencing. 31. When I am browsing in a bookstore or library, I 1 31. Having completed my work on the 3 flip through the pages of mathematics books. number theory problems, I flip through the pages of mathematics books while I am browsing in a bookstore or library. 32. I enjoy thinking of new mathematics problems 2 32. Having completed my work on the 4 to work on. number theory problems, I enjoy thinking of new mathematics problems to work on. 33. I enjoy preparing my own mathematics tests, 2 33. Having completed my work on the 5 quizzes and puzzles. number theory problems, I enjoy preparing my own mathematics tests, quizzes and puzzles. 34. I enjoy finding mathematics problems which 3 34. Having completed my work on the 4 have never been solved. number theory problems, I enjoy finding mathematics problems which have never been solved. 35. When I am with my friends, I discuss new 2 35. When I am with my friends, I discuss 4 mathematics concepts which I have recently new mathematics concepts which I have learned. learned from working on the number theory problems..

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q29’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

380

Appendix II

Student Q30 Responses

381

Table Q30A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 5 3. When I am studying mathematics, I attempt to use study methods that have been used 4 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 4 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 4 6. When I want to improve my understanding of a mathematics concept, I draw 2 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 1 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 2 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 5 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 2 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 5 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 5 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 4 14. In order to help myself understand a mathematics concept, I think of my own 2 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 4 16. After I have read a mathematics problem, I know if I am able to solve it. 4 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 3 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 4 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 5 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 3 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 5 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 5 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 4 my teacher.

382

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 3 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 4 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 4 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 5 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 5 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 5 34. I enjoy finding mathematics problems which have never been solved. 3 35. When I am with my friends, I discuss new mathematics concepts which I have 1 recently learned.

Notes: i.) Student Q30’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 116/150 = 77.33% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 19/25 = 76.00%

383

Table Q30C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 3 concepts. 2. My effort and persistence determined how well I was able to do on the number theory 3 problems. 3. When I worked on the number theory problems, I attempted to use techniques that 4 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 3 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 2 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 1 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 3 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 2 9. After completing my work on the number theory problems, I asked myself if I learned 2 anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 2 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the most 4 important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 3 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 4 help. 14. In order to help myself understand the concepts relating to the number theory 1 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 4 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 5 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 5 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 4 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 2 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 2 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 4 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 2 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear up 3 my confusion. 24. When I was working on the number theory problems, I thought about whether or not 2 I was working on the main questions presented by the problems.

384

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 1 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 2 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 3 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 4 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 5 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 3 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 5 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 2 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 3 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 1 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 1 learned from working on the number theory problems.

Notes: i.) Student Q30’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 88/150 = 58.66% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 12/25 = 48.00%

385

Table Q30A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 7. Before I start trying to learn a new 1 7. Before I started trying to solve one 3 mathematics concept, I decide on clear goals or more of the number theory that I must reach as part of the learning problems, I decided on clear goals that process. I needed to reach as part of the problem solving process. 16. After I have read a mathematics problem, I 4 16. After I finished reading the number 5 know if I am able to solve it. theory problems, I knew if I was able to solve them. 18. If I write the data presented by a problem, 3 18. If I wrote the data presented by the 4 I have a better understanding of the problem. number theory problems, I had a better understanding of the problems. 28. When I cannot find the solution to a 3 28. When I could not find solutions to 4 mathematics problem, I know the nature of the the number theory problems, I knew difficulty I am experiencing. the nature of the difficulty I was experiencing. 29. I believe that mathematical problem 4 29. I believe that the number theory 5 solving techniques vary in their level of problems require problem solving difficulty. techniques which vary in their level of difficulty.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q30’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

386

Appendix JJ

Student Q31 Responses

387

Table Q31A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 5 3. When I am studying mathematics, I attempt to use study methods that have been used 4 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about it. 2 5. If I have a real interest in a mathematics topic, I can learn more about it. 3 6. When I want to improve my understanding of a mathematics concept, I draw sketches, 5 pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 4 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 4 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 5 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 3 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 4 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 5 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 4 14. In order to help myself understand a mathematics concept, I think of my own 5 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 5 16. After I have read a mathematics problem, I know if I am able to solve it. 3 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 4 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 5 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 4 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 4 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 5 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 5 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 4 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 3 my teacher. 27. I believe that mathematics topics vary in their level of difficulty. 5

388

Question from Questionnaire Part A Score 28. When I cannot find the solution to a mathematics problem, I know the nature of the 4 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 3 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 3 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 2 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 4 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 2 34. I enjoy finding mathematics problems which have never been solved. 3 35. When I am with my friends, I discuss new mathematics concepts which I have 1 recently learned.

Notes: i.) Student Q31’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 124/150 = 82.67% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 12/25 = 48.00%

389

Table Q31C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 4 concepts. 2. My effort and persistence determined how well I was able to do on the number theory 5 problems. 3. When I worked on the number theory problems, I attempted to use techniques that 5 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 4 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 5 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 5 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 4 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I learned 5 anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 4 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the most 4 important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 5 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 4 help. 14. In order to help myself understand the concepts relating to the number theory 5 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 5 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 4 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 5 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 5 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 3 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 4 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 4 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear up 5 my confusion. 24. When I was working on the number theory problems, I thought about whether or not 5 I was working on the main questions presented by the problems.

390

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 5 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 3 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 5 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 4 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 5 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 3 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 3 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 4 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 4 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 3 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 1 learned from working on the number theory problems.

Notes: i.) Student Q31’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 132/150 = 88.00% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 15/25 = 60.00%

391

Table Q31A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 3. When I am studying mathematics, I attempt 4 3. When I worked on the number 5 to use study methods that have been used by theory problems, I attempted to use successful mathematics students. techniques that have been used by successful mathematics students. 4. If I already know something about a 2 4. If I already know something about 4 mathematics topic, I can learn even more the concepts referenced in the number about it. theory problems, I can learn even more about them. 5. If I have a real interest in a mathematics 3 5. If I have a real interest in the number 5 topic, I can learn more about it. theory problems and the concepts to which they relate, I can learn more about it. 10. As soon as I solve a mathematics problem, 3 10. As soon as I solved one or more of 4 I wonder if I could have found an easier or the number theory problems, I quicker way to solve it. wondered if I could have found easier or quicker ways to arrive at the solution(s). 16. After I have read a mathematics problem, I 3 16. After I finished reading the number 4 know if I am able to solve it. theory problems, I knew if I was able to solve them. 18. If I write the data presented by a problem, 4 18. If I wrote the data presented by the 5 I have a better understanding of the problem. number theory problems, I had a better understanding of the problems. 25. Once I find a solution to a mathematics 4 25. Once I found a solution to the 5 problem, I try to find some other, different number theory problems, I tried to find solutions to the same problem. some other, different solutions to the same problems. 29. I believe that mathematical problem 3 29. I believe that the number theory 5 solving techniques vary in their level of problems require problem solving difficulty. techniques which vary in their level of difficulty. 31. When I am browsing in a bookstore or 2 31. Having completed my work on the 3 library, I flip through the pages of number theory problems, I flip through mathematics books. the pages of mathematics books while I am browsing in a bookstore or library. 33. I enjoy preparing my own mathematics 2 33. Having completed my work on the 4 tests, quizzes and puzzles. number theory problems, I enjoy preparing my own mathematics tests, quizzes and puzzles.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q31’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

