Technology-Integrated Mathematics Education (Time)
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UNIVERSITY OF HAWAI'I LIBRARY
TECHNOLOGY-INTEGRATED MATHEMATICS EDUCATION (TIME)
A STUDY OF INTERACTIONS BETWEEN TEACHERS AND STUDENTS IN
TECHNOLOGY-INTEGRATED SECONDARY MATHEMATICS CLASSROOMS
A DISSERTATION SUBMITTED TO THE GRADUATE DIVISION OF THE UNIVERSITY OF HAW AI'I IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PIDLOSOPHY
IN
EDUCATION
AUGUST 2008
By Tae Young Ha
Dissertation Committee:
Neil A. Pateman, Chairperson Morris K. Lai Catherine P. Fulford Curtis P. Ho Joseph T. Zilliox We certify that we have read this dissertation and that, in our opinion, it is satisfactory in scope and quality as a dissertation for the degree of Doctor of Philosophy in Education.
DISSERTATION COMMITTEE
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© Copyright by Tae Young Ha 2008 iv ACKNOWLEDGEMENT
I believe there are angels watching over and supporting me as I have encountered countless challenges in my life. I also have been blessed with many wonderful people who encouraged, strengthened, enlightened, and understood me as I went through the entire dissertation process. I wish to express my appreciation by recognizing and acknowledging them.
As one of well-known Confucian teachings in Korea, ~m:x:-e (read as
Goon Sa Boo II Chae in Korean) teaches, I have equivalent respect for all of my teachers in my life and my parents. As the teaching explains, I believe that the collective efforts of the king, the teacher, and the parents would complete my education. Without my teachers, I would not be where I am today.
As I work on this study, I have been honored and blessed to meet and work with five inspiring teachers. They watched over me and guided me whenever I needed direction. I would like to thank Dr. Neil A. Pateman for being my advisor with inuneasurable patience, insights, and inspiration. I would also like to thank Dr. Joseph T.
Zilliox for his insightful guidance and his constant warm encouragements. Both of them guided me to broader visions in mathematics education.
I would like to thank other members of my committee: Dr. Curtis P. Ho for helping me with educational technology part of my study; Dr. Catherine P. Fulford for guiding and reminding me to balance my study between the ideas of mathematics education and educational technology; Dr. Morris K. Lai for his knowledge of academic papers and encouragements. v I would also like to thank people at a public high school in Hawai'i who made this study possible: Mr. Mike Long and Ms. Rebecca Lowe for graciously giving me permission to come into their classes for five months to observe and participating with the interviews; Students in one section of AP calculus and two sections of AP statistics for letting me videotape them while they studied and participating with the interviews.
I am very grateful for my loving parents for being there for me whenever I needed their comfort hands and a source of encouragement. They worked hard and inspired me to work hard to achieve my goals. They are my guardian angles. I remember the countless prayers they offered for me as they raised me and as I go through this study.
Without their kind support and constant prayers, it would be impossible for me to make it through.
I would like to show my appreciation to my two precious daughters, Eura and
&Ira, for being my cheerleaders, unconditional supporters and their understanding when I was not able to be there for them when they needed me. They are my angels and the source of my inspiration, encouragement, and energy to go on. I am so grateful for having them in my life.
Finally, I would like to express my belated gratitude to my loving wife, Eun-Ab, whose immeasurable love, encouragement, and confidence in me helped my journey continue and sustained me until the moment I completed my doctoral study. I sincerely hope she understands that the tears she shed, the stresses and pains she bore, and the innumerable prayers she offered had special meaning and without them this study would not have been completed. vi ABSTRACT
The results of numerous national and international assessments have raised concerns regarding secondary mathematics education in the United States. According to government reports, there has been a significant increase in the use oftechnology in U.S. schools in the last decade. However, student achievement in mathematics has not improved during this time.
A qualitative case study involved observing and interviewing two high school mathematics teachers and 48 high school students in three Advanced Placement (AP) mathematics classes. The study focused on what students and teachers thought about the integration of technology in mathematics education, on how they actually used technology in class, on whether technology helped students to learn cooperatively, and on whether technology helped teachers improve their instruction.
Collective results from questionnaire data, interview data, and class observations helped to build an understanding about how technology was used in the three secondary mathematics classrooms. Before classes were observed, all students completed questionnaires and the first teacher interviews were conducted. Observations of classes were followed by a second round of teacher interviews and student interviews.
The students and teachers extensively used graphing calculators and strongly believed that technology helped them learn and teach mathematics by helping them to visualize the abstract concepts of mathematics. The use of technology prompted students to interact more with each other. Also discussed by the teachers and students were additional reasons for their belief that technology integration positively influenced their vii learning and teaching, how students used technology to learn, and how teachers technology to teach. viii TABLE OF CONTENTS
ACKNOWLEDGEMENT ...... iii
ABSTRACT ...... vi
TABLE OF CONTENTS ...... viii
LIST OF TABLES ...... xi
LIST OF FIGURES ...... xii
CHAPTER 1. INTRODUCTION ...... 1
Focus of Study ...... 1
International and National Studies ofMathematica1 Achievement ...... 3
The Program for International Student Assessment (PISA) ...... 4
Trends in International Mathematics and Science Study (TIMSS) ...... 5
The National Assessment of Educational Progress (NAEP) ...... 7
Hawai'i State Assessment (HSA) ...... 8
Conclusion ...... 10
Research Questions ...... II
CHAPTER 2. LITERATURE REVIEW ...... 14
The Roles of Technology in Mathematics Education ...... 14
Student Interaction with Technology ...... 16
Teacher Interaction with Technology ...... 19
CHAPTER 3. METHODOLOGy...... 24
Participants '" ...... 24
Data Collection ...... 27 ix Data Analysis ...... 33
CHAPTER 4. DATA ANALYSIS AND RESULTS ...... 43
Questionnaire ...... 44
First Teacher Interviews ...... 56
Class Observations ...... 70
Second Teacher Interviews ...... 93
Student Small Group Interviews ...... 97
CHAPTER 5. DISCUSSION ...... 128
Analysis Overview ...... 128
Case Studies ...... 131
The Comparison of the Three Classes ...... 132
The Comparison of the Two Teachers ...... 141
Answers to Research Questions ...... 157
Limitations of the Study ...... 173
Conclusions and Recommendations ...... 174
Recommendations for Teacher Education ...... 176
Recommendations for Professional Development of Experienced Teachers ...... 179
Recommendations for Technology Integrated Mathematics Classrooms ...... 182
APPENDICES ...... 185
Appendix A: Questionnaire ...... 185
Appendix B: Questionnaire Results ...... 190
Appendix C: Student Responses to Item #4 ...... 195 x Appendix D: Student Responses to Item #5 ...... 196
Appendix E: Student Responses to Item #7 ...... 197
Appendix F: Student Responses to Item #78 ...... 198
Appendix G: Student Responses to Item #79 ...... 200
Appendix H: Student Responses to Item #80 ...... 202
Appendix I: Student Responses to Item #81 ...... 204
Appendix J: Student Responses to Item #82...... 205
Appendix K: Student Responses to Item #83 ...... 206
Appendix L: Student Responses to Item #84 ...... 207
Appendix M: Student Responses to Item #85 ...... 209
Appendix N: Student Responses to Item #86 ...... 211
Appendix 0: Student Responses to Item #87 ...... 212
Appendix P: Teacher Interview Questions ...... 214
Appendix Q: Student Interview Questions ...... 215
Appendix R: Human Subjects Approval Letter ...... 216
Appendix S: IRB CertificationlDeclaration of Exemption ...... 216
Appendix T: Parental Consent Form ...... 218
Appendix U: Teacher Consent Form ...... 219
REFERENCES ...... 220 xi LIST OF TABLES
1. Avemge mathematics litemcy score comparison of 15-year-old students ...... 5
2. Comparison of comprehensive mathematics achievement by nation
and grade level: TIMSS 1995, 1999, and 2003 ...... 6
3. WHS Hawai'I State Assessment Mathematics Proficiency Levels
for 10th gmders: Years, 2002 - 2007 ...... 9
4. Comparison of the mix of ethnicities in WHS AP Classes to that of National and
Hawai'i State AP Classes, SY 2004-2005 ...... 25
5. Primary data analysis schedule ...... 35
6. Profiles of the student responses to focus I ...... 47
7. Profiles of the student responses to focus 2 ...... 48
8. Profiles of the student responses to focus 3 (part 1) ...... 50
9. Profiles of the student responses to focus 3 (Part 2) ...... 53
10. Profiles of the student responses to focus 3 (Part 3) ...... 54 xii LIST OF FIGURES
Figure
1. Figure 1. Trends in average mathematics scale scores for 8th -graders: Various years, 1990 - 2007* (Hawaii vs. US) ...... 7
2. Figure 2. AP Calculus Classroom ...... 71
3. Figure 3. AP Statistics Classroom ...... '" ...... 72 1 CHAPTER 1. INTRODUCTION
Focus of Study
This study focused on how teachers use technology for teaching mathematics and on the role of technology in mathematics classrooms. In various fields, technology is integrated to improve the quality of communication. From my teaching experiences, I observed that students were better able to understand concepts when technology was incorporated into mathematics instruction. This observation motivated my decision to investigate the use of technology tools in the teaching of mathematics.
There has been a deep concern regarding U. S. secondary students' low achievement in mathematics (The Program for International Student Assessment [PISA],
2000; Trends in International Mathematics and Science Study [TIMSS], 1995; The
National Assessment of Educational Progress [NAEP], 1969). Although the evidence is ambiguous because the results of the tests can be interpreted in more than one way depending how the evaluator interprets the results (Berliner & Biddle, 1995; Bracey,
2000, 2004), overall, the results indicate that students are not doing well. This situation has persisted for many years, at least since the Nation at Risk (U.S. Department of
Education, 1983) report was released.
Because the public and government evaluate the education system by measurement of the achievement of students on state, national, and international mathematics assessments, improved student scores became the key indicator of the effectiveness of the mathematics curriculum. The public and government look to teachers to be responsible for improving student achievement, and that is why the No Child Left 2 Behind (NeLB) Act was signed into law in an attempt to legislate better teaching. Note, though, that there are other factors that influence student mathematics achievement.
Some of these factors are the physical environment, the emotional and psychological state of the learner, and parental and social involvement.
After the results of U.S. student performance on international assessments were disclosed, there were mixed reactions from many different groups of people. The U.S. government published a series of reports on the achievement levels of U.S. students.
However, according to Pursuing Excellence (1997), a U.S. Department of Education report, the result of the 1995 TIMSS has no significant meaning until subsequent research is conducted to follow up on the findings.
There are no educational characteristics that are present in every high-performing TIMSS country ... Instead, we need to use these findings as an objective assessment of the strengths and weaknesses characteristic of each specific national education system. All countries, including the U.S., have something to learn from other nations, and have something from which other countries can learn. (p. 57)
Even though it is difficult to make direct connections between the quality of instruction and student learning of mathematics, there will always be room for teachers to improve their instruction. Because improved instruction has a positive effect on achievement, I hope to link the effective use of technology with improved instruction.
A government report (Wells & Lewis, 2006) shows technology capacity has grown tremendously in U.S. public schools and classrooms over the last 11 years.
According to Wells and Lewis (2006), public schools have made consistent progress in expanding Internet access in instructional rooms: "In 2005,94 percent of public school 3 instructional rooms in the United States had access to the Internet, compared with
3 percent in 1994" (p. 4).
Even though more technology has become available in schools and classrooms, the results of national and international assessments of mathematics indicate the achievement of U.S. students has not improved. Thus, it is difficult to conclude that technology availability itself has made any impact on student achievement in mathematics. It is therefore important to investigate the role oftechnology in mathematics education to understand whether and how technology is actually used to learn mathematics. Ifwe want to use technology to improve mathematical achievement, students should be able to access technology for study. Do students have access to technology? Do teachers use technology effectively? According to my literature review, the effort is not students' alone: teachers must also know how to integrate technology to improve their teaching and effectively promote student learning. Is there any clear instruction for effective technology integration in the classroom for teachers?
In the rest of the introduction, I am planning to discuss further the results of state, national, and international assessments, and the interpretations of these results. I will also elaborate on the purpose of the study and close with a statement of the research question.
International and National Studies of Mathematical Achievement 4 U.S. government officials, educators, and parents collectively want their students educationally prepared so they can perform competitively with students in other parts of the world. How do U.S. students perform on international and national tests?
The Program/or International Student Assessment (PISA)
The Program for International Student Assessment (PISA) is a system of international assessments that, every 3 years, measures IS-year-olds' capabilities in reading literacy, mathematics literacy, and science literacy. PISA-first implemented in
2000-is carried out by the Organization for Economic Cooperation and Development
(OECD), an intergovernmental organization of 30 industrialized countries (together with candidate countries who engage in OECD activities). According to the OECDlUnited
Nations Educational Scientific and Cultural Organization (UNESCO) (2003), PISA focuses on how well students apply knowledge and skills to tasks that are relevant to their future life, rather than on the memorization of subject matter knowledge (p. 3). In other words, students' high achievement on PISA would indicate that they know how to apply their knowledge to real-life situations. Rather than seeking to indicate how well students can simply memorize a number of facts, PISA is testing conceptual knowledge.
According to several research reports (Lemke & Gonzales, 2006; Lemke et al.,
2004; Organization for Economic Cooperation and Development I United Nations
Educational Scientific and Cultural Organization, 2003; U.S. Department of Education,
2001b), U.S. students did not score well on PISA in comparison with students from
Korea, Australia, the United Kingdom, Japan, and Canada. 5 Table 1 Average mathematics literacy score comparison of15-year-old students
PISA2000 PISA2003
United States 493 482.9
Korea 547 542.2
Australia 533 524.3
United Kingdom 529 508.3
Canada 533 532.5
Japan 557 534.1
OECD Average 500 500
Considering what PISA measures, what multiple reports point out is that U.S. results are not positive and reconceptuaIization of mathematics education in the U.S. needs to be seriously considered (Smith. 2004).
Trends in International Mathematics and Science Study (I'IMSS)
TIMSS was developed to continue the series of international comparative school achievement studies conducted by the International Association for the Evaluation of
Educational Achievement (lEA) and which began in 1959. The First International
Mathematics Studies (FIMS) were implemented in 1964. The Second International
Mathematics Studies (SIMS) were implemented between 1980 and 1982. 6 However, TIMSS is the first attempt to investigate mathematics and science achievement simultaneously. It measures trends in students' mathematics and science achievement. Offered in 1995, 1999, and 2003, TIMSS provides participating countries with an unprecedented opportunity to measure students' progress in mathematics and science achievement on a regular 4-year cycle. Various resources (U.S. Department of
Education, 1998b, 2001 a, 2001 c) indicate that the achievement of U.S. students is not
Table 2 Comparison ofcomprehensive mathematics achievement by nation and grade level: T1MSS 1995.1999. and 2003
TIMSS 1995 TIMSS 1999 TIMSS 2003
Grade Level 4th 8th 12th 4th 8th 12th 4th 8th 12th Country
United States 545 492 461 502 518 504
Korea 611 581 587 589
Australia 546 519 522 525 499 505
United Kingdom 513 498 496 531
Canada 532 521 519 531
Japan 597 581 579 565 570
International 529 519 500 521 495 466 Average
high compared to other participating nations. It should be noted that TIMSS scores for 4th and 12th graders in 1999 are not available because they were not measured. For the same 7 reason, the TIMSS scores for 12th graders in 2003 are not available. Some scores are missing in 1995 and 2003 from Korea, the United Kingdom, Canada, and Japan because in those years they did not participate in the assessment for various grade levels.
The National Assessment ofEducational Progress (NAEP)
NAEP, or "the Nation's Report Card," is the only nationally representative and current assessment of what America's students know and can do in various subject areas.
Since 1969, assessments have been conducted periodically in reading, mathematics, science, writing, history, geography, and other fields.
Now, let us take a look at how our students performed on the NAEP assessment.
Because this is one of the major assessments that the U.S. government administers, there are numerous government reports (National Center for Education Statistics, 2006; U.S.
Department of Education, 1995a, 1995b, 2004) that are available for accountability purposes.
When the mathematics achievements of eighth graders in Hawai'i were compared to the nation average of eighth graders in the United States, there was a clear and very consistent gap between the two scores (See figure 1). Based on the seven NAEP scores resulting from assessment in seven different years, the eighth graders in Hawai'i consistently achieved below the national average.
Figure 1. Trends in average mathematics scale scores for Sth-graders: Various years, 1990 - 2007'" (Hawai'i vs. U.S.) 8
r=;::~;;;:=~~r --- +----''-----~.._"'''=-==------___l ,-+-U.S. '-4-Hawaii
Because the mathematical achievements of average U.S. secondary students on the international assessments were already discouraging, the NAEP scores of Hawai'i secondary students strongly demanded a need for meticulous evaluation of mathematics education in the state. The mathematical achievements ofHawai'i and U.S. secondary students, as indicated in the results of the NAEP test shown in Figure I, were basically very discouraging
Hawai'i State Assessment (HSA)
The Hawai'i State Assessment (HSA) is an assessment initiated by the State of
Hawai'i Department of Education (HIDOE) to meet the U.S. government requirement for compliance with NCLB (U.S. Department of Education, 2002). The assessment is 9 carefully aligned with what is tested on the NAEP since the assessment is mandated by the U.S. government as is described on the HIDDE Webpage (Hawaii Department of
Education, 2006)
The Hawai'i State Assessment results shown below contain the scores of all children tested in the state. Although the data are used as part of the process for determining Adequate Yearly Progress (A YP) for a school, there is a key distinction between the results reported on this page and those reported on the NCLB page. For NCLB reports, results are provided for test participation, percent proficient for subgroups, and graduation/retention. For AyP participation rates, all children tested at a school are included in the calculations. For AyP proficiency, the data are "filtered" for students who have been at their school for a full academic year. This means that only the scores for tested students who have been at their school for a full academic year are used in the AyP proficiency analyses for a school. Thus, please remember that the results reported via the links below are for all children tested in the state. (p. 1)
The achievement of Hawai'i public school 10th graders on HSA for the last six years was tracked and compared with the achievement of West High School (WHS) 10th graders in Table 3.
Table 3 WHS Hawai 'i State Assessment Mathematics Proficiency Levels for 1rI' graders: Years. 2002-2007
Mathematics Proficiency Level (Meets + Exceeds)
Year State (Grade 10) WHS (Grade 10)
2002 19% 13%
2003 18% 9%
2004 21% 18%
2005 20% 21%
2006 18% 14% 10 2007 29% 26%
The achievement of the research school (WHS) students was generally below state average scores except in the year 2005. In 2003, the school achievement was only half of the state average.
Given the fact that the achievement of Hawai'i state students on the NAEP test was not very hopeful, mathematics education at WHS appeared to be dreadful to me when I consider how WHS students achieved on the HSA. As one of the school's mathematics teachers, I believed that the school's mathematics education needed serious structured help to improve the situation.
Conclusion
After considering the achievement of U.S. students on the national and international assessments, one can clearly understand Smith's (2004) belief that mathematics education needed to be changed and possibly reconceptuaIized.
Mathematics education needs reconceptualizing because students know very little mathematics by the time they graduate from high school. Mathematics has become a subject to be feared and dreaded for centuries. [ ... ] Regardless of who is to blame, most students entering high school are not prepared to problem solve nor are they interested in mathematics except as the dreaded requirement needed to graduate. (p. viii)
If the achievements of U.S. students on these assessments are not comparable with their counterparts in other parts of the world, it is important to reevaluate and improve our education system. 11 Research Questions
1. Do teachers and students understand the role of educational technology in
learning mathematics?
2. How do teachers and students use technology in mathematics classes?
3. How does educational technology integration help students learn mathematics
cooperatively?
4. How does educational technology influence mathematics instruction?
The first research question was posed because I have observed numerous situations involving the use of technology in classrooms similar to the situation that
Smith (2004) pointed out in her dissertation.
Naturally, technology has influenced mathematics education tremendously in the last decade. Unfortunately, many mathematics educators use technology as a crutch instead of using it to enhance mathematics education. (p. ix)
Many teachers usually think that they understand how to integrate technology into their instruction. However, I have observed that when some teachers used technology, it was obvious that they did not know how to use it effectively. Some of them were struggling with the idea of integrating technology for instructional purposes. Many teachers used technology mainly for presentation and communication purposes. That is different from using technology for instructional purposes.
That is why I frequently ask myself, "Do we (teachers) really understand the role of technology in mathematics learning?" As I met with in-service and pre-service teachers in an educational technology course I taught for the last few summers, I noticed that the majority of them were not sure how to implement technology tools properly for 12 their curriculum. A significant amount of the class time was devoted to the discussion of the role of educational technology in teaching.
The second research question was posed because I am interested in what types of experiences students and teachers have when they study mathematics with technology.
Even though many researchers (Hughes, Packard, & Pearson, 1999; Tiene & Luft, 2002;
Wilson, 2005) have shown what type of technology tools have been used and what type of instruction has been implemented in what content area, it often is not clear what the participating students and teachers thought about (and what their experiences were) using technology tools in their learning and teaching.
The third research question was posed to examine the relationship between technology integration and student interaction with peers and teacher. This question is posed because I have found from my literature review that learning mathematics cooperatively is a highly recommended way oflearning mathematics. I would like to understand whether students learn more cooperatively when they learn mathematics with technology. Why or why not? More on this idea will be discussed in Chapter 2 since I would like to know what researchers say about technology integration and cooperative learning. Some secondary questions that I posed were as follows:
a. Does technology integration in mathematics learning encourage students to
externalize their internal knowledge and share it with other students and
teachers? Why or why not? 13 b. Does technology-integrated mathematics instruction generate more
meaningful communication among students and between student and teacher
in class?
The fourth research question was posed to seek out the advantages or disadvantages of teachers using technology for instructional purposes. I believe this question is important because teachers will not develop technology-integrated instruction unless it provides some benefits. If technology can assist students in concept development, teachers will have a strong motivation to integrate technology. Associated questions that might be asked are:
a. Does educational technology help teachers individualize their instruction for
students' learning?
b. Does educational technology enable teachers to present mathematical
concepts effectively? 14 CHAPTER 2. LITERATURE REVIEW
My topic of research is a combination of mathematics education and educational technology. I am looking for the integration of the two fields. This chapter is a review of three related areas:
• The roles of technology in mathematics education,
• Student interaction with technology, and
• Teacher interaction with technology.
The Roles of Technology in Mathematics Education
A great deal of research was conducted during the 1970s, 80s, and early 90s on the effects of computer usage on student achievement, attitude, learning rate, and other variables. There is a wide range of topics that we need to cover when we talk about computers in education, from computerized learning activities which supplement conventional instruction, to computer programming and other applications (such as for record-keeping, the development of databases, and word processing).
Regardless of the types of computer application, the demand for computers has increased in numerous fields. The usage of computers in the education field has significantly increased during the last few decades. Cotton (1991) noticed the increasing usage of computers in education. The use of microcomputers expanded rapidly during the
1980s. During that time, American schools acquired over two million microcomputers.
The percentage of schools owning computers increased from approximately 25% to 15 virtually 100%, and more than half of the states began requiring or at least recommending that all prospective teachers have a reasonable background in technology education.
According to Wilson (2005), the roles of students and teachers can be different when computer technology is used in the curriculum. Teachers do not have to teach everything as before, in a traditional teaching situation, because the computer can lead students to practice and help them think the process through gradually. Teachers can even be a part of the learning team with students as new ideas are being tried on the computer.
Students also have different roles in the classroom. They can be self-directed learners without depending on a teacher's guidance to learn new ideas or obtain information.
One of the main ideas about success in teaching with computers concerns successful interaction between students through computer mediation. Moore (1987) explained that the computer could be a very useful tool in education because it can contribute to education through facilitating active learning. Computer technology in instruction can contribute to education through the presentation of information in a variety of sensory modes as well as giving students sustained attention and consistent feedback. Examples for effective computer usage in instruction are the interactive tablet computer in science class (Baptist, 2005; Hughes et al., 1999; Naatanen, 2005; Olivier,
2005; Paper!, 1993), video storybook with interactive screen (Hughes et al., 1999) and the light-wall study (Dugger & Johnson, 1994; Kikin-Gil, 2006). 16 Student Interaction with Technology
Many educators have given us several good reasons for using the computer in mathematics education (Olivier, 2005; Savage, Sanchez, O'Donnell, & Tangney, 2003;
Tall, 2002; Wilson, 2005). One of these reasons is that with the use of computer technologies and software, the possibility of meeting a primary goal of mathematics education is in sight.
For example, Schwartz (1993) says that most people expect students to be creative in most academic subjects, but when it comes to mathematics, students are expected to follow rules, which does not promote creativity. Schwartz believes that students have the right to be creative in mathematics, just as in other subject areas, and the essence of this creativity lies in the making and exploring of mathematical conjectures through problem-posing as well as problem-solving. Schwartz says that appropriately designed software environments can help students understand how to pose problems, which leads to the making and the exploring of conjectures and the opportunity for students to think inductively, solve problems, and generalize-all of which lie at the very heart of mathematics education.
Hussain, Lindh, and Shukur (2006) investigated "the effect of one year of regular
'LEGO' training on pupils' performance in schools" (p. 182). As part of their mathematics learning, Swedish elementary and middle school students used LEGO Dacta
Material that is programmed. This program was written in LOGO, a suitable computer language for children (Papert, 1980). The pedagogical perspectives of this study were 17 Piaget's constructivist theory and the theory of situated cognition. The researchers believed that knowledge is constructed in the mind of the student by active learning. A combination of qualitative (extensive observations, interviews, and inquiry) and quantitative (pre- and post-tests, the control group, and the experimental group) approaches were used for triangulation purposes.
The study by Hussain et al. found four main findings relating to I) different strategies oflearning the material, 2) pupils' learning, 3) the learning context, and 4) the role of the teacher. First, students learned the material in various ways but "tcial-and error" and "cooperative" methods were the most common. There was no difference between different age groups and no difference between students' genders. Second, as
Hussein et al. explained, "It is difficult to confirm the hypothesis that LEGO-generally has positive effects on cognitive development. However, our study indicates certain positive effects can be shown for groups/categories of pupils" (p. 192). Even though student achievement did not improve, students in grades 5 and 9 and with higher ability in mathematics tended to be more engaged and had a positive attitude towards the LEGO material. Third, in terms of the working groups, groups of two to three students are ideal for successful learning. Fourth, "the role of the teacher, as a mediator of knowledge and skills, was crucial for the students' coping with problems related to this kind of technology" (p. 189). I strongly agree with the role of teachers in technology integration for mathematics education, as the role is expressed in this study. Students' interest in technology was strongly influenced by teacher attitude. In addition, teachers also 18 expressed a strong opinion about the advantage of two teachers working together in the classroom.
Focusing their study, Tall & Chae (2001) used "an experimental approach with computer software to give visual meaning to symbolic ideas and to provide a basis for further generalization." They collected "evidence for the ways in which students develop conceptual links between symbolic theory and the visual and numeric aspects of computer experiment" (p. 1). This study is based on two assumptions: 1) students prefer to think algebraically rather than geometrically when they solve problems, and
2) computer-assisted learning which uses graphical representations can improve students' mathematical understanding in general because computer graphic software can provide students with environments, which facilitate intuitive thinking prior to the construction of a formal concept.
In this study, "xlogis" software was used to collect various forms of data. The students took a pre-test to assess their understanding of geometric convergence. In addition to the formal assessments, one of the researchers observed as participant observer to collect "data using audio-tapes with field notes made at that time" (p. 5).
After the course, students were given a questionnaire to "investigate their understanding relating to their visual experiences and symbolic theory" (p. 5).
The researchers found some benefits of visualizing concepts and manipulating symbols. They noticed that students who gained conceptual knowledge in this way were able to develop concepts further than those using procedural knowledge. They considered conceptual knowledge that which was constructed through visualization using interactive 19 graphical software. As part of their investigation, a study was conducted and focused on
"the establishment of the connection between visual orbits of x = f (x) iteration, the numeric information provided by the software and the underlying mathematical theory"
(p. 2). The study showed that eight out of twelve students--including three who did not have the desired pre-requisite knowledge of the notion of geometric convergence-were
able to use flexible links between numeric, graphic, and symbolic representations of
geometric convergence and to successfully use computer software to gain visual insight
and to obtain numerical approximations to link with theory.
Students who did not achieve highly in mathematics could understand the difficult
concepts taught in this study by using appropriate technology tools. As students
interacted with computer software, they could visualize difficult-to-understand concepts
through the conversion of abstract mathematics concepts into concrete, visualized
formats which helped students to understand. In my observations, I have noticed similar
influences of technology in the classrooms.
Teacher Interaction with Technology
Lu & Rose (2003) reported one of the "nine Web-based video case studies that
provide mathematics professional development for elementary and middle school
teachers around the country" (p. I). The Seeing Math Telecommunications Project used
audio, video, and interactive computer tools to teach specific math content that is widely
recognized as difficult to teach. The Concord Consortium and Teachscape developed the 20 case studies. Before the project started, the teacher was interviewed to help the researchers understand her strategies and expectations. The data collection was done through two or three taped class sessions, pre- and post- lesson interviews with the teacher, and a collection of student work.
Video Paper Builder 2 (VPB2) was used to collect and review video data. VPB2 is software that closely links text and video. Since the teaching was videotaped, the teacher could review her expectations that she had envisioned prior to taping the class session and reflect upon "the classroom experience as it actually unfolds" (p. 4). This videotaping and review process would definitely help the case study teacher since listening, watching, and analyzing the way teachers make decisions about their teaching leads participants to make better analyses and decisions about their own teaching. Because VPB2 allows users to add typed/textual commentary to video, the teacher and content specialist could add their views of the featured classroom from different perspectives. This helped the teacher to have additional insights into the lesson she had taught.
This study shows how teaching can be improved when video is used in a case study. One of the advantages of using video in research is that it allows a number of different people to share the same information while also allowing them to simulate their own unique situations. As teachers have the opportunity to review through video what other teachers do in their classroom, they can learn many things by asking themselves, "If this were my class and these were my students, what would I do?"
To align with the idea of teaching mathematics with technology, we need to change the way we currently teach by making necessary modifications so we can prepare 21 all students to function with strong technology and mathematics backgrounds for the future. According to research, there is a set of strong beliefs among mathematics teachers about mathematics education. In her study, Nolan (2004) pointed out that many pre- and in-service teachers believe that basic concepts should be taught before differentiating instructions to accommodate students' learning styles and other circumstances. Nolan
(2004) also believes what the pre- and in-service believe (that basic concepts should be taught first), but she also believes that there are simply too many topics to cover and that the quality of mathematics education cannot be improved simply by covering the basics. -
Believing there is not enough time and there is too much content to cover require deconstructing in order to get at the heart of what is important in teaching and learning. Without such a deconstruction, the barriers to the promises and possibilities ofICT [Information and Communications Technologies] integration, and the barriers to shifts in the practice of teaching mathematics in general, remain too substantial to overcome. (p. 113)
What would be an ideal solution to the problem that Nolan points out? Can we use technology to teach mathematics effectively? To examine the possibility, we need to keep trying to find the best way of integrating technology that helps both new and old teachers to teach more effectively. The ultimate goal of technology integration in mathematics education would be promoting students' mathematics learning by making learning mathematics more interesting, effective, and meaningful to them. There are many technology tools that we can choose from for implementation, and selecting the right one is not easy. However, we still need to find the right combination that will assist teachers in resolving the problems that Nolan discussed.
Naatanen (2005) proposes a possible solution to help teachers to integrate technology into their curriculum. In Finland, a group of mathematics educators created a 22 mathematics Web magazine called Solmu. Naatanen (2005) discusses this magazine in detail.
Very popular are link collections for different levels of school. They need to be well organized and have short descriptions of the contents of the links and how and in which context to use them. Teachers do not have time and energy to work out such collections of links themselves but are happy to use them occasionally if they are well enough structured and worked out for them. Solmu contains mathematical problems, solutions are given, separately, and also a question and answer service is provided. Many files on ideas of teaching mathematics have been col1ected and articles on the results of international mathematics achievement comparisons are also published. (p. 170)
If a well-organized Website is available to instruct teachers in ways to effectively use technology to teach mathematics, more teachers will implement technology in their curriculum.
After the Solmu project was launched, the researcher who represents the participants pointed out one very important issue that must be addressed when we create our own Web database. Naatanen (2005) wrote, "We would like to start a database on good applications of mathematics - not trivial and not too complicated. Here international col1aboration of mathematicians would be useful, to collect good cases"
(p. 171). In fact, developing a useful Web link database for mathematics is what we need for successful technology implementation in mathematics instruction.
Naatanen (2005) discussed some of the benefits of developing we11-organized
Websites for mathematics teachers. First, the Internet is good for accumulating databases.
Second, it is freely available whenever and wherever teachers need it--an easy and cheap distribution channel, once the technology has been purchased anyway. Third, it is 23 relatively easy to update (although this is also a problem, since constant updating is needed and often forgotten).
I strongly believe that a weIl-organized compilation of useful Websites for mathematics education would be a very powerful tool for teaching and learning mathematics. This effort wiIl especially benefit primary and secondary mathematics educators. The organized Web database wiIl save tremendous amounts oftime for teachers looking for meaningful resources for effective instruction. It wiIl also support students and their parents when seeking out dependable resources to study mathematical concepts. 24 CHAPTER 3. METHODOLOGY
In this section I explain why the collected data are relevant to my study. I also set out the procedures by which data was analyzed.
Participants
The subjects of this study were West High School (WHS) students who were enrolled in one of three mathematics classes - one Advanced Placement (AP) calculus class and two Advanced Placement (AP) statistics classes. There were 16 students In the
AP calculus, 14 students in the AP statistics period 2, and 18 students in the AP statistics period 6. These classes were selected because the teachers were both interested in technology-integrated instruction. The two teachers and their students graciously accepted my proposal and agreed to participate in the study voluntarily.
To compare my sample group with the general population of students who took
AP mathematics courses statewide and nationally, I looked at a set of data collected by the College Board (2004). In the school year (SY) 2004-2005, 67.33% of the national test-taking student population was White and 16.74% was Asian. However, the mix of ethnicities in the state of Hawai'i was different from the national population: 74.51% of the Hawai'i population of test takers was Asian and 13.94% was White. Over 80% of both the national and state populations were White and Asian even though White prevailed in the national population and Asian prevailed in the Hawai'i population. (See
Table 4.)
At first glance, the mix of ethnicities in the WHS AP mathematics classes was different from that of the national population, but resembles that of the Hawai'i AP 25 classes (see Table 4): 70.83% of the WHS AP mathematics student population was Asian and 10.42% was White. The number of the Filipino students in WHS AP mathematics classes was very large and, for statistical purposes, the Filipino population was included in the Asian population.
Table 4 Comparison ofthe mix ofethnicities in WHSAP Classes to that ofNational and Hawai'i State AP Classes. SY 2004-2005
Ethnicity U.S. Hawai'i WHS
Not Stated 6249 (2.19%) 44 (3.91%)
American Indian!Alaskan 1052 (0.37%) 4 (0.36%)
Asian!Asian American 47654 (16.74%) 839 (74.51%) 34(70.83%)
Black!Afro-American 10595 (3.72%) 9 (0.80%) 2 (4.17%)
Latino: Chicano, 9365 (3.29%) 6 (0.53%) Mexican American Latino: Puerto Rican 1294 (0.45%) 5 (0.44%)
Latino: Other 8050 (2.83%) 3 (0.27%) 1(2.08%)
Other 8747 (3.07%) 59 (5.24%) 6 (12.50%)
White 191719 (67.33%) 157 (13.94%) 5 (10.42%)
Total 284725 (100%) 1126 (100%) 48 (100%)
One participating class was one section of AP calculus taught by a teacher
(Mr. A), who had been teaching the same course for two years when this research was conducted. The teacher also had a certain level of interest in integrating educational 26 technologies as part of his instruction. Due to the level of work the students were required
to produce, they constantly used graphing calculators (Texas Instruments TI-83) on a
daily basis.
In addition to the graphing calculator, the teacher also used Microsoft PowerPoint
to present to the class the concepts of the lesson along with practice questions via
television monitors connected to a computer provided by the school to every teacher. He
also used television monitors as a presentation and communication tool by attaching a
video camera to them through RCA cables. When he or his students were presenting their
ideas and work to the entire class, that work was displayed by simply placing it under the
video camera.
The other two participating classes were two sections of AP statistics classes taught by another teacher (Ms. 8) who had been teaching the same courses for two years
when this research was conducted. Ms. 8 was also interested in technology integration as
an instructional aid for the classes she taught. However. her background and knowledge
of the implementation of educational technology was rather limited when compared to
Mr. A's. In her classes, she set up a TV monitor and video camera system, which she
used as a presentation tool for the students. She also used a TI-83 graphing calculator to
explain the concepts in class, and most of her students had TI -83 graphing calculators for their own personal use. She also utilized the overhead projector for her lectures and
student presentations. 27 Data Collection
At the beginning of this study, there was a brief introductory session to explain the purpose of the research to the participating students and to give them time to complete a questionnaire-survey form designed to collect information about the participants' attitude toward studying mathematics as an academic subject and toward using educational technology as study tool. Part of the questionnaire designed to gather students' opinions about what helped their learning and achievement in mathematics classes.
A questionnaire similar to the Gressard-Loyd (1984; 1985; 1987), the Computer
Attitude Scale was generated for this study. Some items were open-ended. However, most items used were in Likert scale format with four response categories: "strongly disagree," "disagree," "agree," and "strongly agree."
Multiple videotaped classroom observations were made to provide a source of spoken text (speech) data to provide evidence of how teacher and students interacted.
This set of data was also collected to document the learning activities in each classroom and to document how students and teachers used the technology in each session.
Each session of the three participating classes was videotaped from the beginning to the end of the class period according to the WHS bell schedule. There were 17 videotaped observations per class. In total, there were 51 videotaped classroom observations. I made these videotapes, but I was not a neutral observer. During these sessions, I was not a passive person in the room. On request, I interacted with some of the students and the teachers. 28 During the first few classroom observations, I placed the video camera in front of the classroom, close to the teacher because I thought capturing students' facial expressions was crucial. However, in some cases, the video camera distracted students' attention from the lecture. Therefore, I moved the video camera to the back of the classroom and focused more observing the student-teacher interactions. Depending on the lesson, I also carried the camera among the students to capture more student-student interactions close up while they worked in groups.
Extensive field notes were another source of data. Field notes included explanations of what I observed in each class and of any type of interactions I had with the students. I wrote my field notes as I observed class sessions as another source of data, which will be compared to the Videotaped classroom observations to provide triangulation. I took careful note of events that occurred over and over and that struck me as meaningful events that occurred in any observed videotaped session during or right after each session.
There were two separate sets of field notes for different purposes. The first set was taken exclusively to keep track of my communications with the two teachers regarding clarification purposes for any interesting and unusual instructions or class activities that I observed in class. This set also includes all the communications relating to any educational technology implementation issues regarding the implementation of technology for instructional purposes, including teachers' comments and expressions regarding educational technology in class. 29 The second set offield notes was of the overall classroom observations from a participant-observer's point of view. In this set, all mathematica1learning activities that would either encourage or discourage students' learning were carefully recorded.
Whenever I had any personal questions as to why students performed or behaved in a certain way, the questions were also included as part of this set. Students' unusual reactions and comments about the teacher's instruction were also part of this set In this set, I included many notes about the teachers' interactions with students for mathematics
learning. Whenever I interacted with a student or a group of students, I also used my
laptop to take notes to remember the details of the interactions, including the
circumstance, the purpose and the actual content.
Another source of data was videotaped teacher interviews. There were four interview sessions with the teachers because there were two interview sessions for each participating teacher. Teacher interviews were conducted in a different manner from
student interviews. Each teacher was interviewed individually instead of in joint
interviews because each class was different. Each teacher had a different level of understanding about how to integrate educational technology.
To collect accurate information, prior to the interview, selected interview questions were prepared after an extensive discussion with my advisor. The interview
questions were given to the teachers in advance for more in-depth responses. This allowed the teachers deeper contemplation about how they used technology for
instructional purposes and for promoting interactions with their students. The interview
sessions were held in each teacher's classroom. When an interview was conducted, I 30 asked each teacher the same questions. Their responses were audiotaped rather than videotaped because the interviews were not regarding the classroom teaching sessions.
The first interview was conducted to gain an understanding of how teachers view the integration of technology in instruction and what their technology background was prior to the study. The second interview was conducted to gain an understanding of what the teachers thought about their own instruction in several selected teaching sessions. To select the videotaped classroom sessions to discuss during the interviews, I first read my field notes for each videotaped lesson and chose several sessions based on my impressions. For the second interview sessions, I used the post-lesson, video-stimulated, interview technique (Clarke, 2004; Emanuelsson & Clarke, 2004; Shimizu, 2002; Tomer,
Sriraman, Sherin, Heinze, & Jablonka, 2005; Williams & Clarke, 2002) in which the teachers watched videotapes of themselves accomplishing a task and were then interviewed about their thoughts while they were teaching, as seen on the video segment.
The interviews were focused on how the two teachers viewed their own teaching practice and the role of educational technology in their teaching.
During the interviews, teachers were invited to observe and reflect upon several taped lessons which they had taught. Both teachers were asked to reflect on how they used educational technology in their instruction and how it influenced their instruction and students' learning. As we went through the videotaped sessions, if there was something that needed to be clarified or explained by the teachers, the video records were used as a visual prompt of what happened in the classroom before the teachers started any explanations. 31 Videotaped student interview sessions were an additional source of data. One focus of the interviews was to understand whether technology integration helped the students to better learn mathematics. Another focus was to fmd out whether educational technology helped students become actively engaged in mathematics learning.
During some of the observed lessons, I conducted informal student interviews with a few selected students. Brief, informal interviews were conducted in the classroom area while students were using PowerPoint on computers for their assignments. After the data collection was completed, interviews, which were more formal were held with students to provide follow-up sessions to what I observed.
Student interviews were conducted in my classroom after school or in an assistant principal's office during my personal preparation periods. Students for small interview groups were pre-selected based on my observations. One criteria for selecting members of the interview group was the level of their communication in class. All students who were in more than one participating class were in the same interview group because I wanted to know what they thought about both classes and how they used technology in those classes. Another way to group students for the interviews was to review videotaped classroom sessions to identify those students who frequently interacted with the same group of students. This review allowed me to find their associates, with whom I believed they would communicate better if placed in the same interview group.
I had originally planned to interview 10 small groups based on my observations; however, due to scheduling conflicts I was only able to successfully complete interviews 32 with five. One interview group comprised students who were in both AP calculus and AP statistics while the study was conducted.
The interview sessions were planned to be conducted in two parts. The first part was intended to use meaningful interview questions to determine student perspectives relating to the use of technology for mathematics learning. The second part was designed to allow students to observe a few selected sessions of their class on video, to elicit discussions about how they studied in class, and to establish student versions or interpretations of what they observed on the videos. However, since students had a very limited time to participate in the interview sessions, there was not enough time to accomplish the second part completely. Therefore, the second part of the interview was not done properly for most groups.
For small group interviews, I pre-selected some interview questions after an extensive discussion with my advisor. I used Microsoft PowerPoint software to present the interview questions on a laptop computer. As students answered the interview questions, the entire interview session was videotaped and also audio-recorded simultaneously as a backup of the video record.
In the interview sessions, the questions focused on how students felt about learning mathematics and on what they thought about the role of educational technology in mathematics learning. To understand how students perceived the role of educational technology in mathematics learning, their opinions and experiences relating to learning mathematics with and without educational technology were discussed in depth. 33 Data Analysis
To achieve the planned objectives of this study, 1 used the triangulation method with five different sources of data. 1 applied Grounded Theory (G1) (Glaser, 2002, 2004;
Glaser & Holton, 2004) as a major data analyzing tool because of the type of collected data. 1 believed and sincerely hoped that the findings of the collected data that were analyzed with the GT method would provide a learning experience for readers, as Boaler and Humphreys (2005) mentioned: "Teachers and researchers are finding that analyses grounded in actual practice allow a kind of awareness and learning that has not previously been possible" (p. 4).