392

Appendix KK

Student Q32 Responses

393

Table Q32A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 5 3. When I am studying mathematics, I attempt to use study methods that have been used 4 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 3 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 5 6. When I want to improve my understanding of a mathematics concept, I draw 5 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 4 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 3 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 3 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 4 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 4 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 4 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 3 14. In order to help myself understand a mathematics concept, I think of my own 5 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 3 16. After I have read a mathematics problem, I know if I am able to solve it. 3 17. When I am asked to solve a mathematics problem, I focus on whatever data are 4 presented. 18. If I write the data presented by a problem, I have a better understanding of the 5 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 4 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 5 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 4 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 3 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 3 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 3 my teacher.

394

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 5 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 3 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 5 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 3 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 3 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 1 34. I enjoy finding mathematics problems which have never been solved. 2 35. When I am with my friends, I discuss new mathematics concepts which I have 4 recently learned.

Notes: i.) Student Q32’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 120/150 = 80.00% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 13/25 = 52.00%

395

Table Q32C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 4 concepts. 2. My effort and persistence determined how well I was able to do on the number 4 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 3 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 4 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 5 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 3 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 3 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 4 9. After completing my work on the number theory problems, I asked myself if I 3 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 5 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 4 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 5 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 4 help. 14. In order to help myself understand the concepts relating to the number theory 4 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 3 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 3 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 4 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 5 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 5 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 4 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 4 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear 5 up my confusion. 24. When I was working on the number theory problems, I thought about whether or not 4 I was working on the main questions presented by the problems.

396

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 3 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 4 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 4 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 4 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 2 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 4 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 3 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 3 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 1 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 2 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 3 learned from working on the number theory problems.

Notes: i.) Student Q32’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 118/150 = 78.67% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 12/25 = 48.00%

397

Table Q32A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part Response

4. If I already know something about a 3 4. If I already know something about 4 mathematics topic, I can learn even more the concepts referenced in the number about it. theory problems, I can learn even more about them. 8. As I am studying a new mathematics lesson, 3 8. As I was working on the number 4 I evaluate my progress in learning the theory problems, I evaluated my material. progress. 10. As soon as I solve a mathematics problem, 4 10. As soon as I solved one or more of 5 I wonder if I could have found an easier or the number theory problems, I quicker way to solve it. wondered if I could have found easier or quicker ways to arrive at the solution(s). 12. I try different ways of learning 4 12. I tried different ways of 5 mathematics, depending on the type of approaching the number theory mathematics involved. problems, depending on the type of concepts involved. 13. Whenever I am confused in mathematics 3 13. Whenever I was confused by the 4 class, I ask someone else for help. number theory problems, I asked someone else for help. 19. When I am trying to solve a mathematics 4 19. In attempting to solve the number 5 problem, I try to use solutions of similar theory problems, I tried to use solutions problems which I encountered in the past. of similar problems which I had encountered in the past.

23. If I become confused in trying to solve a 4 23. If I became confused in trying to 5 mathematics problem, I try to clear up my solve the number theory problems, I confusion. tried to clear up my confusion. 24. When I am working on a mathematics 3 24. When I was working on the number 4 problem, I think about whether or not I am theory problems, I thought about working on what the problem is really asking. whether or not I was working on the main questions presented by the problems. 26. I know how I performed on my 3 26. I knew how I performed on the 4 mathematics test before I receive any feedback number theory problems as soon as I from my teacher. finished my work on the problems.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q32’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

398

Appendix LL

Student Q33 Responses

399

Table Q33A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 5 3. When I am studying mathematics, I attempt to use study methods that have been used 4 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about it. 3 5. If I have a real interest in a mathematics topic, I can learn more about it. 4 6. When I want to improve my understanding of a mathematics concept, I draw sketches, 4 pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 5 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 4 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 4 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 4 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 5 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 4 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 5 14. In order to help myself understand a mathematics concept, I think of my own 4 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 4 16. After I have read a mathematics problem, I know if I am able to solve it. 4 17. When I am asked to solve a mathematics problem, I focus on whatever data are 4 presented. 18. If I write the data presented by a problem, I have a better understanding of the 4 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 4 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 4 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 4 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 4 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 4 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 4 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 3 my teacher.

400

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 4 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 4 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 4 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 4 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 4 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 4 34. I enjoy finding mathematics problems which have never been solved. 3 35. When I am with my friends, I discuss new mathematics concepts which I have 5 recently learned.

Notes: i.) Student Q33’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 123/150 = 82.00% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 20/25 = 80.00%

401

Table Q33C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 4 concepts. 2. My effort and persistence determined how well I was able to do on the number 5 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 4 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 3 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 5 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 5 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 4 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 5 9. After completing my work on the number theory problems, I asked myself if I 4 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 4 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 4 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 4 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 4 help. 14. In order to help myself understand the concepts relating to the number theory 4 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 4 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 4 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 4 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 4 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 4 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 4 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 4 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 4 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear 4 up my confusion. 24. When I was working on the number theory problems, I thought about whether or not 4 I was working on the main questions presented by the problems.

402

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 4 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 4 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 4 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 4 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 4 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 4 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 4 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 3 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 3 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 3 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 3 learned from working on the number theory problems.

Notes: i.) Student Q33’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 123/150 = 82.00% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 16/25 = 64.00%

403

Table Q33A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 5. If I have a real interest in a mathematics 4 5. If I have a real interest in the number 5 topic, I can learn more about it. theory problems and the concepts to which they relate, I can learn more about it. 6. When I want to improve my understanding 4 6. When I wanted to improve my 5 of a mathematics concept, I draw sketches, understanding of the concepts pictures or diagrams relating to the concept. referenced by the number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 8. As I am studying a new mathematics lesson, 4 8. As I was working on the number 5 I evaluate my progress in learning the theory problems, I evaluated my material. progress. 26. I know how I performed on my 3 26. I knew how I performed on the 4 mathematics test before I receive any feedback number theory problems as soon as I from my teacher. finished my work on the problems.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q33’s

responses evidenced an increase in scores.

ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

404

Appendix MM

Student Q34 Responses

405

Table Q34A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 3 3. When I am studying mathematics, I attempt to use study methods that have been used 3 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about it. 2 5. If I have a real interest in a mathematics topic, I can learn more about it. 3 6. When I want to improve my understanding of a mathematics concept, I draw sketches, 5 pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 2 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 2 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 3 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 3 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 3 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 3 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 4 14. In order to help myself understand a mathematics concept, I think of my own 5 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 5 16. After I have read a mathematics problem, I know if I am able to solve it. 3 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 3 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 5 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 2 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 2 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 4 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 3 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 5 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 3 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 3 my teacher. 27. I believe that mathematics topics vary in their level of difficulty. 5

406

Question from Questionnaire Part A Score 28. When I cannot find the solution to a mathematics problem, I know the nature of the 4 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 5 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 3 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 1 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 2 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 1 34. I enjoy finding mathematics problems which have never been solved. 2 35. When I am with my friends, I discuss new mathematics concepts which I have 3 recently learned.

Notes: i.) Student Q34’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 105/150 = 70.00% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 9/25 = 36.00%

407

Table Q34C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 2 concepts. 2. My effort and persistence determined how well I was able to do on the number 3 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 3 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 3 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 2 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 4 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 3 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I 2 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 3 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 3 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 2 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 4 help. 14. In order to help myself understand the concepts relating to the number theory 4 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 5 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 3 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 4 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 4 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 3 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 3 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 2 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 4 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear 4 up my confusion. 24. When I was working on the number theory problems, I thought about whether or not 3 I was working on the main questions presented by the problems.

408

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 2 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 2 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 5 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 3 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 5 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 3 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 2 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 2 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 2 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 2 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 2 learned from working on the number theory problems.

Notes: i.) Student Q34’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 96/150 = 64.00% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 10/25 = 40.00%

409

Table Q34A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 4. If I already know something about a 2 4. If I already know something about 3 mathematics topic, I can learn even more the concepts referenced in the number about it. theory problems, I can learn even more about them. 7. Before I start trying to learn a new 2 7. Before I started trying to solve one 3 mathematics concept, I decide on clear goals or more of the number theory that I must reach as part of the learning problems, I decided on clear goals that process. I needed to reach as part of the problem solving process. 8. As I am studying a new mathematics lesson, 2 8. As I was working on the number 3 I evaluate my progress in learning the theory problems, I evaluated my material. progress. 18. If I write the data presented by a problem, 3 18. If I wrote the data presented by the 4 I have a better understanding of the problem. number theory problems, I had a better understanding of the problems. 20. When I am trying to solve a mathematics 2 20. In attempting to solve the number 3 problem, I try to identify the parts of the theory problems, I tried to identify the problem which I do not understand. parts of the problems which I did not understand. 23. If I become confused in trying to solve a 3 23. If I became confused in trying to 4 mathematics problem, I try to clear up my solve the number theory problems, I confusion. tried to clear up my confusion. 31. When I am browsing in a bookstore or 1 31. Having completed my work on the 2 library, I flip through the pages of number theory problems, I flip through mathematics books. the pages of mathematics books while I am browsing in a bookstore or library. 33. I enjoy preparing my own mathematics 1 33. Having completed my work on the 2 tests, quizzes and puzzles. number theory problems, I enjoy preparing my own mathematics tests, quizzes and puzzles.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q34’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

410

Appendix NN

Student Q35 Responses

411

Table Q35A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 4 3. When I am studying mathematics, I attempt to use study methods that have been used 3 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 4 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 2 6. When I want to improve my understanding of a mathematics concept, I draw 5 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 3 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 4 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 4 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 3 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 5 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 3 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 3 14. In order to help myself understand a mathematics concept, I think of my own 4 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 4 16. After I have read a mathematics problem, I know if I am able to solve it. 4 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 5 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 5 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 5 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 4 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 3 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 3 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 5 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 3 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 3 my teacher.

412

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 3 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 5 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 3 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 2 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 2 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 1 34. I enjoy finding mathematics problems which have never been solved. 2 35. When I am with my friends, I discuss new mathematics concepts which I have 1 recently learned.

Notes: i.) Student Q35’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 116/150 = 77.33% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 8/25 = 32.00%

413

Table Q35C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 2 concepts. 2. My effort and persistence determined how well I was able to do on the number theory 4 problems. 3. When I worked on the number theory problems, I attempted to use techniques that 4 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 2 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 4 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 5 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 2 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 2 9. After completing my work on the number theory problems, I asked myself if I learned 4 anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 4 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the most 4 important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 4 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 4 help. 14. In order to help myself understand the concepts relating to the number theory 4 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 4 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 4 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 4 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 4 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 4 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 4 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 4 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 4 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear up 4 my confusion. 24. When I was working on the number theory problems, I thought about whether or not 4 I was working on the main questions presented by the problems.

414

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 4 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 4 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 4 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 4 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 3 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 4 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 4 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 3 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 3 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 3 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 3 learned from working on the number theory problems.

Notes: i.) Student Q35’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 112/150 = 74.67% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 16/25 = 64.00%