According to Glaser (2002; 2004), GT involves comparing one segment of data with another to determine similarities and differences (e.g., comparing one statement from an interview about using calculators in mathematics education with another statement from another interview or observation).
Data were grouped together on a similar dimension. This dimension was tentatively given a name; it then became a category. The overall object of this analysis was to seek patterns in the data-to formulate a theory. This method fit the inductive and concept-building orientation of the study. As Glaser (2002; 2004) wrote,
Remember again, the product will be transcending abstraction, not accurate description. The product, a GT, will be an abstraction from time, place and people that frees the researcher from the tyranny of normal distortion by humans trying to get an accurate description to solve the worrisome accuracy problem. Abstraction frees the researcher from data worry and data doubts, and puts the focus on concepts that fit and are relevant. (p. 1)
As 1 analyzed the collected data, 1 focused on the "embodied mode" that Tall
(2002) described: "I argue that the embodied mode, though it lacks mathematical proof 34 when used alone. can provide a fundamental human basis for meaning in mathematics"
(p. I). In the same research. he explained his "three worlds of mathematics" and their
relationship with the "Rule of Four" for an in-depth, basic, mathematics-learning model;
they will be the focal points of data analysis from the mathematics education perspective
In his research, Tall also explained several stages of learning mathematics: graphic.
numeric. symbolic, and verbal. For my research, I studied the collected data and sorted
the connections between what I observed and what Tall (2002b) explained in his
research.
From the perspective of educational technology, I looked for the "three primary
curricular goals" discussed by Cradler, McNabb, Freeman, and Burchett (2002):_
The Center for Applied Research in Educational Technology (CARET) has gathered compelling research and evaluation fmdings to answer frequently asked questions about how technology influences student achievement and academic performance in relation to three primary curricular goals: I) Achievement in content area learning, 2) Higher-order thinking and problem-solving skill development, and 3) Workforce preparation. (p. 47)
The comparison between the findings of the research and the results of the collected data
analyses were carefully documented to find the similarities and the differences in those
three areas listed above.
I sincerely hoped that the content of this study could serve mathematics educators
by providing meaningful information about how technology can be integrated into their
mathematics classrooms for purposes similar to those, which Boaler and Humphreys
(2005) wished for their research:
We hope our cases of teaching will provide such an example of practice and that the details that are portrayed will serve as sites for both inquiry and learning. Just as the students in the cases are learning through their mathematical inquiries, we 35 hope that such inquiries may serve as a source for teacher learning as teachers pose their own questions about the interactions they see. (p. 4)
Here again are the four research questions that I was interested in answering through my data analysis. Further below, I also explained how the interpreted data from each data source was analyzed to answer these research questions.
1. What roles do students and teachers report for technology integration? Why
do they think those are the roles?
2. How do teachers and students use technology in mathematics class?
3. How does educational technology integration help students learn mathematics
cooperatively?
4. How does educational technology influence mathematics instruction?
Student questionnaires, videotaped classroom observations, videotaped interviews, and field notes were the existing primary data sources that I collected during the time period from October 19, 2004 to June 2, 2005. I interpreted the primary data sets from each primary data source to generate the interpreted data sets that were used to answer my research questions. Table 5 lists and briefly describes the types of interpreted data sets generated from each primary data source.
Table 5 Primary data analysis schedule
Type of Data How Data Was Analyzed and Interpreted
Videotaped • Transcribed the content of the interviews
Student Interviews • Separated out student responses in the interview sessions 36 according to the questions discussed to allow comparisons
• Labeled and categorized separated student responses
according to the content of the answer
• Categorized the main themes/ideas by combining similarly
labeled student responses
• Used observed facial expressions and body language as
needed for more accurate interpretation in association with
the relevant spoken words (because much valuable
information was missing when speech [spoken] data was
converted into text [written] data)
Videotaped • Transcribed the content of the interviews
Teacher Interviews • Separated out teacher responses in the interview sessions
according to the questions discussed to allow comparisons
• Labeled and categorized separated teacher responses
according to the content of the answer
• Categorized the main themes/ideas by combining the
similarly-labeled teacher responses
• Used observed facial expressions and body language as
needed for more accurate interpretation in association with
the relevant spoken words (because much valuable
information was missing when speech [spoken] data was
converted into text [written] data) 37 Videotaped Classroom • Scanned the videos and separated the contents of videos by
Observations the interview questions to allow comparisons.
• Labeled and categorized separated video clips according to
their contents
• Categorized the main themes/ideas by combining the
similarly-labeled student and teacher responses
• Matched up the themes found in videotaped classroom
observations with the themes found in videotaped student
interviews and videotaped teacher interviews for
triangulation
• Explained the connection between the selected scenes of
videotaped classrooms and the main themes/ideas developed
in the videotaped student interviews and videotaped teacher
interviews
Field-Notes • Read and separated the contents of the field notes by the
interview questions to allow comparisons.
• Labeled and categorized selected portions of field notes
• Categorized the main themes/ideas by combining the similar
labels of student responses
• Matched up the themes found in field notes with the themes
found in videotaped classroom observations, Videotaped
student interviews, and videotaped teacher interviews for 38 triangulation
• Explained the connection between the selected portions of
field notes and the main themes/ideas developed in the
videotaped classroom observations, videotaped student
interviews, and videotaped teacher interviews
• Sought out evidence of what I observed and tested the
predictions (which I had made while I was participating in
the class sessions) to support my fmdings from my analysis
of videotaped classroom observations, videotaped student
interviews, and videotaped teacher interviews
Questionnaire • Organized and separated student responses to the
questionnaire by interview questions to allow comparisons
• Labeled and categorized student responses
• Categorized the main themes/ideas by combining the
similarly-labeled student responses
• Matched up the themes found in the questionnaire with the
themes found in the field notes, videotaped classroom
observations, videotaped student interviews, and videotaped
teacher interviews for triangulation
• Explained the connection between the categorized student
responses to the questionnaire and the main themes/ideas
developed in the field notes, videotaped classroom 39 observations, videotaped student interviews, and videotaped
teacher interviews
To analyze the multiple sources of my data, I had to view my data in strict chronological order. The order I used to analyze my data was the same order in which I collected them. Since first I collected the questionnaires as I started to observe the first class session, I analyzed the questionnaire responses according to the steps outlined in
Table 5.
The classroom observations were the next source of data I analyzed. Classroom observations were analyzed according to the steps listed in Table 5. When I was done with the classroom observations, I worked on the field notes by following the steps outlined in Table 5. After the classroom observations and field notes were analyzed, teacher interviews were the next data set collected and were therefore interpreted chronologically. Student interviews were the last source of data to be interpreted.
I made two data comparisons. First, the teacher interview data was compared with the classroom observations and field notes to find common or isolated themes which emerged from each set of data source. Second, the student interview set was compared with the classroom observations, field notes, and questionnaire responses for a similar purpose.
Finding the emerging themes from the first data source was my starting point.
When the second data source was analyzed, I checked to see whether the major themes of the second data source provided support for what I found in the first data source. The comparison between the emerging themes of the other data sources and the themes 40 generated from the interpretation of the first data source were a very important part of the data analysis to understand what actually happened in the study. I continuously checked whether themes of one data source supported themes from another data source. This process was continued until the common themes of the primary sources were clearly identified.
Interpreted primary data sources and identified common themes were used to answer my research questions. My first research question was, "What roles do students and teachers report for technology integration? Why do they think those are the roles?" I looked at teacher interview data to understand their perspective on technology integration in their teaching. What I understood from the interview data was carefully compared with the classroom observations and field notes. If any clarification was needed for this section, I conducted an additional interview with the two teachers to obtain the necessary information.
While I was analyzing data from the two teachers, I conducted a comparison study between them. First, I compared Mr. A and Ms. B's uses of technology in teaching.
Second, since Ms. B taught two different sections of the same course, I compared her technology use in each class.
To understand students' perspectives on the role of technology in their mathematics learning, I started my analysis with the questionnaire. What I found from the questionnaire responses was compared with classroom observations, field notes, and student interviews to take advantage of triangulation. 41 Again, because I observed three different mathematics classes, I compared them to study the similarities and differences of student use of technology in learning. Also, there was a small group of four students who took both mathematics classes from the two participating instructors. I compared their perspectives of using technology in learning mathematics to determine whether they had a different outlook compared to students who took only one mathematics class from one teacher.
My second research question was, "How do teachers and students use technology
in a mathematics class?" This question was closely related to the first research question.
Therefore, the analysis procedure for this question was meant to be identical to that of the
first question. I believed analyzing the two questions at the same time would be time
efficient.
My third research question was, "How does educational technology integration
help students learn mathematics cooperatively?" For this question, I started from the
student interview. What I found was compared with classroom observations and field
notes. I looked at the questionnaire responses to understand what students believed prior
to their participation in the study and compared that with what they said in the interview.
Reponses of students who took both mathematics classes from Mr. A and Ms. B
were carefully analyzed since students may have compared the two teachers' different
ways of integrating technology in their instruction. After comparing the two teachers
from the students' perspectives, the classroom observations and the field notes were
analyzed to fmd evidence of what students believed about the two classes. 42 My fourth and final research question was, "How does educational technology influence mathematics instruction?" I started my analysis of the teacher interviews to understand their perspectives on the issue. Based on the emerging ideas from the interview data, I analyzed the classroom observations and the field notes to support what
I found.
The responses of the two teachers and the observed classroom actions were compared carefully. Any common response from both teachers was identified and discussed. For any conflict or disagreement between the two teachers, I investigated further to understand the cause and discussed that cause.
I analyzed an abundant amount of data, including the videotaped class observations, the field notes, the student interviews, the teacher interviews, and the questionnaire responses to investigate the interactions between students and their teachers in these secondary mathematics classrooms.
The best way to analyze the data I collected in relation to the teachers was as a set of interwoven case studies. I wrote case studies of Mr. A and Ms. B and their AP statistics (2 sections) and AP calculus classes. These multiple case analyses allowed me make comparisons between and within the two different cases. That was how I generated theories of what happened in the integration of technology in the mathematics. In addition, I discussed how the technology was used in different situations and its level of effectiveness. I fed the information emerging from the teacher interviews and the student interviews into the multiple case analyses. 43 CHAPTER 4. DATA ANALYSIS AND RESULTS
To analyze the multiple sources of data, I viewed them in strict chronological order, analyzing them in the same order I collected them. I began by analyzing the questionnaire responses (as described in Table 3) because I collected the questionnaire first. The questionnaire analysis was followed by the analysis of the first teacher interview and then the classroom observations that included two sets of field notes and videotaped class sessions. Analysis of the class observations was followed by analysis of the second teacher interviews and, finally, the student interviews.
Initially, I carefully selected the interview questions. However, as the interview progressed, I modified the questions to collect more meaningful data. Also, after participants responded to the initial question, I added subsequent questions to clarify their answers. All of these interview questions were open-ended questions and the responses were tied up at the end.
I chose to analyze my data sources chronologically because I wanted to report in chronological order any change in student and teacher perspectives on the integration of technology into learning and teaching mathematics. By analyzing chronologically, I was able to compare their thoughts and ideas before and after my observations, and it was easier for me to examine whether there were any changes or differences between their stated thoughts about how technology would be used for mathematics learning and how they actually used technology in mathematics learning.
GT required this chronological order because each analytical step required a prediction of what would happen in the future or of what the future would be like based 44 on current observations. I then had to compare my predictions with what actually happened. Then I had to make a new prediction for the future and repeat the process until the data analysis process was completed.
Questionnaire
As the questionnaire was passed out, students were told that the teachers would not see the individual results of the surveys and that they could opt to not participate if they wished. Each student was given a questionnaire and asked to return it as an extra credit assignment (Ms. B's classes) or with no points attached (Mr. A's class). Students were allowed to take the survey home and complete it so that they had enough time to think about each question. Students who did not wish to complete the questionnaire simply returned their blank surveys as others returned their completed surveys. Hence, the teachers and I were not able to judge who did or did not complete a survey since students did not have to include their names on the questionnaire.
The questionnaire (see Appendix A) contained 87 questions, 70 of which were four-point, Likert-scale items (using a range of "strongly disagree" to "strongly agree" for given statements). The items were developed based on a grounded survey (Klein,
2003). Because I did not have time to develop responses to fix issues arising from preliminary analysis of the observations and interviews for this study as did Klein (2003),
I took his insights and experiences and used a subset of his questionnaire items in my questionnaire. In order to find meaningful information for the purpose of my study, I included some selected questionnaire items to measure the target areas of my study. The four target areas of my study were whether students and teachers understood the role of 45 technology, how they used technology, how technology enhanced student learning of mathematics, and how technology influenced mathematics instruction.
To set the baseline for data analysis, I wanted to understand what students think about their mathematics learning and the use of technology to study mathematics. That is why all questions that related to mathematics and using educational technology were selected and included in my survey questionnaire. In addition to the grounded survey, I used some selected items from the Fennema-Sherman Mathematics Attitude Scales
(Doepken, Lawsky, & Padwa, 2007) which were related to mathematics learning because they helped me understand students by comparing their responses against their scales.
Those researchers used the scales to understand four scales: "The scale consists of four sub-scales: a confidence scale, a usefulness scale, a scale that measures mathematics as a male domain and a teacher perception scale" (p. I). For this study, however, I decided to focus on how confident students were about studying mathematics and what they thought about the usefulness of mathematics.
To analyze the questionnaire data, I developed a comprehensive profile of each questionnaire item by calculating how many students reached agreement for each possible response to each question. I did not segregate the responses according to participating classes. Responses to each questionnaire item were combined to calculate the actual number of students who selected a particular response choice, and that number was converted into an appropriate percentage of the student population. Further care was taken to observe the particulars of the student responses--to try to construct a story out of the responses in conjunction with the other sets of data. 46 The purpose of the survey was to add credibility to (or to contest) themes as patterns were derived from my observations and interviews. The intent was not to
"discover" the "truth" about a population on a given item. Here, each item was
"significant" in the sense that it suggested multiple stories to be analyzed in tandem with other items. Fetterman (as cited in Klein, 2003) confirms,
While "anthropologists usually develop questionnaires to explore a specific concern after they have a good idea how the larger pieces of the puzzle fit together," there remain ''methodological problems associated with questionnaire use--including the distance between the questioner and respondent" and that these problems "weaken its credibility as a primary data collection technique" (p. 84).
Participating students believed that, overall, they had good experiences when using educational technology in their classes. When they were asked to respond to various statements that describe opinions regarding educational technology usage in school, they also strongly agreed with the statement that educational technology was beneficial to those who study mathematics and science. However, they disagreed with the claims that educational technology would benefit only those in vocational education or only college-bound students. Students strongly believed that educational technology would benefit all students as long as they used it to study mathematics or science.
There were 87 items on the questionnaire in total, but not all of them were related to my study focus questions. Therefore, responses that were not related to the focus questions were excluded in this evaluation and discussion.
To see the bigger picture drawn by the questionnaire responses, I clustered its items under three main topics and one special interest topic to evaluate alignment with other main ideas, which gave me a clear baseline for my data analysis. For each cluster, I 47 used the actual question and the profile of it that shows how students responded to each item of that particular cluster. After discussing the relationship between each item of the cluster and the focus questions of my study, I summarized each cluster by explaining any outstanding finding from the cluster.
I generated a comprehensive profile for an entire cluster whenever the combination of individual items was appropriate for understanding how the items were
I interrelated. After I generated the grand profiles, I discussed what I found in the questionnaire data set as a whole.
Focus 1: What roles do students and teachers report for technology Integration? Why do they think those are the roles?
Table 6 shows all the questionnaire items that I expected to be related to the first topic of my study. The table was organized to include the item number, the question as asked in the questionnaire, and its profile. To help in understanding how students responded to each item, I indicated the actual percentage of students' responses in each category of the Likert-Scale (SA=Strongly Agree, A=Agree, D=Disagree, SD=Strongly
Disagree, and NR=No Response).
Table 6 Profiles ofthe student responses tofoeus 1
Item # SA A D SD NR
9 Educational technology should be available to all students who enroll in math and science. 76% 21% 3% 0% 0% 48 25 People who like computers like mathematics. 9% 24% 62% 6% 0%
64 Mathematics and computers go together well. 18% 74% 9% 0% 0%
67 People who like mathematics like computers. 12% 26% 50% 9% 3%
Students considered technology an important mathematics-learning tool (see items 9 and 64). Students considered technology integration to be an essential part of effective mathematics learning. I was surprised to find that more than 50% of students
(items 25 and 67) did not think that people who liked computers liked mathematics and did not think that people who did not like mathematics liked computers.
However, few items asked students to indicate how they thought technology should be implemented into mathematics learning. Therefore, I was not able to fmd any specific way of integrating technology that students considered ideal for implementing technology into their mathematics learning.
Focus 2: Do teachers and students use technology in mathematics classes? All the questionnaire items that were related to the second topic were organized in
Table 7 according to the item numbers, with the actual question as given on the questionnaire.
Table 7
Profiles ofthe student responses 10 focus 2
Item # SA A D SD NR 49 8 I have had experience with educational technology in a different class before. 62% 35% 3% 0% 0%
18 My experience with educational technology was positive. 41% 53% 3% 3% 0%
46 I have used calculators in my previous math classes. 62% 38% 0% 0% 0%
47 I prefer calculators to computers in mathematics class. 38% 53% 9% 0% 0%
58 It is easier for me to learn from the computer. 3% 26% 62% 9% 0%
I noticed students mainly preferred to use calculators rather than computers in class: 91 % preferred calculators (item 47). Students (94%) had positive experiences with educational technology (item 18). However, 71 % of students said they disagreed with the statement that it is easier to learn using a computer (item 58).
Based on their experiences in multiple classes, students had positive experiences with using educational technology. They also have used technology to do their work outside of class. In other words, they do not have any problem with using technology and are willing to use it whenever they need to complete their work.
In a separate item, they reported satisfaction with they way their teachers used technology in mathematics instruction. In most cases, teachers used calculators in mathematics classes. That is why every student in the classes had experience using a calculator. In addition to that experience, they felt comfortable using calculators in class.
In fact, when I observed the class, they used the calculator (the TI-83) extensively. They also used different types of technology tools for presentations. 50 However, students used their calculators for study purposes throughout the class time.
Teachers taught students how to use the calculator by demonstrating it. Once students knew how to use it, they used it constantly for their own learning and when they discussed work with their friends in class. The calculator was part of their daily learning activity.
Focus 3: How does educational technology integration help students learn mathematics cooperatively?
All the questionnaire items that were related to the third topic were organized in tables 8, 9 and 10. The tables were organized in the same way that the other tables in previous sections were organized.
Table 8 Profiles ofthe student responses to focus 3 (Part 1)
Item # SA A D SD NR
26 I am a more active learner in this class versus more traditional math classes since more educational technology is available. 15% 47% 35% 0% 3%
31 I learn less when I have to use educational technology than I would without educational technology. 0% 15% 65% 20% 0%
32 Using educational technology helps me understand mathematical concepts better. 18% 64% 18% 0% 0%
33 When I use educational technology I am motivated to learn mathematics. 6% 50% 38% 3% 3% 51 34 When my teacher uses educational technology to explain a mathematics concept, I can understand better. 12% 71% 15% 3% 0%
36 Educational technology helps me learning mathematics. 12% 74% 9% 6% 0%
57 It is easier to learn the content of this class (mathematics) on the computer than it would be to learn it in a lecture/recitation format. 3% 18% 58% 21% 0%
61 I arn motivated to use more educational technology for learning new ideas in this (mathematics) class. 9% 68% 21% 0% 3%
Do students learn mathematics better with educational technology? Students'
responses to several questionnaire items helped me to understand how students think
about it. Students said that their mathematics learning was improved because of the
availability of educational technology. What made the students feel so confident about it?
Every teacher teaches mathematics in a unique way, and students have to follow their teacher's teaching style, which is not very flexible in many cases. Therefore, when
students have to work with one teacher throughout the year, their chances of having an
opportunity to learn mathematical concepts in a way that best suits their learning style
would be very slim and unrealistic to expect.
With technology integration, however, students can be exposed to various levels
of differentiated instructions when they study mathematical concepts. Technology
integration would enable students to understand mathematics problems in multiple ways.
Using technology would help students analyze flexibly and would also make problem-
solving easier. Simply providing different ways of approaching, analyzing, and 52 understanding mathematical concepts in class would be a great motivation for students.
That might be a possible reason why students strongly believed that they would be able to understand the concepts better when technology was integrated into their learning.
Students' responses simply indicated that using educational technology did not
slow down their mathematics learning. Their responses also could be interpreted as
indicating that educational technology integration did not have any negative influence on
their mathematics learning. Students were convinced that they could understand
mathematics concepts better when their teachers used educational technology than when
teachers just explained mathematics without using educational technology. However, I
frequently observed that the two participating teachers were using technology tools as
presentation tools rather than as experimental learning tools that might allow students to
explore and understand mathematical concepts. Therefore, it was not clear to me why
students felt that they could understand mathematical concepts better with technology.
Students had a positive attitude toward using technology tools for mathematics
learning, and they strongly felt that educational technology integration in their
mathematics classes would benefit their learning. Students' responses were strongly
based on their first-semester experiences in the mathematics classes since the
questionnaire was answered in the second semester.
Why did students strongly believe that technology-integrated mathematics
learning would positively influence their mathematics study? Students said that learning
mathematics with educational technology did not make their learning any easier. If
technology integration did not make their learning any easier, why did students think that 53 they learned better when their teachers integrated technology into their teaching?
Students indicated that technology integration with mathematics learning motivated them. They also believed that they could learn more with technology tools.
When students were motivated, they were able to learn more independently and actively.
It was clear to me that the presence of educational technology motivated students to be more active learners. Students became more active learners because they were highly motivated when using educational technology. Students should learn how to use technology tools in order to be active and independent learners. In other words, students would not learn more actively if they do not know how to use the technology tools effectively.