415

Table Q35A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 3. When I am studying mathematics, I 3 3. When I worked on the number theory problems, 4 attempt to use study methods that have been I attempted to use techniques that have been used used by successful mathematics students. by successful mathematics students. 5. If I have a real interest in a mathematics 2 5. If I have a real interest in the number theory 4 topic, I can learn more about it. problems and the concepts to which they relate, I can learn more about it. 10. As soon as I solve a mathematics 3 10. As soon as I solved one or more of the number 4 problem, I wonder if I could have found an theory problems, I wondered if I could have found easier or quicker way to solve it. easier or quicker ways to arrive at the solution(s). 12. I try different ways of learning 3 12. I tried different ways of approaching the 4 mathematics, depending on the type of number theory problems, depending on the type of mathematics involved. concepts involved. 13. Whenever I am confused in mathematics 3 13. Whenever I was confused by the number theory 4 class, I ask someone else for help. problems, I asked someone else for help. 22. If I am having difficulty with a 3 22. If I had difficulty with the number theory 4 mathematics problem, I read the problem problems, I read the problems several times. several times to make sure I understand it. 23. If I become confused in trying to solve a 3 23. If I became confused in trying to solve the 4 mathematics problem, I try to clear up my number theory problems, I tried to clear up my confusion. confusion. 25. Once I find a solution to a mathematics 3 25. Once I found a solution to the number theory 4 problem, I try to find some other, different problems, I tried to find some other, different solutions to the same problem. solutions to the same problems. 26. I know how I performed on my 3 26. I knew how I performed on the number theory 4 mathematics test before I receive any problems as soon as I finished my work on the feedback from my teacher. problems. 28. When I cannot find the solution to a 3 28. When I could not find solutions to the number 4 mathematics problem, I know the nature of theory problems, I knew the nature of the difficulty the difficulty I am experiencing. I was experiencing. 30. If I encounter difficulty in solving a 3 30. If I encountered difficulty in solving the 4 mathematics problem, I am not embarrassed number theory problems, I was not embarrassed to to ask my teacher for help. ask my teacher for help. 31. When I am browsing in a bookstore or 2 31. Having completed my work on the number 4 library, I flip through the pages of theory problems, I flip through the pages of mathematics books. mathematics books while I am browsing in a bookstore or library. 32. I enjoy thinking of new mathematics 2 32. Having completed my work on the number 3 problems to work on. theory problems, I enjoy thinking of new mathematics problems to work on. 33. I enjoy preparing my own mathematics 1 33. Having completed my work on the number 3 tests, quizzes and puzzles. theory problems, I enjoy preparing my own mathematics tests, quizzes and puzzles. 34. I enjoy finding mathematics problems 2 34. Having completed my work on the number 3 which have never been solved. theory problems, I enjoy finding mathematics problems which have never been solved. 35. When I am with my friends, I discuss 1 35. When I am with my friends, I discuss new 3 new mathematics concepts which I have mathematics concepts which I have learned from recently learned. working on the number theory problems.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q35’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

416

Appendix OO

Student Q36 Responses

417

Table Q36A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 4 3. When I am studying mathematics, I attempt to use study methods that have been used 5 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 3 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 3 6. When I want to improve my understanding of a mathematics concept, I draw 5 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 4 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 3 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 3 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 4 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 5 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 5 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 5 14. In order to help myself understand a mathematics concept, I think of my own 5 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 3 16. After I have read a mathematics problem, I know if I am able to solve it. 3 17. When I am asked to solve a mathematics problem, I focus on whatever data are 4 presented. 18. If I write the data presented by a problem, I have a better understanding of the 4 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 5 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 3 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 2 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 5 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 4 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 3 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 4 my teacher.

418

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 3 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 4 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 5 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 2 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 2 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 2 34. I enjoy finding mathematics problems which have never been solved. 3 35. When I am with my friends, I discuss new mathematics concepts which I have 4 recently learned.

Notes: i.) Student Q36’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 120/150 = 80.00% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 13/25 = 52.00%

419

Table Q36C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 4 concepts. 2. My effort and persistence determined how well I was able to do on the number theory 5 problems. 3. When I worked on the number theory problems, I attempted to use techniques that 5 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 4 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 4 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 5 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 3 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I learned 4 anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 4 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the most 4 important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 5 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 4 help. 14. In order to help myself understand the concepts relating to the number theory 5 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 4 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 5 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 5 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 4 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 4 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 5 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 3 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear up 5 my confusion. 24. When I was working on the number theory problems, I thought about whether or not 5 I was working on the main questions presented by the problems.

420

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 3 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 5 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 5 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 3 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 4 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 5 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 4 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 4 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 5 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 3 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 5 learned from working on the number theory problems.

Notes: i.) Student Q36’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 129/150 = 86.00% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 21/25 = 84.00%

421

Table Q36A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 2. My effort and persistence determine how 4 2. My effort and persistence determined how well I 5 well I perform on a mathematics test. was able to do on the number theory problems. 4. If I already know something about a 3 4. If I already know something about the concepts 4 mathematics topic, I can learn even more referenced in the number theory problems, I can about it. learn even more about them. 5. If I have a real interest in a mathematics 3 5. If I have a real interest in the number theory 4 topic, I can learn more about it. problems and the concepts to which they relate, I can learn more about it. 9. After completing my mathematics 3 9. After completing my work on the number 4 homework, I ask myself if I have learned theory problems, I asked myself if I learned anything new and valuable. anything new and valuable. 15. I have my own ways of remembering what 3 15. I have my own ways of remembering what I 4 I have learned in mathematics have learned from the number theory problems. 16. After I have read a mathematics problem, I 3 16. After I finished reading the number theory 5 know if I am able to solve it. problems, I knew if I was able to solve them. 17. When I am asked to solve a mathematics 4 17. In attempting to solve the number theory 5 problem, I focus on whatever data are problems, I focused on whatever data had been presented. presented. 20. When I am trying to solve a mathematics 3 20. In attempting to solve the number theory 5 problem, I try to identify the parts of the problems, I tried to identify the parts of the problem which I do not understand. problems which I did not understand. 21. When I am trying to solve a mathematics 2 21. When I was trying to solve the number theory 3 problem, I ask myself questions to help focus problems, I asked myself questions to help focus my attention on the problem and my attempts my attention on the problems and my attempts to to solve it. solve them. 24. When I am working on a mathematics 4 24. When I was working on the number theory 5 problem, I think about whether or not I am problems, I thought about whether or not I was working on what the problem is really asking. working on the main questions presented by the problems. 26. I know how I performed on my 4 26. I knew how I performed on the number theory 5 mathematics test before I receive any feedback problems as soon as I finished my work on the from my teacher. problems. 31. When I am browsing in a bookstore or 2 31. Having completed my work on the number 4 library, I flip through the pages of mathematics theory problems, I flip through the pages of books. mathematics books while I am browsing in a bookstore or library. 32. I enjoy thinking of new mathematics 2 32. Having completed my work on the number 4 problems to work on. theory problems, I enjoy thinking of new mathematics problems to work on. 33. I enjoy preparing my own mathematics 2 33. Having completed my work on the number 5 tests, quizzes and puzzles. theory problems, I enjoy preparing my own mathematics tests, quizzes and puzzles. 35. When I am with my friends, I discuss new 4 35. When I am with my friends, I discuss new 5 mathematics concepts which I have recently mathematics concepts which I have learned from learned. working on the number theory problems.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q36’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

422

Appendix PP

Student Q37 Responses

423

Table Q37A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 2 3. When I am studying mathematics, I attempt to use study methods that have been used 3 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 4 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 5 6. When I want to improve my understanding of a mathematics concept, I draw 3 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 2 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 2 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 5 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 5 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 5 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 5 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 3 14. In order to help myself understand a mathematics concept, I think of my own 5 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 5 16. After I have read a mathematics problem, I know if I am able to solve it. 5 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 5 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 5 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 3 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 4 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 4 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 5 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 3 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 3 my teacher.

424

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 4 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 5 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 5 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 2 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 2 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 1 34. I enjoy finding mathematics problems which have never been solved. 1 35. When I am with my friends, I discuss new mathematics concepts which I have 2 recently learned.

Notes: i.) Student Q37’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 124/150 = 82.67% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 8/25 = 32.00%

425

Table Q37C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 5 concepts. 2. My effort and persistence determined how well I was able to do on the number 2 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 3 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 5 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 2 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 4 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 2 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I 4 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 5 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 5 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 4 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 4 help. 14. In order to help myself understand the concepts relating to the number theory 3 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 3 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 5 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 5 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 5 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 5 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 4 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 3 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear 5 up my confusion. 24. When I was working on the number theory problems, I thought about whether or not 5 I was working on the main questions presented by the problems.

426

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 4 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 3 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 5 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 3 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 5 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 5 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 4 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 4 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 1 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 1 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 1 learned from working on the number theory problems.

Notes: i.) Student Q37’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 121/150 = 80.67% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 11/25 = 44.00%

427

Table Q37A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 1. After I have studied a mathematics lesson, I 4 1. After I worked on the number theory 5 know how well I understand it. problems, I knew how well I understood the concepts. 4. If I already know something about a 4 4. If I already know something about 5 mathematics topic, I can learn even more the concepts referenced in the number about it. theory problems, I can learn even more about them. 6. When I want to improve my understanding 3 6. When I wanted to improve my 4 of a mathematics concept, I draw sketches, understanding of the concepts pictures or diagrams relating to the concept. referenced by the number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 8. As I am studying a new mathematics lesson, 2 8. As I was working on the number 3 I evaluate my progress in learning the theory problems, I evaluated my material. progress. 13. Whenever I am confused in mathematics 3 13. Whenever I was confused by the 4 class, I ask someone else for help. number theory problems, I asked someone else for help. 20. When I am trying to solve a mathematics 3 20. In attempting to solve the number 4 problem, I try to identify the parts of the theory problems, I tried to identify the problem which I do not understand. parts of the problems which I did not understand. 23. If I become confused in trying to solve a 4 23. If I became confused in trying to 5 mathematics problem, I try to clear up my solve the number theory problems, I confusion. tried to clear up my confusion. 25. Once I find a solution to a mathematics 3 25. Once I found a solution to the 4 problem, I try to find some other, different number theory problems, I tried to find solutions to the same problem. some other, different solutions to the same problems. 31. When I am browsing in a bookstore or 2 31. Having completed my work on the 4 library, I flip through the pages of number theory problems, I flip through mathematics books. the pages of mathematics books while I am browsing in a bookstore or library. 32. I enjoy thinking of new mathematics 2 32. Having completed my work on the 4 problems to work on. number theory problems, I enjoy thinking of new mathematics problems to work on.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q37’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

428

Appendix QQ

Student Q38 Responses

429

Table Q38A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 5 3. When I am studying mathematics, I attempt to use study methods that have been used 2 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 4 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 3 6. When I want to improve my understanding of a mathematics concept, I draw 5 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 4 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 3 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 4 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 5 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 5 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 5 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 3 14. In order to help myself understand a mathematics concept, I think of my own 3 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 4 16. After I have read a mathematics problem, I know if I am able to solve it. 5 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 5 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 4 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 5 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 3 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 5 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 4 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 3 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 5 my teacher.

430

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 4 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 4 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 5 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 3 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 3 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 3 34. I enjoy finding mathematics problems which have never been solved. 3 35. When I am with my friends, I discuss new mathematics concepts which I have 2 recently learned.

Notes: i.) Student Q38’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 126/150 = 84.00% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 14/25 = 56.00%

431

Table Q38C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 4 concepts. 2. My effort and persistence determined how well I was able to do on the number 5 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 2 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 4 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which 3 they relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 5 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I 4 decided on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I 4 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 5 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 5 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on 5 the type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else 3 for help. 14. In order to help myself understand the concepts relating to the number theory 3 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 4 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 5 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 5 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 5 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of 4 similar problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of 5 the problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions 3 to help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear 5 up my confusion.

432

Question from Questionnaire Part C Score 24. When I was working on the number theory problems, I thought about whether or 4 not I was working on the main questions presented by the problems. 25. Once I found a solution to the number theory problems, I tried to find some other, 3 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 5 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 5 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature 4 of the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques 4 which vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 5 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the 3 pages of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of 3 new mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 3 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 3 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 2 learned from working on the number theory problems.

Notes: i.) Student Q38’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 126/150 = 84.00% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 14/25 = 56.00%

433

Table Q38A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response _ Each Part A response _ Each Part C response matched the matched the corresponding Part C corresponding Part A response. response.

Notes: i.) Corresponding questions from Parts A and C of questionnaire (if any), for which Student Q38’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

434

Appendix RR

Student Q39 Responses

435

Table Q39A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 4 2. My effort and persistence determine how well I perform on a mathematics test. 4 3. When I am studying mathematics, I attempt to use study methods that have been used 3 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about it. 3 5. If I have a real interest in a mathematics topic, I can learn more about it. 4 6. When I want to improve my understanding of a mathematics concept, I draw sketches, 5 pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 1 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 4 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 4 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 5 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 4 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 3 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 3 14. In order to help myself understand a mathematics concept, I think of my own 5 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 4 16. After I have read a mathematics problem, I know if I am able to solve it. 4 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 5 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 5 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 4 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 5 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 4 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 5 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 1 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 4 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 3 my teacher.

436

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 5 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 3 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 1 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 1 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 3 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 1 34. I enjoy finding mathematics problems which have never been solved. 4 35. When I am with my friends, I discuss new mathematics concepts which I have 2 recently learned.

Notes: i.) Student Q39’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 115/150 = 76.67% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 11/25 = 44.00%

437

Table Q39C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 2 concepts. 2. My effort and persistence determined how well I was able to do on the number theory 3 problems. 3. When I worked on the number theory problems, I attempted to use techniques that 4 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 3 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 5 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 5 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 3 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 2 9. After completing my work on the number theory problems, I asked myself if I learned 4 anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 5 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the most 2 important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 3 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 5 help. 14. In order to help myself understand the concepts relating to the number theory 5 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 4 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 2 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 2 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 5 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 5 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 4 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 2 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear up 2 my confusion.