Table 9 Profiles ofthe student responses tofoeus 3 (Part 2)
Item # C H CIH NR
84 Using educational technology in our class was D Confusing, D Helpful in understanding mathematical representations such as graphs, tables, and equations. 3% 94% 3% 0%
According to most items of Table 8, students clearly indicated that using educational technology benefited their mathematics learning. However, it was not clear to me what made students think that using educational technology was helpful. Finding the answer to that issue would definitely be a part of this study. Item #84 was included to gain an understanding of what made students believe that technology helped their mathematics study in general. All of the students' actual responses to this item are 54 organized in Appendix L.
Students said technology integration helped them understand and learn mathematical concepts better (94%). The two most popular student responses were
"helping with understanding mathematics" (30%) and "helping with visualizing mathematical concepts" (20%). Students strongly believed that they understood better because they could visualize mathematical concepts by using technology tools. Students thought that they could understand mathematics better when they used graphing calculators. Visualizing the mathematical concepts was a very important reason for using a graphing calculator.
Table 10 Profiles ofthe student responses to focus 3 (Part 3)
Item # Y N YIN NR
85 Educational technology is valuable when I solve mathematics problems using equations 0 Yes 0 No because 82% 15% 0% 3%
86 Educational technology is valuable when I solve mathematics problems using tables 0 Yes DNo because 79% 18% 0% 3%
87 Educational technology is valuable when I solve mathematics problems using graphs 0 Yes 0 No because 88% 9% 0% 3%
Three items in Table 10 asked students whether technology helped them study mathematical concepts presented in three common mathematical representations: equations, tables, and graphs. Students expressed the belief that technology tools helped 55 them when studying mathematical concepts in equation format. Students used their graphing calculators (the Tl-83) for study and other technology tools for presentation and communication purposes.
How could the Tl-83 calculator help students understand better when they work with mathematical equations? Students said that mathematics problem-solving was easier when they used the Tl-83 in class because using the calculator helped them to complete the calculation quickly and accurately and to visualize their mathematical equations. All of the responses to this item are organized in Appendix M.
How about when students work on mathematical problem in a table format?
Students thought that educational technology helped them to work with tables. There were some common reasons students gave. "Visualizing the ideas" was one of the frequent responses. Some students even mentioned that they could see a graph based on the given table when they used a graphing calculator. Personally, I believe the visualization of mathematical concepts is one of the strengths and benefits students can have when educational technology is integrated into mathematics learning. All of the student responses to item #86 were organized in Appendix N.
How about when students work on a mathematical problem in a graph format?
Frequent responses were "easy to use" (43%), "visualize the concept" (25%), "accurate calculation" (21%). and "fast to calculate" (11%). Graphing calculators (the TI-83) were great visualization tools as they worked on their graph. However, they also used other visualization tools such as computer software, video cameras, and TV monitors. Students could view their graphs clearly and accurately when they used advanced, high-technology 56 tools. All of the student responses to this item were organized in Appendix O.
Overall, students indicated that they felt more motivated to learn mathematics when technology tools were integrated. When the reason was asked, they said the speed and the visualization of the concepts appeared to be the two top reasons. When students were asked to think about how technology could help their learning with mathematics problems in three different formats, a few common responses emerged as follows:
1) Easy to work with technology, 2) Helped them visualize the problems, 3) Helped them have an accurate outcome. I think they clearly summarized how technology integration helps students' mathematics learning.
First Teacher Interviews
After I started my class observations, I interviewed the two participating teachers to discuss how they perceived technology integration into their instruction. To focus on the topics that are relevant to my study, I used a series of questions (see Appendix P) as a guideline for the interviews.
To analyze this set of data, I compared the two teacher's responses to the given questions as I discussed the differences and similarities of their responses to the same topic. Pre-selected questions helped me focus on key ideas that I wanted to fmd out during the interviews, and they also helped me compare the two teachers' responses effectively. 57 Question 1: What are your earliest recollections ofeducational technology as part of instruction in your entire teaching career?
1. Mr. A
The teacher had taught at three different schools. In the first school, he did not have much opportunity to use technology in his teaching. When he moved to the second school, however, he worked with other teachers in a team setting and used more technology in the classroom. He was a part of the team that did a project that was able to incorporate technology a little better. He used a commercially-made computer software called Algebra Animator to teach mathematics and had many good memories about this particular software. He remembered using the software was kind of unique because students could physically see what was happening and what their number meant in the equations.
The teacher pointed out that visualization of the concept was a strength of the software they incorporated to teach algebraic concepts. He learned how to visualize the concept by using this particular software. He added, "And kids got something out of it. I mean, they got to see how the equations worked or not. And it was a visual thing, which I liked."
2. Ms. B
Ms. B's earliest recollection of using technology was when she was in college.
She primarily used a graphing calculator to teach mathematics. She used different models of calculator each year. She attended some workshops to learn how to use graphing calculators to teach mathematics. 58
Question 2: What are your professional experiences using educationallechnology as part ofyour research and teaching?
1. Mr. A
Mr. A learned by accessing infonnation on the National Council of Teachers of
Mathematics (NCTM) website and by subscribing to a magazine called High School
Mathematics, an NCTM publication. He also searched the Internet for mathematical infonnation in general. He had been reading both online and offline to understand effective ways of teaching mathematics because he believed that reading what other people do for better teaching through the Internet was helpful.
Then he talked about his communication through email with other calculus teachers across the nation, as he needed some tips to be a more efficient calculus teacher.
He joined a listserv and constantly received emails from other Calculus teachers throughout the nation. He believed communicating through email helped him greatly. He was even able to communicate directly with the author of the textbook he used to teach his class.
When he taught the laptop program, his students did homework assignments on their computers. He incorporated Microsoft PowerPoint and Excel into his instruction.
Students used PowerPoint to present their works and Excel to do their assignments.
Students were required to prepare PowerPoint presentations for selected assignments.
Students' solutions were shared, in class, when they had time. However, if they did not have enough time, their work was posted on a class Website. 59 Mr. A thought passing out and collecting assignments became more manageable with technology integration. When the teacher wanted to give lengthy information to students, he simply put the information on a disk and passed it around the class. When students received the information on a disk, they copied and pasted right into their hard drives. From the rubrics to descriptions and more, they received information. Students just needed to type it out. Mr. A thought technology integration was a neat experience.
Mr. A used Web pages to instruct and to communicate with the students and their parents. Another way to use Web pages was to inform and initiate meaningful communication between parents and teacher. He used a Palm Pilot to send out multiple emails with a very simple process. He explained, "With Palm, all you need to do is one or two buttons. It creates 180 emails in a few minutes and I connect to the computer and send it out. And I can do it in a short time."
2. Ms.B
Ms. B was somewhat experienced with graphing calculators, the Web, and an overhead proj ector in the past. When she started to teach the AP statistics classes, she began to incorporate the Web and the overhead projector. Therefore, her use of technology up to that point was very limited, to the graphing calculator.
She was using educational videos and a video camera to show examples. She also used her laptop computer with a TV monitor. One day, she wanted to use her computer and video camera alternatively by connecting to a TV monitor. She did not know how to use them. I explained to her how she could connect them by using a simple tool. I gave her a switch box and an extra RCA cable and showed her how to set it up. 60
Question 3: What are your personal experiences as educational technology was integrated into the curriculum? How well did you think it fit? What would have some of the benchmarks, milestones, and memorable events been? Why?
1. Mr. A
In the previous section, the teacher said, "The laptop program was where I really started to use technology in the classroom." Until he participated in the program, he used technology to take care of personal teaching duties such as grades and attendance. He said that it was an eye-opening experience for him to start to figure out how to integrate technology into his teaching.
Another experience he remembered about integrating technology in his teaching was a professional development opportunity he had. He believed that attending the conference helped him significantly and that it helped him understand how to integrate technology efficiently.
He also talked about working with the school technology coordinator and getting support from the school administration. When teachers needed to install computer software for instruction purposes, they had to request that the school technology coordinator install the software on their computers. Frequently, the installation was not done on time for various reasons. Without positive and proactive support from the school administration, effective technology integration would not be successful.
However, he used other types of technology because he thought they would also help to improve the quality of instruction and increase student learning. Students in the 61 AP calculus class used their graphing calculator (TI-83) extensively on a daily basis. The progression of technology integration was slow but steady.
Mr. A had been trying to use various technology tools for his students, especially for communication purposes. Sometimes, students in some fairly large classes were not able to understand what they were taught because they could not see or hear what the teacher explained. So it was his challenge to use technology tools to communicate with the large size of the class and focus on teaching mathematics at the same time. He declared, "I tried to incorporate as much technology as 1 can and still maintain a certain integrity for the mathematics program."
2. Ms.B
To start off, the teacher talked about her experience with the TI-83 graphing calculator. She learned and used the calculator as she began to teach. Whenever a calculator fit with her lesson, she used it and let her students use a calculator. She considered the school year 2003-2004 to be her milestone for her incorporation of technology. Up to that point she mostly used a graphing calculator for her teaching. She started to incorporate different technology tools (such as the video camera and TV) into her classes in that school year. During the same year, her video camera and VCR were stolen. Then she had to bring her own to teach her classes.
Question 4: What are some ofyour memories ofhow another person (to be discussed anonymously, ofcourse) at any school where you have taught that was affected by the 62 introduction ofeducational technology into their research and curricula? How well did they think it fit?
1. Mr.A
The teacher started with the laptop program team he worked with more than five years ago. He mentioned that teachers teaching different school subjects had different definitions of technology integration. He also talked about a workshop he attended to learn how to use the graphing calculator for science and mathematics education.
He shared how computers helped him to do his job better and more effectively.
He used a computer to create different versions of a test and to modifY his instruction to make it more meaningful for his students each year by altering the instructional materials he used previously. He could modifY instructional materials much faster and with less stress because he could save time by using previously created instructional materials that were saved on his computer.
2. Ms.B
Ms. B started with her current school. She said that there was a person in the mathematics department who really used technology and promoted it. One of the mathematics teachers used technology all the time. She thought the teacher was all for it, but he did not push it on other people because it would take time for teachers to learn and feel comfortable using technology. She thought that incorporation of technology was not difficult, but learning how to use it was-especially challenging would be finding time to apply what was learned. 63 Question 5: What is your personal vision for the jitture ofeducational technology at High
School/District/State DOE level?
1. Mr. A
Teacher A emphasized the importance of preparing students for their future with technology skills because most professionals nowadays use computers to some extent.
Therefore, he believed that the more we had it in the hands of the students, the better prepared they would be for their future. The teacher thought that using the Web would be an excellent way to give students more opportunities to access technology.
Teacher training would be one of the key factors for successful technology integration into curriculum. He believed that if teachers first get used to using technology tools for teaching, helping students to use technology tools for their learning would be much easier. Mr. A believed that it was important to train all the pre-service teachers at the college level. He was very skeptical about the situation because he had not seen many new incoming teachers who were ready for technology integration. He noticed that if teachers were not ready by the time they get to their classroom, it was not going to happen any time soon.
He thought that even though teachers might collaborate with each other in teaching, if they were not comfortable using technology, they would not integrate technology. He believed that many teachers were still not comfortable with using technology.
2. Ms.B
The teacher started with the idea of having her own mini computer lab in her 64 classroom. She wanted to her students to be able to go online and "research" whenever they needed to. In addition to that, she envisioned all of her students having their own
TI -83 graphing calculators.
She mentioned what teachers had in their teaching area. There were about four to six computers in each instruction area. The teacher thought that most students and teachers were using those computers to type up documents. To use the computers more meaningfully, she said she would use them as a search tool. She wanted students to look online for the concepts they were studying for better understanding while they were in class.
She believed that providing more training for teachers would be necessary for effective technology integration. However, she also thought that motivating teachers to participate in training was another very important key for effective technology integration. Ms. B wanted to learn how to use technology for teaching, but she hesitated because she did not know how to start. She was discouraged by the amount of time and efforts she thought she had to put in for effective technology integration.
To use technology more meaningfully, ample training and support were crucial.
However, there were two technology coordinators at the school at that time. Therefore, all teachers and students could not receive proper assistance from them when they had questions or simply needed help. She believed that with some help from a technology expert, technology integration would be much more manageable. 65 Question 6: How does your technology background influence your perception ofusing educational technology while you teach mathematics?
1. Mr. A
This topic was covered in the previous discussion. Therefore, we skipped this portion. After he learned how to use technology tools, he had been using any available technology for his teaching. He used various software, Palm Pilots, and Web pages to teach students mathematics. However, he pointed out that teacher training was vital for meaningful and effective technology integration in class.
2. Ms.B
Ms. B was feeling comfortable with using the calculator as an aide as part of her lesson. She was using a video camera to show students what she did and what was on her calculator. She liked to use a video camera in class because she could show the calculator buttons while she explained how to use calculators.
She pointed out some of our math teachers don't know how to use the calculator.
She believed that stay away from it because they don't have to use it. She thought that being able to see how calculators and other technology tools can help their lessons would encourage other teachers to integrate technology into their instruction.
She thought that teachers had to invest much time in learning how to integrate technology, and the amount of time they had to spend would influence teachers' perception of technology integration. She believed that spending too much time was one of the critical factors of and a major obstacle for technology integration in class.
I asked Ms. B why she was willing to use calculators and other technology tools 66 even though she had to spend a significant amount of time with them. She told me some of her good experiences with using technology in her class as her answer. Technology integration would be more challenging for teachers who did not have a positive experience like she had. Ms. B expressed her subtle frustration and disappointment for the negative experiences that many teachers had had and how that prevented student
learning in mathematics classes.
She thought that providing more professional development (PD) opportunities would help teachers learn how to use technology. Ms. B mentioned that she had more PD opportunities in her previous school. She wanted to attend more workshops to learn how to use a graphing calculator and other types oftechnology.
Ms. B believed that there were too many meetings for teachers and many of them did not serve their purposes. She talked about the meetings, " ... you know make a department meeting instead of millions of meetings, none of them work or productive."
She envisioned having a department meeting with a technology person in which teachers would learn how to use different technology tools. She believed each meeting should not be longer than an hour per session. Even though I believed it would be more appropriate to discuss this issue in another research, I wanted to mention it in this study because this issue would directly affect a teacher's attitude toward technology integration.
At this time, I felt that Ms. B firmly believed that teachers at this school were burned out because they had too many meetings to attend and very limited time to improve their instruction. She thought that if she knew how to use technology, she would use it more often and that would change the way she would instruct her students. 67
Question 7: How does your technology background influence on students' learning mathematics?
1. Mr. A
The teacher strongly believed that making technology available for each student was very crucial and that would be the beginning point of effective technology integration. He believed that students would have stronger ownership for their own work when they use technology tools. He picked the laptop program as a good example of using technology for learning mathematics.
He also had positive experiences with PowerPoint presentations. If he required students to use PowerPoint when they presented a problem, they thought about their problems differently and gave more thought before they presented their problem to the class. Even students who normally did not want to study mathematics began to think about mathematics problems more seriously and focused on problem solving when they used PowerPoint presentations.
He believed that the PowerPoint integration motivated students to complete the work thoroughly and they had a positive learning experience. He recognized the influence of technology when the quality of student work went up. He said, "You know, so, it was really unique to see that they are thinking at a different level."
I noticed that students spent more time preparing their work when they presented their work with technology tools. In addition to that, it would be challenging for students because they had to learn how to use new technology tools as well as the mathematical 68 concepts they were working on. Students would learn mathematics better if they focused and persevered with solving given mathematics problems. If the learning process could be fun, students would learn more effectively since they focused on the problem longer.
When students used technology tools for learning mathematics, they enjoyed learning more since they were doing work in unconventional ways.
2. Ms. B
When I asked if students would learn more because they knew how to use technology, she started to explain that students would learn better when students have technology in their hands. First, she said mathematics learning would be enhanced by the
speed and accuracy when students use technology tools. She explained,
Yes, because like discovery functions, how do you discover different functions unless you have some kind of graphing tool? To type in different functions, there is no other way to do it. I mean, yeah, you can do it by hand. We have worksheets we use but it takes forever and it's not instantaneous and as quickly as if they can see how the function is moving. It was opening and closing up and down, and of course they are going to graph that quicker than when they took them, and you know, an hour to just graph four different functions. And then they say, 'Oh, now I can see it.' So, I definitely think it enhances it, and it would help improve their retaining of the information.
Ms. B strongly believed that technology would enhance the communication between a teacher and students. She said students would ask as they work on problems.
Some questions they frequently asked were "What did you put in?" "What's that?" "Is the function shifted?" She added that students would discuss with each other about the different functions, the way they are moving if they work with their graphing calculators.
She strongly believed that technology would definitely increase the communication between them because it is instantaneous. 69
Question 8: What do you believe motivates or would motivate mathematics teachers to use more educational technology for their instructional purpose?
1. Mr.A
Mr. A believed having teachers visit and learn from each other was an important part of technology integration because it would motivate teachers to integrate technology into their instruction. He believed that, by watching good examples, teachers would use more educational technology for instructional purpose. When a teacher could visit and see how other teachers integrate technology, he or she would have a better idea of technology integration.
2. Ms.B
Ms. B' s idea for motivating teachers to integrate technology into their instruction was to provide them with clear and simple instructions with many good examples. She emphasized the flexibility and the practicality of the application to accommodate the differences of teachers. She believed teachers would be more interested in incorporating more technology if they could learn how to use the tools without spending too much time.
Ms. B thought that not many teachers would change the way they taught even after an expert explained to them how they could incorporate technology in their curriculum and what kind of technology was available for them because they were simply too busy to integrate their own. Therefore, giving practical training with applicable examples would motivate them. She said that she had some of the technology tools in her classroom because not many teachers were interested in using them. She thought that 70 having not enough time was a possible reason why not many teachers were interested in technology integration.
When I asked Ms. B how teachers could motivate students to use technology in their learning, she replied immediately,
Well, having it available in classroom. These kids seem really interest in technology anyway. They all have a MP3 player; they all know how to do all these crazy things on computer already. They already got motivated by TV, you know, wanting to be upon current technology but I just think that having it available. A lot of them will get on there and figure things out without having needing an instruction and without it they can do it, you know.
She strongly believed that student were ready for technology integration in their mathematics learning. If tools were ready and available, students would use it for their study and if teachers were ready to use technology, students would be willing to as well.
When technology was incorporated, the quality of teaching and learning mathematics improved.
Class Observations
Classroom The two classrooms observed were located in two different buildings. They looked very different because the AP statistics classroom was one of the resource rooms and the AP calculus classroom was part ofa larger pod area. However, the space that was occupied by the students was about the same.
The AP calculus class was sharing a large traditional classroom space with another class. There were no dividing waIls between the two classes. Therefore, three rolling whiteboards (B) divided 45-50 students into two classes. Everyone in the pod area 71 could hear various levels of noise from the other side. The entire class was in a 12 feet by
15 feet open space. The teacher generally moved around the class space during the class time but when he gave a lecture, he used the whiteboards that were used as a divider. The
TV and VCR unit was placed close to the entrance next to the students (A). In this class students sat so that they faced each other during the class time. o o o o A ...... 8 ( J
Figure 2. AP calculus Classroom
The AP statistics classroom, however, was more traditional. The room had four
walls and two doors. A teacher's desk and a TV and VCR unit were in one corner of the
room (A). Ms. B put her overhead (C) in the middle of the front section of the classroom.
Students normally stood in front of a whiteboard (B) or in the teacher desk area (A) when
they gave presentations. In this class students generally faced the front but when they
discussed problems they situated themselves differently to accommodate their discussion. 72
Figure 3. AP statistics Classroom
Week 1 (October 19,2004 - October 28,2004)
1. Lessons
In the AP calculus class, students were learning about Limits, Derivatives, Anti
Derivatives, and Definite Integrals. In the AP statistics class, students were learning about Designing Samples, Designing Experiments, and Simulating Experiments.
2. Summary
As I started my videotaping, I was careful because I did not understand how students were interacting with each other and with their teacher without interrupting what was going on. That was why my first day video recording was done in the back of the classroom.
In Mr. A's class, on the second day, I recorded the class among the students and my camera was set much closer to the students compared to the first day. I asked the teacher for a copy of the student assignment. I also asked Mr. A for a copy of his warm up questions on the PowerPoint to follow what they were doing and also because I wanted to use them for small group interview sessions. 73 In Ms. B's classroom, I also changed where I was videotaping to the front of the class in order to capture students' facial expressions and interactions during the class. I also asked for student assigmnents from the teacher because the actual student works helped me to remember what they did when I had student interview sessions.
On October 22, 2004, in Ms. B's period 2, students were working on problems with their graphing calculators on their desks. I noticed that many students began to use their calculators after the teacher mentioned their use. I also noticed that students did not actively use their calculators to solve problems until their teacher reminded them to use it. Some students, however, still did not use their graphing calculators even after the teacher encouraged them to use it. I became very curious as to why students did not use their calculators until their teacher reminded them. Did they know that they could use their graphing calculators to do work in class? Did they know how and when to use their graphing calculators for problem solving?
On October 26, 2004, Ms. B was absent and a substitute teacher gave students a quiz and a worksheet to complete in class. While they were working on the worksheet, I had a mini interview session with the teacher's permission. In the videotaped interviews, students answered a few questions about learning mathematics and using technology. In the students' responses to the questions, I found very strong opinions regarding the purpose oflearning mathematics and their defInition of technology. I noticed that many students expressed philosophical defInitions when they were asked what technology was.
In Ms. B's period 6, most of the class activities were identical to her period 2. I also had a mini interview session while students worked on their worksheet. When I 74 compared the responses of this period's students with that of students in period 2, I noticed that their answers were more practical.
I was wondering why students' behavior in period 6 was generally poorer than students in period 2. Was it because there were more students in this period? Was it because this period met after lunchtime? More students of this period also seemed to rely on their teacher as well.
On October 29, 2004, I noticed Ms. B used multiple technology tools to make her lesson more presentable and appealing to students. She used a video carnera and TV monitor unit, an overhead projector, and the whiteboard. I thought those tools were very suitable to communicate with students visually. However, at the same time, I was wondering why students did not use the technology resources more to participate in class and learn actively instead of being a passive learner.