438

Question from Questionnaire Part C Score 24. When I was working on the number theory problems, I thought about whether or not 2 I was working on the main questions presented by the problems. 25. Once I found a solution to the number theory problems, I tried to find some other, 1 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 2 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 5 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 3 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 3 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 2 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 1 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 5 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 2 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 5 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 3 learned from working on the number theory problems.

Notes: i.) Student Q39’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 100/150 = 66.67% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 16/25 = 64.00%

439

Table Q39A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 3. When I am studying mathematics, I attempt 3 3. When I worked on the number 4 to use study methods that have been used by theory problems, I attempted to use successful mathematics students. techniques that have been used by successful mathematics students. 5. If I have a real interest in a mathematics 4 5. If I have a real interest in the number 5 topic, I can learn more about it. theory problems and the concepts to which they relate, I can learn more about it. 7. Before I start trying to learn a new 1 7. Before I started trying to solve one 3 mathematics concept, I decide on clear goals or more of the number theory that I must reach as part of the learning problems, I decided on clear goals that process. I needed to reach as part of the problem solving process. 13. Whenever I am confused in mathematics 3 13. Whenever I was confused by the 5 class, I ask someone else for help. number theory problems, I asked someone else for help. 22. If I am having difficulty with a 4 22. If I had difficulty with the number 5 mathematics problem, I read the problem theory problems, I read the problems several times to make sure I understand it. several times. 24. When I am working on a mathematics 1 24. When I was working on the number 2 problem, I think about whether or not I am theory problems, I thought about working on what the problem is really asking. whether or not I was working on the main questions presented by the problems. 30. If I encounter difficulty in solving a 1 30. If I encountered difficulty in 2 mathematics problem, I am not embarrassed to solving the number theory problems, I ask my teacher for help. was not embarrassed to ask my teacher for help. 32. I enjoy thinking of new mathematics 3 32. Having completed my work on the 5 problems to work on. number theory problems, I enjoy thinking of new mathematics problems to work on. 33. I enjoy preparing my own mathematics 1 33. Having completed my work on the 2 tests, quizzes and puzzles. number theory problems, I enjoy preparing my own mathematics tests, quizzes and puzzles. 34. I enjoy finding mathematics problems 4 34. Having completed my work on the 5 which have never been solved. number theory problems, I enjoy finding mathematics problems which have never been solved. 35. When I am with my friends, I discuss new 2 35. When I am with my friends, I 3 mathematics concepts which I have recently discuss new mathematics concepts learned. which I have learned from working on the number theory problems.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q39’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 440

Appendix SS

Student Q40 Responses

441

Table Q40A

Question from Questionnaire Part A Score 1. After I have studied a mathematics lesson, I know how well I understand it. 3 2. My effort and persistence determine how well I perform on a mathematics test. 3 3. When I am studying mathematics, I attempt to use study methods that have been used 4 by successful mathematics students. 4. If I already know something about a mathematics topic, I can learn even more about 3 it. 5. If I have a real interest in a mathematics topic, I can learn more about it. 3 6. When I want to improve my understanding of a mathematics concept, I draw 5 sketches, pictures or diagrams relating to the concept. 7. Before I start trying to learn a new mathematics concept, I decide on clear goals that I 3 must reach as part of the learning process. 8. As I am studying a new mathematics lesson, I evaluate my progress in learning the 4 material. 9. After completing my mathematics homework, I ask myself if I have learned anything 4 new and valuable. 10. As soon as I solve a mathematics problem, I wonder if I could have found an easier 4 or quicker way to solve it. 11. After I complete my work on a mathematics problem, I think about the most 4 important elements of the problem to make sure I really understand them. 12. I try different ways of learning mathematics, depending on the type of mathematics 4 involved. 13. Whenever I am confused in mathematics class, I ask someone else for help. 3 14. In order to help myself understand a mathematics concept, I think of my own 5 examples or illustrations of the concept. 15. I have my own ways of remembering what I have learned in mathematics 5 16. After I have read a mathematics problem, I know if I am able to solve it. 4 17. When I am asked to solve a mathematics problem, I focus on whatever data are 5 presented. 18. If I write the data presented by a problem, I have a better understanding of the 5 problem. 19. When I am trying to solve a mathematics problem, I try to use solutions of similar 5 problems which I encountered in the past. 20. When I am trying to solve a mathematics problem, I try to identify the parts of the 5 problem which I do not understand. 21. When I am trying to solve a mathematics problem, I ask myself questions to help 3 focus my attention on the problem and my attempts to solve it. 22. If I am having difficulty with a mathematics problem, I read the problem several 5 times to make sure I understand it. 23. If I become confused in trying to solve a mathematics problem, I try to clear up my 4 confusion. 24. When I am working on a mathematics problem, I think about whether or not I am 4 working on what the problem is really asking. 25. Once I find a solution to a mathematics problem, I try to find some other, different 4 solutions to the same problem. 26. I know how I performed on my mathematics test before I receive any feedback from 4 my teacher.

442

Question from Questionnaire Part A Score 27. I believe that mathematics topics vary in their level of difficulty. 5 28. When I cannot find the solution to a mathematics problem, I know the nature of the 4 difficulty I am experiencing. 29. I believe that mathematical problem solving techniques vary in their level of 4 difficulty. 30. If I encounter difficulty in solving a mathematics problem, I am not embarrassed to 3 ask my teacher for help. 31. When I am browsing in a bookstore or library, I flip through the pages of 3 mathematics books. 32. I enjoy thinking of new mathematics problems to work on. 3 33. I enjoy preparing my own mathematics tests, quizzes and puzzles. 1 34. I enjoy finding mathematics problems which have never been solved. 1 35. When I am with my friends, I discuss new mathematics concepts which I have 2 recently learned.

Notes: i.) Student Q40’s responses to Part A of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part A Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 121/150 = 80.67% iv.) Part A Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 10/25 = 40.00%

443

Table Q40C

Question from Questionnaire Part C Score 1. After I worked on the number theory problems, I knew how well I understood the 4 concepts. 2. My effort and persistence determined how well I was able to do on the number 5 theory problems. 3. When I worked on the number theory problems, I attempted to use techniques that 5 have been used by successful mathematics students. 4. If I already know something about the concepts referenced in the number theory 3 problems, I can learn even more about them. 5. If I have a real interest in the number theory problems and the concepts to which they 3 relate, I can learn more about it. 6. When I wanted to improve my understanding of the concepts referenced by the 5 number theory problems, I drew sketches, pictures or diagrams relating to the concepts. 7. Before I started trying to solve one or more of the number theory problems, I decided 5 on clear goals that I needed to reach as part of the problem solving process. 8. As I was working on the number theory problems, I evaluated my progress. 3 9. After completing my work on the number theory problems, I asked myself if I 5 learned anything new and valuable. 10. As soon as I solved one or more of the number theory problems, I wondered if I 3 could have found easier or quicker ways to arrive at the solution(s). 11. After I completed my work on the number theory problems, I thought about the 3 most important elements of the problems to make sure I really understood them. 12. I tried different ways of approaching the number theory problems, depending on the 3 type of concepts involved. 13. Whenever I was confused by the number theory problems, I asked someone else for 2 help. 14. In order to help myself understand the concepts relating to the number theory 4 problems, I thought of my own examples or illustrations of the concepts. 15. I have my own ways of remembering what I have learned from the number theory 5 problems. 16. After I finished reading the number theory problems, I knew if I was able to solve 4 them. 17. In attempting to solve the number theory problems, I focused on whatever data had 5 been presented. 18. If I wrote the data presented by the number theory problems, I had a better 5 understanding of the problems. 19. In attempting to solve the number theory problems, I tried to use solutions of similar 4 problems which I had encountered in the past. 20. In attempting to solve the number theory problems, I tried to identify the parts of the 4 problems which I did not understand. 21. When I was trying to solve the number theory problems, I asked myself questions to 4 help focus my attention on the problems and my attempts to solve them. 22. If I had difficulty with the number theory problems, I read the problems several 5 times. 23. If I became confused in trying to solve the number theory problems, I tried to clear 5 up my confusion. 24. When I was working on the number theory problems, I thought about whether or not 4 I was working on the main questions presented by the problems.

444

Question from Questionnaire Part C Score 25. Once I found a solution to the number theory problems, I tried to find some other, 4 different solutions to the same problems. 26. I knew how I performed on the number theory problems as soon as I finished my 3 work on the problems. 27. I believe that the concepts referenced by the number theory problems vary in their 4 level of difficulty. 28. When I could not find solutions to the number theory problems, I knew the nature of 4 the difficulty I was experiencing. 29. I believe that the number theory problems require problem solving techniques which 5 vary in their level of difficulty. 30. If I encountered difficulty in solving the number theory problems, I was not 3 embarrassed to ask my teacher for help. 31. Having completed my work on the number theory problems, I flip through the pages 2 of mathematics books while I am browsing in a bookstore or library. 32. Having completed my work on the number theory problems, I enjoy thinking of new 4 mathematics problems to work on. 33. Having completed my work on the number theory problems, I enjoy preparing my 3 own mathematics tests, quizzes and puzzles. 34. Having completed my work on the number theory problems, I enjoy finding 2 mathematics problems which have never been solved. 35. When I am with my friends, I discuss new mathematics concepts which I have 2 learned from working on the number theory problems.

Notes: i.) Student Q40’s responses to Part C of questionnaire ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1 iii.) Part C Metacognition Score = (Σ questions 1 to 30 scores)/(5x30) = 121/150 = 80.67% iv.) Part C Mathematics Attitude/Curiosity Score = (Σ questions 31 to 35 scores)/(5x5) = 13/25 = 52.00%

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Table Q40A/C

Question from Part A Question from Part C Questionnaire Part A Response Questionnaire Part C Response 1. After I have studied a mathematics 3 1. After I worked on the number theory 4 lesson, I know how well I understand it. problems, I knew how well I understood the concepts. 2. My effort and persistence determine how 3 2. My effort and persistence determined 5 well I perform on a mathematics test. how well I was able to do on the number theory problems. 3. When I am studying mathematics, I 4 3. When I worked on the number theory 5 attempt to use study methods that have been problems, I attempted to use techniques that used by successful mathematics students. have been used by successful mathematics students. 7. Before I start trying to learn a new 3 7. Before I started trying to solve one or 5 mathematics concept, I decide on clear goals more of the number theory problems, I that I must reach as part of the learning decided on clear goals that I needed to reach process. as part of the problem solving process. 9. After completing my mathematics 4 9. After completing my work on the number 5 homework, I ask myself if I have learned theory problems, I asked myself if I learned anything new and valuable. anything new and valuable. 21. When I am trying to solve a mathematics 3 21. When I was trying to solve the number 4 problem, I ask myself questions to help theory problems, I asked myself questions focus my attention on the problem and my to help focus my attention on the problems attempts to solve it. and my attempts to solve them.

23. If I become confused in trying to solve a 4 23. If I became confused in trying to solve 5 mathematics problem, I try to clear up my the number theory problems, I tried to clear confusion. up my confusion. 29. I believe that mathematical problem 4 29. I believe that the number theory 5 solving techniques vary in their level of problems require problem solving difficulty. techniques which vary in their level of difficulty. 32. I enjoy thinking of new mathematics 3 32. Having completed my work on the 4 problems to work on. number theory problems, I enjoy thinking of new mathematics problems to work on. 33. I enjoy preparing my own mathematics 1 33. Having completed my work on the 3 tests, quizzes and puzzles. number theory problems, I enjoy preparing my own mathematics tests, quizzes and puzzles. 34. I enjoy finding mathematics problems 1 34. Having completed my work on the 2 which have never been solved. number theory problems, I enjoy finding mathematics problems which have never been solved.

Notes: i.) Corresponding questions from Parts A and C of questionnaire, for which Student Q40’s responses evidenced an increase in scores. ii.) A = Always = 5; F = Frequently = 4; S = Sometimes = 3; R = Rarely = 2; N = Never = 1

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Appendix TT

Preliminary Trial Summary

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Preliminary Trial Summary Preliminary Trial Student Preliminary Number of Percentage of Category Trial Students Preliminary Trial Preliminary Trial in Category Students in Student Sample Space Category Questionnaire Part C Metacognition Score - 0 0/3 = 0.00% exceeded Questionnaire Part A Metacognition Score. Questionnaire Part C Mathematics Q3 1 1/3 = 33.33% Attitude/Curiosity Score exceeded Questionnaire Part A Mathematics Attitude/Curiosity Score. Responses to specific corresponding Q1; Q2 2 2/3 = 66.67% questions on Questionnaire Part A and Part C revealed some positive changes in metacognitive functioning. Responses to specific corresponding Q3 1 1/3 = 33.33% questions on Questionnaire Part A and Part C revealed some positive changes in mathematical curiosity and general attitudes towards mathematics.

Interview responses revealed some Q3 1 1/3 = 33.33% positive changes in metacognitive functioning.