Week 2 (October 29. 2004 - November 04. 2004)
1. Lessons
During this week, AP calculus class covered Anti-derivatives, the Riemann Sums
Theorem, Linear Approximate Differentials, and the Quotient Rule. In the AP statistics class, students learned The Idea of Probability, Probability Models, and General
Probability Rules.
2. Summary
In AP statistics classes, I noticed that students were more attentive when there was a fonn of visual representation of the idea they were talking about. When the teacher explained a problem using the video camera to show the same problem in the textbook, 75 students paid more attention to the problem and they seemed to understand the problem better. A similar change occurred when the teacher used an overhead projector to present a problem.
When students were working on an assignment called "free response," their interest level seemed to drop as they worked on the particular problem compared with the time when their teacher was using a visual aid. Was the visual aid making students more focused and pay attention to the content, or was it just because their teacher gave important instructions that they needed since the information would be important for the future?
On October 29, 2004, when students asked some questions about the previous homework assignment, Mr. A explained on the whiteboard. I was curious as to why he did not use available technology tools such as the computer or video camera and the TV monitor. As I was wondering, the teacher began to use the technology tools with a graphing calculator. When students could see the visual representation of the problem. they instantly grasped what Mr. A was talking about regarding the problem. It was great contrast between using the whiteboard and using the TI-83, video camera, and TV monitor. When Mr. A used the whiteboard, students had a hard time understanding what he was talking about. However, when he presented the same problem using technology tools, the students' understanding was significantly improved.
After I observed this session, I thought that a clear and accurate visual representation of the mathematical concept would be the key to improving students' understanding. Would technology integration help classroom teachers achieve the ideal 76 environment for teaching mathematics? There were some challenges that the teachers faced in using technology in his teaching of mathematics. Teaching students with various levels of understanding how to use the technology tools was one of them. There was a wide range of understanding of how to use the TI-83 to figure out a problem among students. For example, one student needed help to understand the right steps to create a simple graph on her TI-83, whereas another student already knew how to create a far more advanced graph. I was curious to know if there would be any difference in a student's learning and understanding mathematics depending on the level of their understanding of how to use their graphing calculators.
On October 29, 2004, in period 2, Ms. B used the whiteboard and the video camera and TV monitor combination to explain the problems. Students, in contrast, used the overhead projector to present their problems. I thought that it would be even more meaningful and beneficial if students could use the technology tools for learning. Is there any way we can promote having students experiment with their ideas with new technology so they can enhance their own leaming and generate more communication among them?
On October 29, 2004, in period 6, it took longer for the students to settle down and get ready for their lesson at the beginning of the class. I believed the class schedule might be a part of the reason because the class always met in the afternoon right after the lunch hour. This class seemed to be more challenging to teach as well. I would like to see if technology integration would benefit the students' learning in this class.
When Ms. B explained and had students discuss the key steps to solving 77 problems, the students were fully engaged in the discussion and actively participated in it.
Their attitude toward working on the assignment was much better than the way they spent their individual work time. Students spent most of their individual work time talking about nonacademic topics, but they were more focused on what they were studying during the discussion time.
I observed that students were excited about what they were learning and more attentive to their teacher when she used a graphing calculator to explain problems. More students were paying attention and did their work during the class time compared to when she used the whiteboard or explained problems orally only.
Overall, I noticed that direct instructions were benefiting more students since they
seemed to be more focused with direct instructions. Students seemed to understand better when the teacher gave them direct instruction instead of allowing them to explore the ideas. I would like to study if technology integration would be beneficial in this situation.
Week 3 (November 05. 2004 - November 11. 2004)
1. Lessons
During this week, the AP calculus class covered Indefinite Integrals, Integral of a
Function, and Upper and Lower Sums. In the AP statistics class, the students learned
Foundations ofInference, Probability Models, and General Probability Rules.
2. Summary
Ms. B planned a special assignment (Special Problem 5D) that required students to use PowerPoint to present their solution. The assignment was planned after I discussed
with her promoting students' use of technology in their learning. It indicated that there 78 was a sound communication between us. Ms. B was willing to implement different ways of teaching with technology integration.
Ms. B wanted to implement technology in her teaching and I worked with her closely to achieve that objective during my observation. I hoped that this initiation could give her a good idea of how to use technology in her regular teaching in the future. It was interesting to watch how students and Ms. B used technology tools for communication in class because she used different technologies to present different ideas.
On November 05, 2004, Ms. B used the whiteboard to explain how to use a graphing calculator to generate a random number. The idea of generating a random number included multiple steps. I was wondering if she could use a PowerPoint to explain the same information. How would students react if this concept were presented on PowerPoint?
Ms. B planned an after-school session to help students who did not know how to make a PowerPoint presentation. Students also could get help on their special problem from her. I was excited for an opportunity to observe students while they used technology to work on a mathematics problem after school. However, no one showed up after school, even though Ms. B and I waited for over half an hour. I thought that students did not , show up because of bad weather.
On November 08, 2004, Ms. B gave students another opportunity to work on the problem with their teacher. This time the weather was much better than the previous session but, again, not a single student showed up. According to Ms. B, most of the students had an access to a computer at home for the project. Does that mean students are 79 ready to use more technology for their learning if they are given the right opportunity to use it in their learning?
When students worked on a problem as group, they used poster papers and all students were participating actively. The class ran very efficiently because students used the time wisely and were able to complete the given assignment within the given time. It was a very positive experience for me to observe a class full of positive attitude towards their work.
When the teacher explains the importance of understanding the AP exam requirements by using a problem, she used the camera and TV monitor to demonstrate the concepts and the way to start the problem. I thought that was a very efficient way to communicate with students for clearer instruction. When technology tools were integrated effectively, the transitions between different class activities were seamless and well organized. I noticed that giving students multiple opportunities to practice the same ideas to motivate them was important.
In the AP calculus class, students could ask or discuss any problem they needed help with throughout the class period. There was no set time to discuss a particular assignment as in the AP statistics class. They had warm-up problems from the beginning until the end of the class. While they worked on the warm-up problems, they also had an extensive discussion over their previous homework problems. Mr. A also introduced a new concept in the middle of their discussions of warm-up and homework problems.
From an outsider's point of view, it seemed a quite a bit confusing. However students did not appear to be confused and they were following their teacher's instructions and 80 participated with the discussions.
One thing that was vel)' clear to me was students' extensive usage of graphing calculators. Most of the students already had enough experience with their graphing calculators before they entered this class. Therefore, it seemed like using graphing calculators was not an option for most of them.
When a student showed how he could use his graphing calculator and other technology tools to explain how he drew his graph and found the area under the line of the graph, the level of the student's knowledge on how to use his graphing calculator impressed me. It was also clear to me that students could use the same technology tool for sharing and teaching purposes when they had the opportunity.
Even though Mr. A had great knowledge of using various technology tools, the students in the class did not seem to be much motivated by the usage of technology by their teacher. I wondered if students would react differently when they became active technology users.
Week 4 (November 12, 2004 - November 18, 2004)
I. Lessons
During this week, the AP calculus class covered Exploration, The Intermediate
Value Theorem, The Mean Value Theorem, and Rolle's Theorem. In the AP statistics class, students learned Discrete and Continuous Random Variables and Means and
Variances of Random Variables.
2. Summai)'
Mr. A had more experience using technology and integrated technology into his 81 teaching more than Ms. B. However. the integration was limited to showing what was on his calculator for most of the time. I believed that he could teach more effectively if he used some software such as the Geometer Sketchpad or Fathom.
Mr. A let his students work on the warm-up problems by themselves for a very extensive time. I questioned the way students started the class because I could not see any immediate benefit of letting students work on a set of problems more than 20-30 minutes without proper guidance from their teacher. I was curious about the long-term influence of this type of study pattern on students' learning mathematics and if they were correlated.
I frequently noticed that the basic mathematics ideas were reviewed in this class because many students were lacking some basic concepts. Could Mr. A use software to give students practice with the basic concepts instead of using class time when they were supposed to use it for advanced concepts? With the integration of academic software, the class would have more time to talk about the appropriate calculus topics because students would use academic software to review any non-calculus topics during non-class time.
I noticed that students lacked Algebra 2 concepts because an Algebra 2 course was not offered in this school. Students had to take AP calculus without understanding
Algebra 2 concepts that were vital for success in the class. Would the situation be improved and students be properly prepared for this class if an Algebra 2 course was offered?
Students were very energetic and willing to share their thoughts and ideas with each other. They seemed to enjoy working together and communicating with each other. 82 If a student asked the teacher a question, the entire class discussed the problem. Students' communication would be more effective if they could illustrate their ideas in more than one way. Technology integration in mathematics learning would enable students to accomplish it.
From close observations of interactions between teacher and students, I found an interesting pattern. The teacher seemed to interact and communicate more with students who were active and vocal. They asked most of the questions during the class time. They, therefore, naturally received the teacher's attention and feedback more. On the other hand, quiet students who were not expressing their ideas in class did not communicate actively with the teacher, nor did they receive any feedback from the teacher.
Some students were active learners and the others were passive learners depending on their personality. I believed that students' personality should not interfere with their learning. Did technology integration help students who were introverted and lacking in motivation to learn better and minimize the achievement gap between active and passive learners?
I also noticed that students who were actively participating in class activities tended to have more discussion and conversations with their peers about nonacademic topics during the class time or in between class discussions. Was there any correlation between students' verbal skill and their mathematics achievement? Was there any correlation between students' verbal skill and their tendency or attitude toward using technology in their learning?
In the AP statistics class, in period 2, students spent the time more efficiently 83 when they had an opportunity to work on the computers to complete the special problem and they achieved more by the end of the period compared to students in period 6.
I thought the direct instruction was essential for learning, but I had a few questions regarding direct instruction when I saw that students were simply copying the information off the board without deeper understanding. What impact did it have on students' learning and how did it prepare students for their AP exam at the end of the school year? Was there any other way this could be done to encourage students to be more independent learners? Students could share their answers and ideas with each other to come up with reasonable solutions and the teacher could complete it at the end of the discussion instead of the teacher providing the correct answers.
When Ms. B demonstrated one problem by drawing a diagram on the whiteboard.
She was teaching a very vital idea, for which had to prepare the diagram beforehand, and explained how to come up with the diagram. If she used computer software such as an
Excel program, she could save the time of preparing the diagram manually.
On November 16,2004, in period 6, students turned in their special assignment on a disk. Students completed the assignment without attending any help session. According to my observations, students already knew how to use technology to complete the assignment.
After the special assignments were collected, Ms. B explained what methods should be taken to understand the problem better and to come up with a reasonable answer and explanations. The way she explained to students was by drawing a diagram on the whiteboard. I thought that using the right technology tools could help her to do the 84 job more efficiently. If Ms. B chose to use the whiteboard over other technology tools, what could be her reason? What were possible advantages of using the board over the other technology tools?
In this class, presentation was a very valuable learning tool for students.
Presentation was used for individual attention and correction for any misunderstanding students had. Through presentations students, also clarified things they were unsure of.
Presenter actively communicated with the teacher and the rest of the class and received very valuable feedback from them. Unfortunately, not all students presented. Would technology integration motivate students to present more?
Week 5 (November 19, 2004 - November 25, 2004)
1. Lessons
During this week, the AP calculus class covered Application of Definite Integral,
Properties of Definite Integral, and Derivation of Simpson's Rule. In the AP statistics class students learned Binomial Distributions and Geometric Distributions.
2. Summary
Mr. A did not use everything he had in the classroom for his teaching. An overhead projector was one of them. He had an overhead projector in a comer of the classroom. He did most of his explanations and communications with the video camera and TV monitor for some reason. The Internet connection was another one. There were 6
- 8 computers with Intemet connection in his classroom. However, the teacher did not use it to enhance his teaching or students' learning.
Students had a problem that required some trigonometric theorems to solve it. 85 Few students remembered the theorems without any help. Most of the students could not solve the problem because they did not remember the required trigonometric theorems. In this situation, I was not sure if students had to memorize the theorems to solve the problem. What would be changed if the theorems were given to use for problem solving?
Would making theorems available for students prevent them from learning the main idea?
I believed the usage of the Internet would allow students and teacher to access necessary information, as they needed it. I noticed that the Internet was available but was not used for study. Why not?
After students had enough time to do warm-up problems, Mr. A went over the problem with them. I wondered why he did not allow students to present their solutions instead of giving his own ideas.
Was there any correlation between students' participation and their usage of the graphing calculator? From my observation, I noticed that they were somewhat related.
When students frequently expressed their own ideas and participated whenever they had opportunities, they seemed to use their graphing calculator more frequently than those who did not speak much or did not participate during the class.
Students seemed to use technology equipment more frequently when their teacher explained how to use it with a clear example. For example, students paid attention and learned how to generate a graph of a function when their teacher showed how to manipulate the graph by adding points through the TV monitor.
Many students used their graphing calculators because a graphing calculator was allowed for the group test. However, for students who did not understand when they 86 needed to use their graphing calculators, it was useless. It was the time for students to remember as much as they could to answer the test items correctly. I was not sure if using a graphing calculator helped students retrieve the essential information they learned in class. Did students use their calculators for simple calculations?
Students had to use their graphing calculators to solve certain problems because they learned many different functions on their graphing calculators and certain problems required use of one or more features on the graphing calculator.
Some students asked the teacher to clarifY the meaning of the problem that they were working on. The teacher responded to the group of students immediately since she was walking around the classroom. It was an instant communication between teacher and the students. The communication between students and the teacher looked seamless since they stopped working when they had a question.
Many times, secondary students expect an instant response from their teachers. I wondered if the communication would flow with the same level ofsearnlessness if students used technology instead of poster paper. Instant communication would be one of the important factors for successful teaching at secondary level. The implementation of technology improved the communication dynamics in this classroom.
Using technology in mathematics teaching possibly influenced student learning and the communication between teacher and students. Dynamic communications between teacher and students was the key to success in secondary teaching. Giving effective communication tools would be one way to providing dynamic communication for them.
On November 23,2004, I pondered about students' information retention. To 87 achieve high scores on a test, students had to remember what they learned as they studied. What helped students retain the information so they could remember the information longer with better clarity? What would be the correlation between technology integration and information retention rate?
I thought that motivation was one of the burning issues in this period. Students who did well with their presentations were not necessarily those who paid attention more than other students in class. In other words, they did not normally pay attention when instructions were given. Also, when multiple presentations were given, some of the students did not pay attention to what was presented to them.
When students work collaboratively in a group setting, they seemed to be more motivated and focused. They seemed to enjoy interaction among group members. They were especially motivated when they had a challenging assignment. How would technology integration accommodate these conditions and keep students alert and excited?
The two periods had very similar approaches for instructional purpose, except that the teacher took a more experimental approach with the period 2 class while she took a more teacher-controlled approach with the period 6 class because she had already taught the same lesson with period 2 by the time she taught her period 6.
On November 24,2004, the class had a "Casino Activity" day. Instructions on how to play three major gambling games were given on the PowerPoint. After the teacher's brief instructions, students played "blackjack, crap, and roulette" games as group. A worksheet was prepared for students to complete as they were playing each 88 game. Students took turns according to the time limit given by the teacher.
Week 6 (November 26, 2004 ~ December 02, 2004)
1. Lessons
During this week, the AP calculus class covered Finding Approximate Integral by using Trapezoid, Application of Simpson's Rule, and Application of Riemann Sums
(Midpoint) Rule. In the AP statistics class, students learned Sampling Distributions,
Sample Proportions, and Sample Means.
2. Summary
In the AP statistics class, on November 30, 2004, Ms. B used the TV and camera more frequently for explaining and demonstrating purposes. Ms. B used to present problems on the overhead projector. Students were still using mini whiteboards to present their problems. I wondered why students did not use the same tool that the teacher was using. Does using technology help students understand the questions better? I also asked if educational technology could facilitate better communication between students and teacher?
Students presented the problems that were assigned before the Thanksgiving weekend. They used the mini whiteboards to present their problems. I wondered, "If they have a choice in selecting presentation tools to present the problem, would they do their presentation differently? What would be the difference?"
I also noticed that using mini whiteboards might cause some presentation issues.
Students were talking either to the board or to the teacher instead of facing toward their audience. I thought that was one of the problems students had to overcome. When 89 students present, they used multiple whiteboards to include all the information. At the same time they have to look at their work and do not face front toward the class.
On November 30, 2004, I had noticed some differences between period 2 and period 6 since those periods were at very different times of the day. In this period, the set up and the problem the students had for warm-up were about the same as those period 2 had. However, the way students performed and their general behaviors were different.
Students learned the concept of "infinity" as part oftoday's lesson. Students in both periods 2 and 6 could not visualize the concept as it was introduced to them. The teacher attempted to explain what it meant by using the whiteboard only. Many of the students could not understand the concept since they could not visualize it in their mind.
However, as soon as they picked up their graphing calculator and tested the concept by using the graphing feature, they could understand the concept by visualizing it. Students loved what they could do with the graphing calculator.
Week 7 (December 03, 2004 - December 10, 2004)
1. Lessons
During this week, the AP calculus class covered Derivatives and Logarithmic
Differentiation, Symmetric Difference Quotient, Trapezoidal Rules, and Relationship between Continuity and Differentiability. In the AP statistics class, students learned
Sampling Distributions, Introduction to Inference, Estimating with Confidence.
2. Summary
In the AP calculus class, the students were clearly divided into two groups. One group of students was actively participating in class discussions. This group of students 90 was the ones who were actively involved with the class activities and contributed the most to the discussions. They expressed their ideas and actively communicated with their teacher as much as they could.
Another group of students were the ones who did not communicate with other students or their teacher. They generally observed instructions without communicating with their teacher. They usually paid attention to the instructions and did their work but did not express their ideas or ask any questions. Which group would perform better when they used educational technology in learning mathematics? I would like to have them talk about this part when I have small group student interviews.
On December 07, 2004, the students worked today as they did when they took their group test. Students communicated if they wanted to have an open discussion for any topic. However students did not interact as much as they did on their group test.
Students had two possible reasons for less interaction on this day. One possible reason was the fact that they were not required to submit any work that needed to be graded.
Another possible reason was that students did not understand the topic they were studying well enough to discuss with others.
If students did not interact with each other because they did not understand the topic, I thought that technology integration could possibly generate more communication and interactions among students. I read many studies talking about what type of instruction would fit best with technology integration. Technology integration would best fit and enhance self-discovery learning. It would fit well with the constructivists' view of learning and teaching. Would educational technology motivate students to interact more 91 when they had challenging problems?
Technology integration would help the students in this class significantly because they were self-regulated leamers. However, students did not use technology to extend their understanding of mathematical concepts that were introduced in the lessons. They only used their graphing calculator to confirm their answers.
Students' communications would be enhanced by technology integration because technology would allow them to illustrate the necessary steps of problem solving visually. For example, a student who did not quite understaod how to solve a problem was encouraged to present the problem. Because he was not sure, he attempted the same problem in several different ways. He used the whiteboard to explain his solutions. The class had a hard time understanding what he was talking about for most of the time. Ifhe used the video camera and TV monitor to show what he had on his graphing calculator, the class would have understood his explanation much better.
Most of the time, students were busy copying the information that was presented by the teacher instead of paying attention to the instructions and learning the meaning of what they were copying. If the same information were available on the Internet, students would just pay attention to the instructions because they could download the necessary information from the web when they needed it.
In the AP statistics class, using technology generally improved the efficiency of teaching. However, if the teacher did not know how to use it, technology integration might cause him or her to spend more time in planning for an activity. For example, the teacher covered up a part of the screen in the middle of her presentation and told students 92 not to look at the infonnation on their handout because she wanted students to attempt to solve the problem before they looked at the additional information. This type of task could be done simply using the "hide screen" feature built into PowerPoint. Since the teacher did not know that, I suggested to her that she use it next time.
In this class, there was a student who constantly achieved highly. I observed that she was also constantly using her graphing calculator throughout the class period. I began to wonder if students' achievements and their usage of a graphing calculator were correlated somehow.
On December 03, 2004, in period 6, students usually had more interaction with each other than in period 2, even though they had the same lesson content. Students here tended to use graphing calculators more frequently compared to period 2. What could be a possible explanation for the differences?
On December 03, 2004, as I watched what Ms. B was doing to demonstrate how to use a graphing calculator to students, I thought that she could videotape what she was explaining and show the recording to both classes since she taught two classes with the same information. She could save time for the demonstration and she could repeat the information if students did not understand.
I thought using technology effectively would be more important than using more technology for teaching. Ms. B taught the same concept differently in her period 2 without using much of technology. She made mistakes while she was demonstrating and that was why she decided to change the way she demonstrated in this class. In period 2, the teacher's error came out because she did not use the technology to do the calculation. 93 From the experience. she changed her demonstration with educational technology. It worked better and she corrected her mistake by using technology effectively. Awesome!
Also when students present their work, they gave the presentation to the teacher or to themselves instead of interacting with the class. Therefore, I thought deeply about whether technology would be able to help students to present properly and have more interaction with the audience while they made presentations.
On December 07, 2004, more interactions were generated between students and their teacher when the teacher used word processing software to communicate with the class. I knew that educational technology promoted more interaction and communication among students because they could see what they were talking about. Using graphing calculators allowed students to have instant feedback, and they had a meaningful discussion and clarified what they did not understand. However, how would I know if students really understood how to visualize the concept of the problem by using graphing calculators?
Second Teacher Interviews
This is the second interview with the two teachers (Mr. A and Ms. B) after the entire class observations were done. In this interview, there was not a set of pre-selected questions for the teachers to answer. However, the main theme of the discussion was technology integration in mathematics instruction.
According to my plan, they had an opportunity to view videotaped classroom sessions on a laptop computer. The post-lesson video-stimulated interview technique 94 (Clarke, 2004; Emanuelsson & Clarke,2004; Shimizu, 2002; Tomer et a1., 2005;
Williams & Clarke, 2002) was used for the second interview sessions in which the teachers watched videotapes of them accomplishing a task and were interviewed about their thoughts while doing it.
The interviews were focused on how the two teachers viewed their own teaching practice and the role of the educational technology in their teaching. As they watched their own teaching, the teachers made any comment that came to their mind for any issue.
Mr. A In the first interview, I noticed that Mr. A had more experience and knowledge of using technology for teaching mathematics. He also understood that technology integration into instruction might enhance mathematics learning. As we discussed, he still talked about various kinds oftechnology integration he attempted in class, such as using a video camera, the Internet, and his PDA. He was more comfortable with technology integration into his instruction than Ms. B was. He carefully planned technology integration to improve students' learning.
He also believed that technology helped his students visualize mathematical concepts. In this interview he continuously emphasized the use of technology to visualize mathematical concepts. He put emphasis on using a graphing calculator in calculus class because he believed that it was an effective way to visualize the process of graphing a function. He considered using a graphing calculator second nature for calculus students.
He made sure all students had access to a graphing calculator and he let them borrow a calculator if they could not buy their own. 95 He used a video camera extensively to present parts of his instruction and when students shared their ideas. Video camera was used to maintain clear communication between the presenter and the audience in class. For example, Mr. A could explain a very hard to explain theorem effectively without recreating the graph because he could transfer the actual graph from a graphing calculator to TV monitor by using a video camera.
Mr. A incorporated various types of technology tools into his teaching because he believed that technology integration would help students learn in his class. He also thought that technology integration would enable him to differentiate his instruction for students with different levels of understanding. For example, he had online tutoring sessions with students who did not understand the concepts in class. He used the computer to generate multiple versions of a test for students with different levels of understanding.
He searched the Internet for practical knowledge that he needed to improve his instruction and theoretical knowledge to understand what research said about mathematics teaching. For example, he frequently accessed the NC1M website to understand available mathematics instruction. However, he did not make decisions on what to teach based on what he found from his search. He did not mention how information he searched on the web improved his instruction.
Mr. A even mentioned using email to communicate with other calculus teachers in the nation to learn how to teach his class more effectively. However, he did not talk about how it helped his teaching while he was watching his class observation video.
What he discussed with other teachers may not be related to what he found in the video- 96 recorded sessions.
He believed technology integration would be successful when a teacher actively used technology in class because a teacher's positive attitude toward technology integration would have an influence on student's attitude toward the use of technology.
Mr. A said he became a more active technology user to prepare lessons, to teach, and to communicate with students and parents after he taught the laptop program.
Ms.B In this interview, Ms. B showed deeper understanding of technology integration in her instruction. She talked about how technology would influence students' learning mathematics more than in the first interview. She had more experiences of incorporating more technology into her teaching.
Especially, Ms. B had a strong opinion on using graphing calculators to learn AP statistics. She also believed that students could understand mathematical concepts better when they could visualize them and a graphing calculator helped with visualization.
Ms. B still maintained her motivation of using technology for her teaching and to improve student learning. Training teachers so they understood how to incorporate technology in their instruction and finding time for it were still a major issue for successful technology integration. Even though she thought time remained an important factor, Ms. B suggested that using part of the current school meeting time would be an effective way of having more teachers get involved with technology training.
Ms. B continuously increased her incorporation of technology as she learned how to use them. She realized more how she actually integrated technology into her teaching after she watched her own teaching sessions. She was open-minded to suggestions on how to use different technologies for teaching. She seemed to enjoy using technology for teaching more because she became familiar with certain types of technology tools by the end of my observations.
Ms. B believed technology integration would help her students understand and learn mathematics better. Especially, using a graphing calculator helped students visualize mathematical concepts and retain the information longer. She also strongly believed technology integration would generate more communication and change the way she would teach and students would learn positively.
Student Small Group Interviews
I conducted student small group interviews while I was teaching because my sabbatical leave was limited to one semester. I originally plarmed 10 small group interviews. I picked three groups from the AP calculus class, three groups from the AP statistics class (period 2), and four groups from the AP statistics class (period 6).
However, only half of them were able to participate as scheduled. The other five groups could not participate due to scheduled conflicts.
Groups I, 2, and 3 were the groups of students who took AP statistics and not AP calculus. Group 4 was a group of students who took both AP calculus and AP statistics at the same time. Group 5 was intended to consist of three students who were taking AP calculus and not AP statistics. When I selected interview groups, I carefully selected three students to put in the same interview group after I completed my observations. 98 The students in the same group were usually sitting and working together collaboratively in class. Therefore, I believed, they would share their honest thoughts and opinions when I interviewed them because putting them together could generate a dynamic and worry-free atmosphere. However, in the actual interviews for each of groups 1,3, and 5, only one student was available and interviewed because of some schedule conflicts.
Three interviews (groups 2, 4, and 5) were conducted in my classroom after school. I interviewed groups 1 and 3 in an assistant principal's office because these students were only available during the school hours. Even though I interviewed them during my non-teaching period, I could not use my classroom because I shared my classroom with another teacher and he had to teach a class during that time.
Lastly, I prepared twelve questions to conduct the student interviews. However, not all questions were directly related to my research questions. Therefore, I reported student responses to those questions (questions 8, 10, I I, and 12) that were directly related to my research questions with direct quotes of what students told me in their interviews. However, I summarized student responses to those questions that were indirectly related to my research questions without quoting students' actual words.
Question 1: What is involved in learning mathematics?
I. Summary
Students believed having basic mathematical knowledge was an important part of
learning mathematics. They thought that without having a strong foundation, they would 99 not be able to make progress. Most of the points were matters of individual responsibility.
Along with having a strong foundation, some of them also talked about being able to apply their knowledge in more practical situations. By applying the basic knowledge in more complex situations, they thought that they could strengthen their basic knowledge by understanding how they work in various situations. I personally thought that student 2-
1 gave the most comprehensive definition of learning mathematics. She said,
Yeah. So, I guess, overall math is that you to remember the main concepts and after that you can apply to any situation that you have to deal with in your life or school in that matter.
Question 2: How do you learn mathematics?
1. Summary
When student learn mathematics, most of them told me that they learned by looking at a teacher's example and practicing examples from their textbook. However, some students preferred to learn by visualizing the concepts as they approached the problem. Groups 2 and 4 gave one interesting idea. Group 2 students talked about applying the idea they learned in a practical situation and that was how they could learn best.
In the discussion with group 4, they mentioned that teaching the ideas they were learning helped them understand the ideas better. It was clear to me that as students applied the ideas in a given situation, they would understand the ideas more clearly and deeply. 100 Question 3: What do you think are some benefits oflearning mathematics?
1. Summary
Many students compared learning mathematics with learning a universa1language and having common knowledge that society shared. By learning mathematics, students believed that they would function effectively in society. Several groups of students also shared a common idea of learning mathematics. Students believed that learning mathematics would prepare them to become more competitive and give them an advantage when they study in many related fields, because they would understand those fields better compared to those who did not have mathematical knowledge.
A few students pointed out that they would use more mathematical knowledge as they grow older simply because they would have to comprehend and use more information around them to be able to be employed.
Question 4: Which ofthe following aspects ofmathematics have you learned in your high school mathematics classes? (Basic mathematical skills, Series ofmathematical concepts,
Social skills such as how to communicate, and How to apply mathematical skills in life)
1. Summary
Overall, students recognized they learned and used the basics of mathematics because they needed them to learn higher-level mathematical concepts. Some students strongly believed that the IMP curriculum taught the basic mathematical skills. For series of mathematical concepts, many students went back to discussing how they did in IMP classes. 101 They strongly believed that they had good opportunities to practice social skills.
They all had extensive experiences in working collaboratively and presenting to the class.
When they talked about applying mathematical skills in life, students shared many good and practical examples. I strongly felt that students clearly understood the importance of learning mathematics. I hoped that the majority of the students could experience the positive experiences that were shared here.
Question 5: How do you know when you have learned something?
1. Summary
There were a few general ideas among interviewees about how to confirm when they learned mathematical concepts. It was rather simpler than I anticipated. Most of the students said that when they could solve a given problem or explain it to someone else, they would know that they understood the concept clearly. Another way to check was to apply the idea in a situation. They also thought that they had learned it if they could successfully apply any concept to a given situation.
Question 6: How do teachers help you learn mathematics?
I. Summary
In general, students perceived the teacher's role in their learning mathematics to be a person who possessed the necessary knowledge and who showed good examples of what they were supposed to learn. Whenever they needed help on any mathematical problem, the teacher could help them by explaining problems by using meaningful 102 visuals.
Students also thought that their mathematics teachers could help their learning by simplifying the ideas they needed to learn. Careful planning and designing lessons were also included in their opinion about the teacher's role in mathematics classes. In other words, if their teachers made their process of learning mathematics less difficult, they would consider that their teachers were helping them.
Question 7: What is the student's role in the mathematics classroom?
1. Summary
The students strongly believed that they should be self-motivated and self directed learners. They said that students had to do all given assignments faithfully and seek help if they did not understand any part of the assignments. The students also believed that having a positive attitude was one of the student's roles for successful mathematics learning.
Question 8: What is the role oftechnology in learning mathematics?
1. Group Responses
Group 1 (1 student) Student 1 believed that technology was a learning aid and study tool that helped her retain information longer because she could focus more on the actual problem solving. When she used technology, she did not have to worry about making any mistake such as miscalculation or skipping any steps of problem solving.
Student 1: It helps when you do something quickly, like calculators really help 103 because even with all the equations ... instead of having to take halfan
hour to enter the data to find the mean or solve it, you can solve. If you
canjust enter it in a calculator .. .it helps you solve it. So you can be more
efficient.
When I asked for an example, she shared how she used a computer and
PowerPoint to prepare her presentation on a given problem.
Student I: We had to solve for a presentation on ... a survey-a statistical test. We had
to write and we had to present on a PowerPoint. We had to put all of our
data and write out what we did.
Group 2 (2 students) The students pointed out that technology tools generated great visual representations of most mathematical concepts. They also talked about the effectiveness of having visual representations of abstract mathematical concepts.
Student 2-2: But also shows how far we are able to teach people any type of subject.
Technology ... is such a powerful tool in mathematics because, you know, it
provides the visual that students need to be able to process to understand
the concept and be able to apply that.
Researcher: Give me an example.
Student 2-2: You know, in class, you have PowerPoint that showed us ... urnm .. .it helps
us with our notes but not only that she showed us that step by step
processes that has to do with certain formula.
Student 2-2: It is like helping students ... I mean if you just have a teacher andjust
students. And the teacher is just talking and talking ... I guess you can 104 have board to work with. You can only focus so much. You can only
retain information so much. And teacher can only .. .it limits teacher's
ability to teach because if you just write it and translate it and "okay, you
do this" also go to another board and write ... Technology allows and it
makes it easier for teachers. They should try to get their
concepts ... especially do PowerPoint presentations. Creating visuals for
students to focus on the notes and on.
Students really liked their graphing calculators and they were talking about it.
They saw many benefits of using graphing calculator while they were studying mathematics. They recalled that Ms. B's use of a video camera to show how to use a graphing calculator was a great way to explain it, because they could understand very clearly when they saw what Ms. B was talking about on the TV monitor.
Student 2-3: I think calculators are ... graphing calculators are great tools to have in.
Umm ... especially in this class because it did a lot of operations for us. I
am pretty sure that most of us bombed that test without calculators ... honest
with God.
Student 2-2: Yeah. I mean. We saying, "Oh, how to figure this out?" Oh, you know, set
the end "VARS" and to buy and you press "enter." Yeah ...
Student 2-3: You forget how to do it. But you don't have to do it by hand. But it is just
made it faster and easier. And it is more efficient because you do not have
to do it by hand.
Student 2-2: Because teachers teach you how to get power through the calculator. 105 Umm, also teacher tells you whether calculator is used as a process to get
that number.
Student 2-2 brought up an interesting way of integrating technology in mathematics learning. She mentioned how a technology tool could be used as an evaluation tool. She pointed out that teachers could measure the progress of a student by recording the student's perfonnance at two different times. I thought that was a brilliant idea.
Student 2-2: You know. It can also show the progress. Teacher records students in the
beginning of the year in math and records after you see the progress in the
students. For technology, you have the video available, like even
recording videos ... videos we were watching on statistics.
All: (laugh)
Student 2-2: It stilI helps me because it shows .. .it gives another way to learn things to
learn different concepts.
There was a discussion about the teacher's use of technology tool in class. We talked about one particular incident. Ms. B was using a video camera to explain how to set up a particular function on a graphing calculator. She explained as she showed what she was doing through the video camera. Students recalled that it was a very clear instruction and that after the instruction was given, all of them could set up that particular function on their own calculators. When I asked if showing what the teacher did through a video camera helped them understand what was going on, they mentioned the following. 106 Student 2-2: Because she showed us ... Yah!
Student 2-3: We saw what buttons show pressed ... It is like not this one but the yellow
one.
Student 2-1: It is like ... you press second "vars." Where is the "vars?"
Student 2-3: It makes it easier since you can see how it looks like on the screen and
compare to with what we have.
Student 2-2: And it makes easier for her as a teacher. Her job is to teach us how to do
certain things. And that visual not only allows us to see how to do
something but also helps and it makes it easier for her to actually show
what is going on. Having her to go to each person and say, "Ok, press this
button." It is a time saver because you do not want to show us "Ok, do
something ... " how to do with a calculator. As a math teacher, you are
teaching the math instead of teaching us how to press a button.
Researcher: What do you mean by save time?
Student 2-1: Save time ... Vh ... Let say, you have a bunch of students and you know,
maybe some of them get it and some of them don't. You are not sure
which ones are getting. You are going to each table and tell each student
"okay, this is how you do it." That's good. You have one-on-one time.
When you have that one-on-one time, it is very time consuming because
you have to repeat yourself constantly over and over.
Student 2-2: And it means teacher's availability to everyone with us because teacher is
one person and class can be about up to thirty people. And when we try to 107 teach one concept or a function, you lose that the time that interaction with
the entire class. So, technology expands the teacher's availability.
Group 3 (1 student) Student 3 started to talk about how wonderful her experience was with her graphing calculator. She believed that without a graphing calculator she could not perfonn in mathematics classes as well as she perfonned.
Student 3: OK, if it weren't for graphing calculators, we would have been doomed,
OK? Because in math, you know, it is nonnal to plug in the equations on
the graphing calculators and get the answers, so it is not just sounds easy,
you know, some problems you need to manually fmd out the equation, you
know, they just give you the infonnation and you have to make an
equation and then you plug into the calculator. Other than that, technology
is very important in learning math. I am listening everyday you came and
videotaped us Mr. C, we always had a calculator right?
Researcher: Right.
Student 3: Yeah, that's now normal thing in math right nowadays, you know. It is just
we have to have a calculator because even for graphing we have to have a
calculator, you know, graphing needs help. So, yeah, it is very
important because if it were not for them, we would have to do everything
manually that can't be for our day. I know they did back in the days, but
calculators help a lot.
I asked her if technology helped her to do more work or made her lazier. She said that she became lazier because she did not have to think intensely for her calculations. 108 When I asked why she thought so, she explained that she did not think much when she could use a graphing calculator on her tests.
Group 4 (3 students) In this group interview, I asked if technology integration in learning mathematics helped students think more or if it made them think less. Students thought that they did not have to think as much as before because of their graphing calculators. They recognized it as a negative impact of technology integration.
Student 4-3: That is ... It says on the calculator, the name and what you are doing, that
you have to know. But in the process, you might not know, like, for
example, if you are doing a probability thing. You do defme the name of
what property you are doing, enter the numbers how many things and how
many sections and stuff to figure out how much percentage. Then, you
know how to do that part, plugging into the calculator. If you have to do it
by pencil and paper, you will know how. Because this whole time you
learn how to do it on the calculator and you don't know how to do it by
your own self. Using your own pen.
Researcher: OK, that's a downfall for using technology.
Student 4-3: Yeah. It is a negative side.
Student 4-2: Negative side.
Student 4-3: You don't know how to do the procedure. Yab.
Student 4-2: We do learn how to do it by hand just in case we do but we don't stay all
the times and rarely anybody does it by hand anymore anyways because
computers are so common. 109 Group 5 (1 student) Student 5 thought that graphing calculators helped him understand how to graph a function.
Student 5: They really helped, like graphing calculator shows and draws graph of an
equation, so that helps you understand how the graph looks like of a
certain equation.
Researcher: Did it help?
Student 5: Unun-hmm
Researcher: Can you give a little bit more into detail on how it helped?
Student 5: It helps me understand the concept.
Researcher: Understand the concepts better.
Student 5: Yeah, like a graph of an equation.
Researcher: So, then without a calculator, would it be impossible for you to understand
the same concept or...
Student 5: No, but it will take longer time.
Researcher: Longer time. So, it saves you time, right?
Student 5: Unun-hmm
When I asked him if his teacher's use of technology in teaching helped him to learn mathematics better, he said, "No." He explained further that what his teacher explained was found in his textbook. He thought that no matter what type of technology his teacher used, if his teacher taught exactly the same information as found in the textbook, the teaching would not make any difference to him. Therefore, his teacher's use of technology did not make any difference in his learning. 110 2. Summary
The students pointed out retaining information for longer periods and better understanding due to visual representation of mathematical concepts as a few benefits of technology integration in mathematics learning. Using graphing calculators in problem solving made learning mathematics more understandable and using a video camera helped the teacher give the instructions more clearly and students to better understand the instruction.
Question 9: What has been your biggest challenge or obstacle in learning mathematics?
1. Summary
Students who took AP calculus and AP statistics expressed their challenge of taking two highly demanding courses in a year. Not only were the classes demanding but preparing for two AP exams was a very difficult task for them to execute successfully.
Most of the students who were interviewed said that preparing for AP exams was a very difficult thing to do because they had a hard time retaining information they learned throughout the school year. In other words, information retention was the main challenge that was discussed by most interviewees.
One student had very different opinions on this topic. He thought that staying on focus was difficult for him because of the noise level he had to deal with in his mathematics class. On top of the noisy environment, he also had an issue about understanding what was going on in class. Sometimes he could not understand what his teacher was explaining to the class. He attempted to figure it out by himself in most III cases. He decided to figure it out by himself and that was very challenging.
Challenges that most students had were closely related to how they learn the information discussed and taught by their teacher and by participating with various learning activities. What they were learning was not as big an issue as how to learn it.
Question 10: What is your experience ofcooperative learning in mathematics class?
1. Group Responses
Group 1 (1 student) Since the school was a project-based learning school, student
1 felt comfortable with the idea of cooperative learning by the time she took AP statistics.
When I asked her if she liked her experiences of working cooperatively in a group setting, she replied positively. She liked the idea of working with other students in class.
She seemed to enjoy the group setting and did not mind asking her teacher when her group members could not help each other.
Group 2 (3 students) Students talked about how learning through hands-on activities and working in groups helped their learning. Both of them clearly influenced their learning positively. When students talked about why group work helped their learning, they shared some of their positive experiences when they had special assignments they did at home in their own time.
I wondered how they formed a study group to learn together? I asked students how they chose the members of their study group. They explained how they selected their group members and what was the most important thing they had to consider when they did group work. Basically they got into a group natura1ly by compatibility. When they 112 came to know each other in class, they knew with whom they could work better and with whom they could not work well. They formed a group of those with whom they could study well together.
Group 3 (1 student) Student 3 had a positive experience about cooperative and collaborative learning. She said that she enjoyed working with other students and there were many benefits of working together in a group. When I asked her if technology integration would help students do more group work or prevent students from working as a group, she said it would discourage students from doing work as a group. She believed that students would work together in a group to get the answer to a problem, but if she could get her answer with her graphing calculator, she would not need to discuss the problem with other students because she already got what she was looking for.
When I asked her if technology integration promoted collaborative learning or not it, she replied that technology did not promote cooperative learning in class because students would depend on their technology tools to find their answers. Therefore, as long as students could fmd their answers from their technology tools, they would not want to talk to someone else. She pointed out that students would talk to someone if they got an
"error" on their calculator.
When we discussed how Ms. B used technology in her instruction influence students' interaction in class, she answered that it could help them interact more, but in some situations, it isolated students because they did not interact with each other since their teacher could communicate with the entire class by using technology tools.
Group 4 (3 students) Students said they had good experiences of working 113 cooperatively and collaboratively. When I asked them what they called a good experience of working together, they said working with good members who were willing to work together and help each other was a good experience. Then I asked if they would have a better experience when they worked in a group with good members. They said it would make a big difference. My next question was whether choosing their own group members made any difference in their experience in their mathematics classes.
When I asked student 4-2 why he thought that who he was sitting with did not make any difference because he had said having the right group members would be the key to success for doing group work, be replied that the only time it would matter would be when they had to do a group project. When his grade would be heavily affected by the other group members' performance, he said that he would rather have certain people in his group. However, as long as be did not have to do any group work, he did not mind sitting with anyone.
Group 5 (1 student) When I asked this question, student 5 did not understand the meaning of cooperative learning. Therefore, I had to explain to him before he answered this question. After he understood the question, I asked him ifhe thought that cooperative learning was a good idea for him. He thought that working in a group helped him only if his group member could help him.
When I asked him why he was not actively participating with group discussion or any other group activities in class, he said he did not have any particular reason for that.
As we discussed several different things, he shared with me that he had many opportunities to sit in a group setting and to work with other students. However, he did 114 not want to interact with other students without any particular reason. He helped other students in one of his mathematics classes but he chose not to ask for any help from his group members, even when he knew that he needed help from them.
2. Summary
Many students felt comfortable working with other students in a group. By the time they took AP classes, they had numerous experiences of working with other classmates collaboratively. In their mathematics classes, they had multiple opportunities to work as a group. Most comments were positive.
When I asked one student about whether technology integration may help students work together, she said, "No." She said that using technology such as a graphing calculator might provide students convenience but students would focus on their own work instead of collaborating with other students because they would have their own answers by using their own graphing calculators.
When I asked if students and teacher had more communication when their teacher used technology for instructional purpose, she agreed with that idea. She said that when the teacher showed her graphing calculator through the video camera, they watched it and discussed what they saw on the TV monitor after her teacher explained how to solve a problem with her graphing calculator.
However, students did not want to work in groups when they were not evaluated as a group. Therefore, they did not mind sitting with anyone in class. They only wanted to select specific group members for cooperative and collaborative work when they had to submit a group project that would be graded as a group. llS
Question 11: What is your experience ofusing technology in mathematics class?