Interview responses revealed some Q1; Q2; Q3 3 3/3 = 100.00% positive changes in mathematical curiosity and general attitudes towards mathematics.

Interview responses and questionnaire Q1; Q2 2 2/3 = 66.67% responses were not consistent relative to changes in metacognitive functioning.

Interview responses and questionnaire - 0 0/3 = 0.00% responses were not consistent relative to changes in mathematical curiosity and/or general attitudes towards mathematics.

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Appendix UU

Main Study Summary

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Main Study Summary

Main Study Student Main Study Students in Number of Percentage of Percentage Category Category Main Main of Study Study Student Interviewed Students Sample Main Study in Space Students Category

Questionnaire Part C Q6;Q7;Q8;Q12;Q13;Q17;Q20;Q21; 15 15/37 = 40.54% N/A Metacognition Score Q22;Q23;Q24;Q25;Q28;Q31;Q36 exceeded Questionnaire Part A Metacognition Score. Questionnaire Part C Q4;Q5;Q6;Q11;Q14;Q15; 21 21/37 = 56.76% N/A Mathematics Q16;Q20;Q21;Q22;Q23;Q26;Q28; Attitude/Curiosity Score Q29;Q31;Q34;Q35;Q36;Q37;Q39;Q40 exceeded Questionnaire Part A Mathematics Attitude/Curiosity Score. Responses to specific Q4;Q5;Q6;Q7;Q8;Q9; 36 36/37 = 97.30% N/A corresponding questions on Q10;Q11;Q12;Q13;Q14; Questionnaire Part A and Part Q15;Q16;Q17;Q18;Q19; C revealed some positive Q20;Q21;Q22;Q23;Q24;Q25;Q26; changes in metacognitive Q27;Q28;Q29;Q30;Q31;Q32;Q33;Q34; functioning. Q35;Q36;Q37;Q39;Q40 Responses to specific Q4;Q5;Q6;Q8;Q9;Q10; 29 29/37 = 78.38% N/A corresponding questions on Q11;Q12;Q13;Q14;Q15; Questionnaire Part A and Part Q16;Q17;Q19;Q20;Q21;Q22; C revealed some positive Q23;Q25;Q26;Q28;Q29;Q31;Q34; changes in mathematical Q35;Q36;Q37;Q39;Q40 curiosity and general attitudes towards mathematics. Interview responses revealed Q5;Q6;Q7;Q8;Q9;Q10; 9 N/A 9/10 = 90.00% some positive changes in Q11;Q12;Q13 metacognitive functioning. Interview responses revealed Q4;Q5;Q6;Q7;Q8;Q9; 10 N/A 10/10 =100.00% some positive changes in Q10;Q11;Q12;Q13 mathematical curiosity and general attitudes towards mathematics. Interview responses and Q4;Q7;Q8;Q9;Q10;Q11 6 N/A 6/10 = 60.00% questionnaire responses were not consistent relative to changes in metacognitive functioning. Interview responses and Q7;Q8;Q9;Q10;Q12;Q13 6 N/A 6/10 = 60.00% questionnaire responses were not consistent relative to changes in mathematical curiosity and/or general attitudes towards mathematics.

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Appendix VV

Questionnaire Part A and Part C Scores

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Questionnaire Part A and Part C Scores

Student Part A Part A Part C Part C Metacognition Mathematics Metacognition Mathematics Score Attitude/Curiosity Score Attitude/Curiosity Score Score Q1 74.67% 56.00% 72.00% 36.00% Q2 68.00% 20.00% 67.33% 20.00% Q3 70.00% 28.00% 66.00% 56.00% Q4 66.67% 40.00% 62.00% 72.00% Q5 68.00% 48.00% 59.33% 60.00% Q6 72.00% 24.00% 74.67% 44.00% Q7 62.00% 32.00% 66.67% 20.00% Q8 67.33% 36.00% 70.67% 36.00% Q9 72.00% 60.00% 64.00% 56.00% Q10 75.33% 44.00% 70.00% 40.00% Q11 73.33% 36.00% 71.33% 56.00% Q12 70.67% 32.00% 72.67% 20.00% Q13 76.67% 56.00% 80.00% 52.00% Q14 70.67% 48.00% 58.67% 68.00% Q15 74.67% 64.00% 58.67% 84.00% Q16 78.00% 48.00% 74.00% 72.00% Q17 78.67% 60.00% 80.67% 60.00% Q18 91.33% 64.00% 82.00% 60.00% Q19 70.00% 48.00% 54.00% 48.00% Q20 69.33% 36.00% 84.00% 44.00% Q21 79.33% 28.00% 81.33% 32.00% Q22 66.67% 40.00% 71.33% 64.00% Q23 90.00% 92.00% 100.00% 100.00% Q24 71.33% 48.00% 73.33% 48.00% Q25 72.67% 32.00% 81.33% 32.00% Q26 70.00% 36.00% 48.00% 44.00% Q27 68.00% 48.00% 64.00% 40.00% Q28 77.33% 32.00% 81.33% 36.00% Q29 80.67% 40.00% 79.33% 80.00% Q30 77.33% 76.00% 58.67% 48.00% Q31 82.67% 48.00% 88.00% 60.00% Q32 80.00% 52.00% 78.67% 48.00% Q33 82.00% 80.00% 82.00% 64.00% Q34 70.00% 36.00% 64.00% 40.00% Q35 77.33% 32.00% 74.67% 64.00% Q36 80.00% 52.00% 86.00% 84.00% Q37 82.67% 32.00% 80.67% 44.00% Q38 84.00% 56.00% 84.00% 56.00% Q39 76.67% 44.00% 66.67% 64.00% Q40 80.67% 40.00% 80.67% 52.00%

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Appendix WW

Questionnaire Part B Credits

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Questionnaire Part B Credits

Student Credits for Part B Problems Correctly Solved (* => student rewrote one of the proof requirements) Q1 4 ½ Q2 3 Q3 1/3 Q4 1 1/2 Q5 3 Q6 3 1/2 Q7 2 1/2 Q8 4 Q9 1 1/2 Q10 3 Q11 2 Q12 2 Q13 2 Q14 2 Q15 5 ½ * Q16 7 * Q17 6 Q18 6 * Q19 4 Q20 6 * Q21 4 1/2 Q22 5 ½ * Q23 6 Q24 4 1/2 Q25 4 1/2 Q26 4 1/2 Q27 6 * Q28 6 Q29 3 Q30 5 1/2 Q31 4 1/2 Q32 7 * Q33 2 Q34 3 Q35 1 Q36 2 1/2 Q37 2 1/2 Q38 7 * Q39 2 Q40 3

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