1. Group Responses
Group 1 (1 student) Student 1 believed that technology integration helped her with solving mathematics problems.
Student 1: It has been good. I understand how to use the calculator and it really helps
solving the equation and problems we have.
She also mentioned that using technology was more effective if she used it after she learned the same concept manually. Especially when the teacher demonstrated how to use, that helped her a lot.
Student 1: I think it helps when Ms. B makes these equations at first and she was
telling us use calculator because we a kind of understand the process. And
then you understand how much easier it is to use a calculator.
Group 2 (3 students) Students thought that technology integration was a good thing that helped them learn mathematics. They believed that technology would have a positive impact on their learning. They, however, thought that if they depended on technology too much, the technology integration would have a negative impact on their mathematics learning.
Student 2-2: The only con I would think about technology is if we depend on it too
much.
Student 2-3: Yeah.
Student 2-2: If we depend on PowerPoint to learn things and if we depend on video to 116 learn something that shows that we were not able to apply or understand
because it seems like to teach, you know. It is supposed to allow you
enhance teaching process. It is not supposed to be just teaching process in
general. Not everything do videos, it is just there to enhance the learning.
Student 2-3: That is what being depend upon is. It is supposed to teach us so we can
understand it. ...
Student 2-2: I would think it would be with certain technology so I would think that we
should not depend on it too much.
Group 3 (1 student) Student 3 thought technology integration was beneficial for learning mathematics. Especially when she used a graphing calculator, she completed work quickly without making many mistakes. However, she believed that she was getting lazier since she did not have to worry about showing each step of the problem solving with her calculator. She thought that that was a downside of technology integration.
Student 3: Yeah, because I know Ms. B teaches us manual way, you know, just like
writing it out and then she teaches it us on calculator, it just seems so fast,
you know, "Uh, it seems really easy." I learn from writing it out because
that way it will be in my head. But then when I do it on a calculator, it is
like "Oh, that's it?" You know, it's like we spent half an hour trying to
solve it then just plugging into the calculator? So, both experiences, I a
kind of like those experiences in Stats class. It is just be like "Oh, wow!"
we took a whole page to try to solve it this problem, and then she teaches
us how to put it in calculator. Oh, it is done. You know? Like if I knew at 117 the beginning, I probably wouldn't have done that manually writing it out
because obviously it took longer. But. .. Yeah, I think it was a good
experience, though. Because without it we would have been doing, you
know, we can write so much, right Mr. C?
Group 4 (3 students) Students talked about how they used technology in their class. They mentioned mainly how they used their graphing calculators on a daily basis.
In the AP statistics, they used the graphing calculator according to their textbook.
Therefore, they used their graphing calculator when it was suggested in the textbook and the instruction was effective according to students.
Student 4-2: Using technology everyday in statistics.
Student 4-3: I loved it.
Student 4-2: It was easy.
Student 4-3: It made everything easier.
Researcher: Can you go through what kind of technology you use each day?-How
you used it?
Student 4-2: TI-83. (Showing his own calculator) You see this calculator, Texas
Instrument, TI-83 Plus calculator, in statistics and calculus. Mr. A
suggested TI-89. Right? Yeah, whatever, in our class, statistics the book is
specifically made for this calculator ... the statistics book. So, it was easier.
Researcher: Oh, so, the book tells you when to use a calculator, how to use a
calculator, and so on?
Student 4-2: In statistics, if the book, the textbook we used made for this type of 118 calculator. So, that is what made it easier also. When I punch in the
numbers, the answers will be out. Done.
Researcher: You guys use that everyday, yah? Practically.
Student 4-2: Yes we use it everyday.
Students shared one of the special problems they did on computer. They used
PowerPoint to report their answers to the problem. Students liked the way they could organize and present the information that was neatly typed
Student 4-2: In statistics class, you have first the quarter project, special problems,
which we use the computer to type that out our...
Student 4-3: Report.
Student 4-2: Yeah. That's because computer is so neat to type it out. Awesome.
When I asked if they would use more technology tools when they were available.
Students said, "Yes." They wanted to have more and different technology tools for their learning. However, they worried about getting lazier when they have more technology tools for their learning mathematics.
Researcher: Good? Good experiences?
Student 4-2: Yeah, very good experiences.
Researcher: If you have more choices ... I mean more technology provided to you guys,
would you use them or... ?
Student 4-2: Yes.
Student 4-3: Yes.
Student 4-2: However, I think if we are given any more technology, we might get lazier 119 because computer might do it all for us.
Group 5 (1 student) I asked student 5 if he believed that he could learn more In the AP calculus because the teacher implemented more technology tools. He said that the way Mr. A approached teaching helped him to understand the mathematical concepts.
Student 5: I think I learned better with Mr. A's stuffwith technology because
calculus is a lot more difficult to understand. Now you can see and now
you can understand it. The other classes, you don't really have to. It is just
a bunch of numbers. I just used my head.
I asked him if more technology were available like he used in Mr. A's class, would he learn better, the same, or worse? He said that it would make the learning worse because students would rely on technology tools too much.
Student 5: I think it would be worse because like a calculator, if you punch in the
numbers and then it will give you the answer. Then you cannot learn the
concept that you had.
Researcher: So, you are basically saying that, in those classes, more technology would
make it worse.
Student 5: You know, like in elementary school. It is hard to learn the basic concepts
then technology will help you understand it better and faster.
Researcher: Right. Technology only ... It is making the process faster. You see that as
the only benefit of using technology, right?
Student 5: Pretty much. 120 2. Summary
Students thought that technology integration helped them do better when they learned mathematics. They shared that the technology integration would be more effective if their teacher showed and explained what they could do with their technology tool, such as a graphing calculator after they learned it manually first.
They also thought that technology integration had positive influences on learning mathematics as long as students did not depend on it too much. In the AP statistics, students used their graphing calculators when they were instructed to in their textbook.
Students believed their experience In the AP statistics was positive and helped them considerably.
Even though students wanted to have more technology in their classes, they were worrying about getting too lazy because they might not have to do much of anything with the right technology tools such as the graphing calculator.
Question 12: Did using technology help you understand and learn mathematical concepts better?
1. Group Responses
Group 1 (1 student) Student 1 believed that she could understand mathematical concepts better so she was able to learn it clearly when she used technology tools. When I asked her why she believed that technology integration helped her mathematics learning, she said she was exposed to technology when she was very young. That early age exposure helped her to use more technology comfortably. She said that she learned better 121 naturally with different technology tools In the AP statistics because that was how she studied her mathematics throughout her life.
Student 1: For me, because I think my whole life was getting use technology,like
you work with computer and video games and all other stuffs so like, it is
a kind of mentality to use computers and technology and calculators. So it
is kind of easier to do it that way than to be like trying to do it by hands
just because of the background. That's the way that I always do it.
Student 1: Yeah. We, every now and then, used the calculators and then middle
school we could use the calculators too. And I think, in high school, right
in freshmen year we started using graphing calculators.
Researcher: Umm. So you are kind of so used to using those kinds of technology tools.
Student I: I just use it to learn that way.
Group 2 (3 students) Students strongly believed technology integration in their learning was a very efficient way of learning mathematics. They believed that students would be able to learn best because technology accommodated the way they would learn the best. Therefore, when they studied with technology, they would learn the most.
Students also believed technology would allow them do their work more efficiently because it saved their time when they solved a mathematics problem.
Another reason why technology made learning mathematics more efficient was because technology allowed students to approach a mathematics problem in multiple ways. Students thought that when they could see a problem in more than one way, they would understand the concept better. 122 Student 2-2: Yes. It did help to understand because it gives us that visual we kind of
like better than, I mean, we can still learn it through paper, pencil, book,
and the teacher. But what technology does is beats up the process ...
Student 2-3: More efficient.
Student 2-2: Yeah, it becomes efficient because we are not wasting time for our teacher
chooses certain way because each person has different learning process
but it becomes more efficient in that, you know, it helps and it enhances
the learning process for students who do learn better with technology.
Student 2-3: That also allows learning process faster than if the learning process was by
hands vs. using the calculator. Just plugging in the numbers. It does not
process as we normally do. It does not mean it does different process.
Student 2-2: Technology shows us ways to learn different way to do things and it
shows power of doing from hands to calculator, you know. This is how to
do it. This is how to do it through a calculator, you know. It shows you
different ways.
Group 3 (1 student) Student 3 did not have much of the benefits of using technology In the AP statistics. Her experience was heavily based on the use of a graphing calculator. She thought that using a calculator to answer a question did not require very much thinking. She said she would get the answer if she entered a few numbers without understanding the whole process behind it. She said she would learn more clearly if she wrote down what she was doing and corrected any mistakes she made manually. 123 Student 3: Umm, I think I kind of answered to this earlier, it kind of did but mostly it
didn't because I kind of did not understand it. Like "Oh, we just plug in
and then we get the answer" kind of thing? I think writing it out is much
better than actually having technology because ... it is just the whole
learning thing. You know because if you were to type it in the calculator,
it will just give you the answer. You wouldn't be able to be like "Oh, I
know why that happens." At least, if you wrote it out, you made a mistake
and you look through all of your work and you will be realize what you
did wrong. For calculator, you cannot do that! Unless you have one of
those special ones but you don't need those special kinds of things for this,
so ...
Researcher: No, that's not good ...
Student 3: So, yah, I am, kind of, understanding it. I think in a way it kind a mean
what work you used to because then like I know binomial and
geometric, I still kind offorget which one to use. You know, like why do I
use geometric instead of this one, binomial. In calculator it lists as a
binomial geometric so you can just go to it. I don't know overall, it is a
kind of more confused because... Yeah, because even for all the different
types oftests we had to learn throughout the year like the two prop and
stuffs. You know, it's just like we are so used to just plugging in the
infonnation getting into the calculator like the two pops test. We do not
take time to write it out. You know, we just depend on the calculator. 124 That's kind of "bad" cause ... Yeah.
Group 4 (3 students) Students of group 4 did not think that learning with technology would help them learning mathematics better. They agreed that technology would help them do calculations fast and accurately but it would not explain why the answer would be what it was.
Student 4-3: When you plug in the problem into a calculator, you press enter and you
get the problem .... I mean you get the answer. But you don't see how they
got that answer, the process and procedures how to get it. Like what
equations do they use and what ... how to plug it in and where to get the
answer.
Student 4-2: As I said earlier, if we are given too much technology, we will get lazy
and we have to learn the basic concepts to understand how to use the
calculator also. I mean the technology also. Technology didn't really help
me understand the mathematical concepts.
Researcher: Did not?
Student 4-2: Did not help that much.
Researcher: OK.
Student 4-2: They help a little bit but they do not really help you understand.
Student 4-3: They just help you get the answer faster and accurately. Not how ...
Student 4-2: Like you said, Mr. C, it is a tool. It is another tool.
The students said that technology would help them learn mathematics better if it could show all the steps they were supposed to go through for any given mathematics 125 problems. So I asked them if they could study better when they had any educational software that was designed to teach mathematics. Some of them had experiences with it.
However, they pointed out that they needed someone who could guide them through the steps it presented to the user.
Student 4-3: I think that ifwe... they want us into technology that is what they should
make calculators that show the procedures. That's only thing.
Student 4-1: Oh, you mean, like self-taught, like I mean that's not related. It teaches
you?
Student 4-3: Yeah, you know like when you plug in whatever you are trying to learn,
Geometric or whatever, equation and then your information, and then
press, "enter" and you get the answer. But it doesn't show you how they
got the answer. It should ...
Group 5 (1 student) Student 5 thought that the graphing calculator helped him with graph problems. I asked him ifhe thought the use of a calculator would be beneficial when he studied non-graph problems. His answer was "no." He thought that he could do most calculations in his head without using a calculator.
Student 5: Umm-hmm. Like on a graph on a graphing calculator, then visual so I can
actually see and be able to understand it.
Researcher: How about non-graph related mathematical concepts? Like how to factor
out ... Or how to multiply two fraction-numbers ... all these things ... Can
you see that technology also has benefits for learning those things, too?
Student 5: Umm-, not really because it does not show you how it is actually done. 126 Like we write down on the paper. Or do it in my head.
Researcher: OK, something that you can do in your head, you don't necessarily want to
see on a graphing calculator or any technology tool.
Student 5: Like it really cannot show how exactly how you are doing the division and
multiplication procedures.
2. Summary
Students were exposed to technology in their daily life from their early ages.
Therefore, they felt comfortable in using technology when they studied mathematics.
Therefore, for many of them, the use of technology was part of them already. They felt that they could learn the most when they used technology tools.
Students felt that using technology was a more efficient way of learning mathematics because technology would allow students to study in a way they could understand best. They also mentioned that technology worked faster than studying with pencil and paper. They also believed that they could approach a mathematics problem in different ways without wasting time when they studied with technology.
However, not everyone was crazy about using technology in their mathematics learning. Some of them thought that using their graphing calculators made them lazy and they learned less because they did not understand what was happening when they could simply enter few numbers to get an answer to a problem that required very complicated calculation.
They also pointed out that using a graphing calculator did not help them as much as they wanted it to, because it did not show any steps that would help them understand 127 why they had a certain answer for their problem. Without understanding steps in between, they believed, it would not help them to learn mathematics. They did not like software that explained reasonable steps because they wanted to be able to communicate with someone who could clarify what they did not understand. 128 CHAPTER 5. DISCUSSION
Analysis Overview
After data analysis was completed, I used the basic mathematics-learning model that Tall (2002) described to revisit the focal points of my data analysis from the point of view of teaching and learning calculus and statistics. Tall's model discusses the
"embodied mode" and the "three worlds of mathematics" and their relationship with his four stages of mathematics learning (p. I).
When students were working on representations and processes that lead to solving mathematical problems by hand, they were in the embodied mode, a term which was used by Tall (2002) to refer to "thought built fundamentally on sensory perception as opposed to symbolic operation and logical deduction" (p. 4). When they discussed and tried to figure out what they were doing, students were in symbolic-proceptual mode, which
"begins with local deduction (meaning 'ifI know something ... then I can deduce something else') and develops into global systems of axioms and formal proof' (p. 5).
When students used their graphing calculators, they were moving from the iconic stage to formal-axiomatic mode in which,
Formal proof comes into play, first in terms oflocal deductions of the form 'ifI know this, then I know that.' What distinguishes the formal mode is the use of formal definitions for concepts from which deductions are made. (p. 7)
In order to present their thoughts and ideas to other students, those presenting needed their knowledge to be at the symbolic-proceptuaI stage.
When Tall (2002) explained how students could build an embodied understanding of calculus through the use of local straightness and visual ideas of area, he mentioned two concepts--a generic organizer and a cognitive root-which were useful in building an 129 embodied approach to mathematics (p. 12). According to Tall (2002), when students used graphing calculators they experienced the two concepts. If students experienced a generic organizer, it enabled students to manipulate examples and nonexamples ofa specific mathematical concept. In this study, students said that they could visualize problems and manipulate their problems and saw different outcomes when they used graphing calculators for problem-solving.
Tall (2002) said a cognitive root was a concept which was potentially meaningful to the student at the time, but which contained the seeds of cognitive expansion to formal definitions and later theoretical development. In this study, for example, when students used their graphing calculators to zoom in and then zoom in further, they were examining local straightness. Students were able to both confirm their ideas and verify their solutions. As they solved given problems, each and every confirmation they had helped them understand the theorems and axioms they were learning as they completed their assignment.
I believe the similar experiences and understandings would be possible without technology, but technology integration made the process more understandable and clearer to students for several reasons, including, in particular, making it easier to visualize and facilitating cooperation.
In Tall's research, he pointed out four main stages ofleaming mathematics: graphic, numeric, symbolic, and verbal. I studied the collected data and sorted the connections between what I observed and what Tall (2002) explained in his research. As he pointed out, students started formalleaming by beginning with the initial deductive 130 stage. In the three classes, I observed that students used their graphing calculators to begin the process by looking at the correct and accurate graph of the function or relation that is related to their problems. When they understood the problem by using graphical representations, they were able to process the problems symbolically or numerically.
When the numeric and symbolic process was completed, they were able to verbalize what they understood. At this point, students were able to communicate with each other and share their thoughts and ideas in their presentations or through an informal cooperative learning process.
From the educational technology point of view, I looked for the "three primary curricular goals" discussed by Cradler et aI. (2002): achievement in content-area learning, higher-order thinking and problem-solving skill development, and workforce preparation.
The comparison between the research findings and the results of the collected data analyses indicated that the way teachers implemented graphing calculators and other educational technology tools helped students achieve the first goal; their graphing calculators helped students achieve better test scores in mathematics. Unfortunately, I found little evidence that technology integration in the three classes promoted higher order thinking and problem-solving skill development. Technology integration did not appear to give students workforce preparation opportunity either. This was not unexpected because these students are mostly college bound.
Next, I explain how the interpreted data from each data source was analyzed to answer the research questions. I interpreted the primary data sets from each primary data source to generate the interpreted data sets that were used to answer my research 131 questions. The types of interpreted data sets I generated are listed with brief descriptions in Table s.
Case Studies
Case studies of Mr. A and Ms. B and their AP Statistics (I), AP Calculus, and
AP Statistics (2) classes were written to draw comparisons between and within the two different cases. I generated theories about what happened and discussed how the technology was used in different situations and discussed its effectiveness.
After analyzing each data source, I made two major comparisons. First, all teacher-related data were compared to understand what teachers thought about technology integration. The first and the second teacher interviews were compared with the classroom observations to find common themes. Second, all student-related data were compared to explore what students thought about technology integration. Questionnaires and the student interviews were compared with the classroom observations for a similar purpose.
My analysis started with a search for emerging themes from the first data source.
The findings of the first data source were compared with the findings of the second data source to discover any major themes in common. I kept tracking all the themes of the second data source that were similar to the themes of the first data source. I continuously checked to determine whether there were similarities between the themes of one data source and the themes of another data source. This process continued until I identified the common themes from five primary data sources. 132 To IUlderstand what I had to analyze from each source, I summarized textual information in five tables to compare multiple data sets and to find common themes among them. I used certain key words to identify and organize what was collected. Then
I combined and organized according to the research questions all the information that was given by students. I used each research question as a focus for the vast amolUlt of information that was collected from the students. In addition, I organized teacher responses to answer the research questions in the same manner.
After student responses and teacher responses were organized as mentioned, I finalized the answers to the four research questions by using both student responses and teacher responses. After I answered all the questions, I attempted to compare the two teachers and the three classes as appropriate.
The Comparison ofthe Three Classes
My original intention for this section was to compare what students believed regarding technology integration in their mathematics study. I wanted to compare what they learned using technology tools in the three mathematics classes. However, all of the student data were organized and analyzed as one set for several reasons.
First, there were only 43 students all three classes combined. Five students took both AP calculus and AP statistics. Two of these were in one of the AP statistics classes and three of these were in the other AP statistics class. To sort out these interactions was too difficult because the students submitted two copies of the questionnaire.
Consequently, the data were partially duplicated. Therefore, I combined the three classes 133 to understand what students believed about technology integration in learning mathematics.
Second, for student small group interviews, I originally selected ten interview groups from the three classes to collect comprehensive student opinions on how they used technology for mathematics learning in the three classes. Three groups of three students were selected from each class for fairly and evenly represented points of view. I selected an additional group of students who took both AP calculus and AP statistics.
However, because of schedule conflicts, I ended up interviewing only five of planned groups. I had fewer students in each group than planned.
Of the five groups, I interviewed one group of three students who took AP calculus and AP statistics at the same time, one group of one student from AP calculus, two groups of one student from AP statistics period 2, and one group of three students from AP statistics period 6. Therefore, I combined the student data analysis as one data analysis instead of comparing the three classes because the collected student interview data did not represent the three classes fairly and equally.
I organized student responses to each of four research questions based on what I found from the questionnaire, student interviews, and class observations. I listed the main categories found in student responses and synthesized the overarching ideas of the three data sources.
There was a small group of students (five students) who took both mathematics classes from the two participating instructors. I compared their perspectives about using technology in learning mathematics to determine whether they had a different outlook 134 compared to students who took one mathematics class from one teacher only. After I analyzed and compared the student interviews, it was clear to me that the group of students who took both AP calculus and AP statistics did not have a different perspective about using technology in learning mathematics compared to students who took one mathematics class.
Question 1: What roles do students and teachers report for technology integration? Why do they think those are the roles?
When I looked into the questionnaire and the student interviews, there was a common finding. The visualization of the mathematical concepts was a common theme.
Students believed that technology would help them visualize abstract mathematical concepts. Students thought that technology integration helped them learned better and retain the information longer because technology visualized the abstract mathematical concepts.
There were also some finding that were particular to individual data sources. In the questionnaire, students indicated that they believed that technology would help them in their project-based learning activities and that it also motivated them to learn mathematics. In the student interviews, students thought that technology integration helped them learned better and retain the information longer. They also believed that technology integration would allow them to learn mathematics in multiple ways.
While most student comments about the role of technology in mathematics learning were positive, they also expressed some concerns. Students pointed out that the 135 use of graphing calculators in problem-solving caused them to become lazy, think less, and communicate less with their peers.
Question 2: How do teachers and students use technology in mathematics class?
Some common findings between the responses and observations from the questionnaire, the class observations, and the student interviews were subjects reporting understanding mathematical concepts better, learning more actively, visualizing mathematical concepts, experimenting with different ways of learning ideas, conceptualizing their own ideas, confirming answers quickly, and analyzing mathematical concepts better.
The questionnaire, the class observations, and the student interviews commonly pointed out that students used their graphing calculators and other technology tools for two main purposes. First, students used technology as an effective mathematics learning tool that would help them understand mathematical concepts better. Second, students learned more actively when they used technology (such as a graphing calculator) for their mathematics learning. They tended to complete and check their work effectively when they used their graphing calculators.
Students had alternative ways of approaching, analyzing, and understanding mathematical concepts when they used technology to study mathematics. They thought that they solved problems in equation form better when they used their calculators than when they solved them manually because they could confirm their answers quickly. 136 Students enjoyed using their calculators because they could experiment with different ways ofleaming mathematical ideas and their experiments helped them conceptualize their own ideas and visualize mathematical concepts. Students mentioned that they learned mathematics better and retained what they learned longer because they could visualize mathematical concepts when they used their graphing calculators. They also believed that using graphing calculators simplified their calculations and helped them more quickly check different problem-solving methods because of the instant feedback that the calculators provided. That experience explained why I witnessed many students using their graphing calculators extensively for their tests.
There were some uncommon ideas shared in the questionnaire and the student interviews. The class observation findings were similar to that of the other two data sets; therefore, I focused on the differences between the questionnaire and the student interviews. The questionnaire also indicated that students used their graphing calculators and various other technology tools to study mathematics, to communicate with their peers, and to accurately perform complex calculations.
During the student interviews, students said that they generally used their graphing calculators whenever their textbook suggested it. During the student interviews,
I noticed that they used their graphing calculators because they believed that their graphing calculators helped them learn better because they learned the basic concepts faster when they used their graphing calculators 137 Students also seemed to understand better when instruction was presented on a
TV monitor. They especially paid closer attention when the teacher used a graphing calculator on TV to teach how to solve a problem.
However, students expressed a concern regarding the use of a graphing calculator for problem-solving. They thought that using calculators did not help them comprehend the entire problem-solving process because they did not understand the skipped steps in the problem-solving when they used their graphing calculators.
Question 3: How does educational technology integration help students learn mathematics cooperatively?
The results of the questionnaire provided limited information regarding students' thoughts about cooperative learning. Students clearly indicated that they used technology for learning. They thought they would discuss more with their classmates when they used their graphing calculators as they studied mathematics. They said they used technology tools to explain, present, and discuss their ideas with their classmates.
As I analyzed the student interviews, I could understand what students believed about cooperative learning more comprehensively. WHS promoted project-based learning in every class, and its students generally felt comfortable when they studied cooperatively because, from the time they started high school, they had had many experiences with working in a group setting.
Students said that they constantly discussed their ideas with their peers whenever they could not understand what they were working on. They believed that they especially 138 had more discussions when they used their graphing calculators to solve a problem.
During my observations, I also noticed that when they could not get help from their group members, they communicated with their teacher to receive further support.
Students mentioned that when their teacher used technology for instructional purposes, they noticed that their communication increased. For example, when their teacher explained how to use a graphing calculator by showing it via live video, they discussed what they saw on the TV monitor with their group members. Therefore, I believed, teachers' integration oftechnology into their instruction clearly promoted student interactions in class.
When students used PowerPoint and a computer for their special assignments, they discussed more because they focused on how to find and present their solutions. I believed the use of computer technology had a positive impact on students' attitude toward learning mathematics cooperatively because students believed the computer motivated them to work and to discuss their assignments more while they were working on the problem as a group.
In the AP statistics class, students practiced for their upcoming AP exam. After working on a problem as a group, they summarized and presented their solution on a poster paper after discussing it with their group members extensively. When I asked about this experience, students said that using poster paper helped them work cooperatively, and they found the solution of the given problem effectively when they worked cooperatively. 139 Even though students had positive experiences with their graphing calculators because the calculators were easy to use and helped them to better study mathematics, they had a few concerns about some of the negative impacts of technology on cooperative learning. Students said that they focused more on their own work instead of collaborating with their group members because when they used their graphing calculators, they could fmd the answers without working with their fellow group members. As soon as they had the correct answers, students knew that they did not want to work with their group members cooperatively because they believed that the ultimate purpose of working with their group members was to find the correct answers.
Students said their teacher's use of technology increased their interaction with other students in general, but they also believed that technology integration limited their interactions because they did not have to interact with other students when their teacher was able to communicate with them. In addition, they received the necessary help directly from their teacher when technology enhanced the communication between their teacher and themselves.
While analyzing the class observations, I found somewhat different ideas related to this question. I noticed that the use of educational technology saved the student presentation time, and that allowed students to have more time for discussion after their presentation. When students worked on a computer for their special assignment, they actively discussed their assignment, the use of computer and other tecimology, and more.
When they used their graphing calculators to study mathematics, students discussed more and learned more because their calculators enabled them to visualize 140 mathematical concepts. Students seemed to understand their problems better and, therefore, were able to share their ideas more frequently when they were able to see the visual representation of a mathematical problem. The use of graphing calculators definitely generated more discussions between the teacher and the students and helped students to understand mathematical concepts with more depth and clarity.
Question 4: How does educational technology influence mathematics instruction?
In the questionnaire, students thought the use of educational technology would reduce the teacher's instruction time significantly because they could see what the teacher was talking about. They said that they were able to better understand mathematical ideas because they visualized the ideas when they used their graphing calculators and other technology tools.
The responses during the student interviews reinforced the idea of using technology to visualize mathematical concepts. Students believed that visualization of mathematical ideas improved the way teachers instructed. Students believed that technology made it easier to develop visual aids for instruction and that visually showing instructions helped them to leam better. They also believed that technology made the teacher's job easier when students could visualize the concepts because it not only allowed students to see steps and procedures, but it also helped teachers to clearly show their ideas. Students recalled, for example, that their teacher used a video camera to show how to use a graphing calculator. They said that using a TV monitor and a video camera 141 to explain how to use a graphing calculator was a good way to explain mathematical concepts, and it made the instruction clearer to them.
Students thought that when the teacher integrated technology, the teacher could teach with more hands-on activities and give students more practice. For example, they described the Casino Day activities which Ms. B planned to give students more opportunities to practice selected statistical concepts.
Students remembered and discussed their experiences in learning with and without their graphing calculators. In one instance, they were introduced to a new problem in a longer way. First, they were instructed to solve the problem without using their graphing calculators. Then Ms. B taught them how to solve the same problem by using their graphing calculators. The difference was clear. They learned firsthand the effectiveness of technology integration in mathematics learning. Students believed that using graphing calculators in problem-solving made learning mathematics more interesting, doable, and understandable.
The Comparison o/the Two Teachers
I studied three different instances (classes) of technology usage in mathematics education taught by two different teachers. Their teaching styles were different, and they integrated technology into their instruction slightly differently. I thought the best way to analyze the types of data I collected was to write a set of interwoven case studies.
I completed this part of my research by analyzing three sets of data sources: the class observations, the first teacher interviews, and the second teacher interviews. I 142 reviewed the three data sources carefully to understand what the teachers thought about using technology integration to improve the quality of instruction and student's learning and to understand what teachers thought about technology integration in mathematics education.
Analyzing the three sets of data sources helped me understand the interactions between the students and teachers in the three mathematics classrooms. What follows is the summary of teacher responses from the three data sources for each research question.
Question 1: What roles do students and teachers report for technology integration? Why do they think those are the roles?
In the first teacher interviews and in the class observations, I found a few common ideas that helped me understand the two teachers' positions about the role of technology in learning mathematics. The two teachers thought that when they integrated technology into their instruction, technology would enable them to clearly communicate with their students. However, the teachers had different levels of understanding relating to how technology would impact students' learning.
It was clear to me that Mr. A understood the role of technology and how technology integration would help students learn better. He was able to explain his thoughts about the impact of technology integration on student achievement, and he extensively used technology with high levels of proficiency. He believed that technology helped students visualize the concepts they were studying. He also thought that technology integration would enable him to provide more meaningful instruction. 143 On the other hand, in the beginning of the interview, Ms. B did not quite seem to understand how technology influenced student learning of mathematics. She began to incorporate technology into her teaching and wanted to incorporate more technology tools into her instruction because she wanted to learn how technology would help students study mathematics. In the later part of the interview, however, she discussed how technology would help students visualize mathematical concepts.
Ms. B wanted to integrate more educational technology into her teaching, even though she believed that technology integration would require more time and effort to set up, because she thought that students better understood when they visualized mathematical concepts. She also considered letting students use technology to learn mathematics because she believed that it enabled students to better organize and present their ideas.
The second round of teacher interviews helped me to see the progress the teachers had made since the first teacher interviews. First, the difference between Mr. A and
Ms. B had narrowed because Ms. B understood more about how technology would impact students' learning. Second, Mr. A reemphasized that technology integration was very effective for teaching mathematics. He knew that technology integration would help
students better learn mathematics because integrated technology allowed them to visualize what they were talking about.
In the second interview, Ms. B had a stronger opinion about technology
integration for mathematics education, felt more comfortable using different types of technology, and understood how technology improved her teaching. She also wanted to 144 expand her knowledge of integrated technology and use more technology in her instruction. She defmitely incorporated more technology into her instruction and experimented with how different types of technology would help students study mathematics. Ms. 8 believed that technology integration improved her teaching and that it helped students visualize mathematical concepts. She especially considered a graphing calculator as an essential mathematics-learning tool for students.
Question 2: How do teachers and students use technology in mathematics classes?
From the class observations, I found that the two teachers used graphing calculators in the same way. They considered a calculator an idea-checking tool and let students use graphing calculators to explore their ideas and check their own work. Mr. A even incorporated a set of special programming codes that enabled students to perform the specific steps on their calculators to complete selected assignments.
If students learned the logic behind the problems, Mr. A let them use their graphing calculators to save time on tedious calculations. Students used their graphing calculators to check their work and solutions after they solved their problems manually.
The checking process helped students better understand the role of a graphing calculator in problem-solving when they solved the same problem in two different ways.
Mr. A used PowerPoint to present daily warm-ups at the beginning of every class.
The problems were presented on a TV monitor as students walked in. Students used their graphing calculators when they worked on the warm-up problems. Mr. A also used 145 a video camera to show students how to solve the warm-up problems using a graphing calculator.
He pointed out that the visualization of mathematical concepts was another benefit of technology integration in his class. He encouraged students to use their graphing calculators because it helped students to visualize when they study different types of mathematics functions. He used a video camera and a TV monitor to show the graph on a graphing calculator. He used graphing calculators to introduce new ideas to enhance students' understanding, to explain the complicated steps of a problem, and to test different ideas with students as part of iustruction.
Mr. A also used a PDA to collect the assigmnents, to check daily attendance, to maintain the grade book, and to maintain other administrative records. As needed, he also sent his emails to students and their parents to disseminate essential information and to send students' grades to them regularly.
Ms. B also used technology at a different level to present warm-ups. to visually illustrate abstract concepts, and to help students share their ideas. She used an overhead projector and the PowerPoint to present daily warm-up problems. She also used a video camera and a TV monitor to go over the warm-up problems.
She used technology to provide a form of visual representation of the concepts while she was teaching because she believed that the visualization of a concept would encourage students to be more attentive, which would help them understand mathematics better. She also used a video camera and TV monitor to support student presentations.
Students used graphing calculators to test the given problems and visualize them for 146 better understanding. They used a video camera and a TV monitor to show and explain
their work on their graphing calculators.
I found some similar findings between the class observations and the first teacher
interviews. Mr. A used more technology for his instructional purposes because he clearly
understood the impact of technology integration on mathematics learning. He used
educational technology to vary his instruction. For example, he extensively used the
infonnation on the Web to find an effective way to vary his instruction. He accessed the
National Council of Teachers of Mathematics Website to learn different types of methods
for mathematics instruction. He believed that he could assess effectively because
producing different versions of assessment tools and instructional materials became
easier when he used a computer.
In addition, Mr. A used email and other online tools to communicate with other
teachers across the nation to expand his content knowledge. He exchanged questions with
other teachers to discuss and learn different teaching strategies for calculus. He also used
technology to promote mathematics learning after school. He held regular online tutoring
sessions with students every weekday. During the tutorial sessions, students discussed
any challenging problem they faced while they worked on their assignments.
He also integrated technology to help student learning. He said students used the
Tl-83 calculator extensively on a daily basis because using these graphing calculators
helped them learn and understand mathematics better. Mr. A noticed that students
focused on problem-solving longer when they used their graphing calculators. He also
mentioned that educational technology enabled students to experiment with different 147 mathematical ideas and to visualize how numbers and letters in an equation were interrelated. He also said students could learn mathematics more effectively when they understood the meaning of each part of an equation.
Mr. A thought that students organized their work and assignments better when they used technology to do their assignments. He noticed and thought it was interesting that, when they used a computer, students exhibited stronger ownership and exerted more thought before they fmalized their work.
Ms. B recognized educational technology as an important teaching and learning tool. She believed that students would complete more work, learn better, and retain more information when they used their graphing calculators because of its speed and accuracy.
She also thought that students would understand the concepts better because they could focus more on the process itself without worrying about tedious calculations when they used their graphing calculators.
She believed that the technology integration enhanced her communication with students because the communication was instantaneous and lively when she used technology for communication purposes. She also said that using computers as an online search tool would help students learn efficiently because they could look up the information they needed instantly.
When I conducted the second teacher interviews, it was clear to me that the two teachers integrated more technology in their instruction than when they had participated in the first interviews. Mr. A integrated more technology for improving his teaching and
Ms. B developed a deeper understanding of technology integration for teaching 148 mathematics. For example, Ms. B also used technology more effectively in her instruction and let her students use technology to present in class. She let her students do assignments in PowerPoint and share their work in class. On the other hand, Mr. A used online information to gain knowledge, but it was not integrated into his classroom teachings.
Mr. A posted homework assignment lists for his classes on the Internet to help students who were absent. He reported that he updated his assignment lists quarterly. He also offered online tutoring sessions by using the Instant Messenger (1M) software. He held the tutoring sessions daily on weekdays.
Both teachers believed a graphing calculator to be an essential study tool in mathematics class. Mr. A let students use graphing calculators as if were second nature to them. Ms. B thought that a graphing calculator improved student learning because it gave instant feedback, which allowed them to confirm their ideas.
Mr. A used a couple of other kinds of technology equipment to improve his teaching. He used a video camera and a TV monitor to help students who sat far from the boards to be able to watch presentations and to allow them to participate in the discussions. He also used a scanner and Optical Character Recognition (OCR) software to digitize the answer keys for homework assignments. Mr. A used his PDA to maintain grades and record attendance on a daily basis. He also used his PDA to communicate with the students and their parents for various purposes.
In the second interview, Ms. B recognized the importance of using technology for teaching and learning more than in the first teacher interviews. She now believed that 149 students understood mathematics better when they visualized the concepts by using a graphing calculator.
She also believed that students better confirmed and trusted in the theorems they learned when they used their graphing calculators because these tools provided students with valuable opportunities to apply the theorems to solve their problems and to visually check how solutions worked.
Question 3: How does educational technology integration help students learn mathematics cooperatively?
As I observed the classes, I understood the way the two teachers thought about the correlation between cooperative learning and technology integration. Mr. A believed that students were energetic and willing to share their thoughts and ideas with each other when they had the opportunity. He intentionally included an unfamiliar question on the team test because he believed that it promoted interaction and cooperative learning.
Mr. A also altered the seating arrangement to promote cooperative work between students. The student seating arrangement was occasionally changed because, when they sat with different groups, it would allow them to see different points of view from different classmates.
Ms. B said that educational technology promoted more interaction and communication among students because they could see what they were talking about. She believed that using their graphing calculators allowed students to have instant feedback, meaningful discussions, and clarifications of what they did not understand. Ms. B also 150 reported that students seemed to be more motivated and focused on their work when they worked collaboratively and when they had a challenging assignment.
Ms. B began the class by providing instructions and warm-up problems on the overhead projector. Students typically began their class by working on and discussing warm-up problems as they walked into the class. From one of my observations, I noticed that students were working on an assignment that required using randomly-generated data. Students used their graphing calculators to generate random numbers to solve the problem. Students discussed how they could determine the type of random data they needed as groups because they needed to generate different types of data depending on the design of the problem.
When I conducted the second teacher interviews, Ms. B discussed how she believed technology integration influenced students' cooperative learning. She told me that students would learn more if they were allowed to use technology to explain their own thoughts and discuss them with their peers. Ms. B also thought that students would be engaged and learn more dynamically if they used a computer in a group setting because that would motivate students to discuss more, and they would be able to present their solution or ideas without much delay.
Question 4: How does educational technology influence mathematics instruction?
Analyzing the class observations, I have noticed that the way technology influenced mathematics education in the three classes was different. Mr. A showed the graph of a function on the TV monitor by using his graphing calculator and a video 151 camera. He then marked and added lines and shapes to the graph on the TV screen with a regular dry-erase marker to show the step-by-step process. The lines and rectangles on the TV screen helped students understand what he was talking about. It proved to be an interesting and effective way to use a TV monitor for instructional purposes.
The technology-integrated instruction could be a very simple process. For example, when students asked for help on one of their homework problems, Mr. A explained that they needed a set of programming codes on their graphing calculators to complete the specific problem. He then passed his graphing calculator around to let students copy the specific set of codes from his calculator.
In AP statistics, students used technology to verifY the solutions and the steps for problem-solving. For example, Ms. B once made a mistake as she was presenting a problem, but she could not identifY and locate her mistake. When she figured it out later on, she prepared a PowerPoint presentation to go over the problem. In the same presentation, she pointed out what her mistake was. As the class reviewed the problem, students followed the explanation carefully and used their calculators to check the process.
Ms. B used a TV monitor and a video camera to present problems and used the whiteboard to explain key concepts for the problems and to illustrate how to approach each problem. At another time, she presented some problems on the PowerPoint and had students discuss them. In that presentation, she used the hidden slide feature to make the presentation more interesting. After students discussed each problem, she showed the correct answers that were hidden so students could check whether their responses were 152 correct. Students loved the feature because of the instant feedback from the teacher as she used technology.
In AP statistics, students also had a special activity day to review some of the concepts they learned. Ms. B called it casino day and prepared three game stations for students to play simplified games to practice some of the statistical concepts they had learned. At the beginning of the activity, she used the PowerPoint to explain the rules and the expectations of the games. Ms. B thought the day was successful.
Based on the class observations, I found three common practices between the two teachers. First, they continuoUsly used their video cameras to give instruction and to get students' attention. Second, they practiced more student-oriented teaching and let students present their ideas through technology integration. Third, they had more interactions with students as they instructed when they used educational technology.
In the first teacher interviews, the following key ideas were shared. Mr. A thought passing out and collecting assignments were done more efficiently when technology was integrated. He gave lengthy instruction to students, for example, by simply putting the information on a disk and passing it around in class. Students then simply copied it into their computers.
Since he started using technology to instruct, he has relied on technology so much that he could not teach normally without technology. For example, when he taught a fairly large class, students could not understand the instructions because they could not see or hear it, so he used a video camera and a TV monitor to show the written instructions and the presentations to help students receive proper instruction. However, it 153 was challenging for him to teach mathematics with the large class when the video camera was broken.
Mr. A searched the Internet for information. He said, "The Internet is a great resource for me both in terms of finding lessons, things .. .! can do, and also investigating what the alternatives are out there." He said he joined a listserv and constantly exchanged emails with other calculus teachers across the nation. He believed email communication helped him a lot when he prepared his lessons and planned his instruction. He even once communicated directly with the author of the textbook he used for AP calculus.
He was willing to integrate technology for instruction because he believed that students would have stronger ownership of their own work when they used technology for learning. He also thought that encouraging students to use technology to solve and present their problems was an excellent way to use technology in mathematics learning.
Ms. B increased her technology incorporation as she learned how to use different technology tools. She attempted to prepare more lessons by integrating PowerPoint, graphing calculators, and other technology tools. She was willing to learn and use different types of technology as much as she could to help her students understand the concepts and learn the essential skills required for the AP exam.
She was interested in understanding how to prepare teachers to integrate technology into their instruction. She thought that if she knew how to use technology, she would use it more often, and that would change the way she would teach her students.
She strongly believed that technology integration would definitely change the way she instructed and that students would learn better by using technology. 154 Ms. B felt comfortable using graphing calculators as part of her lesson. She was using a video camera to show how she used her calculator and what was on it. She liked to use a video camera in class because she could show the calculator buttons while she explained how to use the calculators.
She talked about some of the benefits of technology integration in mathematics education. She said, "Using technology would allow more visual .. .it is an aid. It is not that you cannot do it without it, but the argument is that it makes it better. It makes them retain the information better because they are seeing it and feeling it and doing it, versus just practicing the long way on paper and they are not really getting the idea for it."
Analyzing the second teacher interviews, I found the teachers' ideas of technology integration were a little different from the ideas in their first interviews. A possible reason might be that they were allowed to watch and talk about what they did in their classes because this interview was conducted as they watched some of the videotaped class sessions.
Mr. A noticed he used a video camera extensively to present instruction and to help students share their ideas. A video camera was used to maintain clear communication between the presenter and the audience in class. For example, he could explain a difficult-to-teach theorem effectively without recreating the graph because he could transfer the actual graph from a graphing calculator to the TV monitor by using a video camera.
When he saw himself using a video camera to present lessons, he talked about the usefulness of a video camera in his teaching. He explained that, when he demonstrated 155 under a video camera, he easily showed students which button of a graphing calculator they needed to use. By using a video camera, he could also show the process of problem solving as well as what the graph of a function looked like.
He mentioned one exciting teaching experience he had with technology integration. He was teaching an important theorem by using a graph of a function that he generated on his graphing calculator. Instead of drawing the same graph on the board, he
showed it tbrough a video camera. Then he drew lines and different shapes directly on the
TV screen by using a dry marker to explain more precisely. Teaching by displaying the graph on a TV monitor was successful and a great discovery for him. He said he used the
same technique more frequently thereafter because it helped students understand better.
One benefit of technology integration in mathematics class that Mr. A believed in was visualizing mathematical concepts. That is why he emphasized using graphing
calculators in class and showed the graphical representation of the function by using a video carnera and a TV monitor. Mr. A believed using technology would help students be
self-directed learners. He prepared answer keys for students to self-check their own
assignments and to improve their understanding. He presented ideas and mathematical
concepts to class by using PowerPoint.
As Ms. B watched her own teaching, she recognized the way she was using an
overhead projector to post instructions for students to start the class as they walked into
the classroom. She thought the use of an overhead projector to present daily instructions
was the most effective way of giving the instructions at the beginning of each class
period. 156 She also believed technology integration helped students understand and learn mathematics better. Using a graphing calculator especially helped students visualize mathematical concepts and retain the information longer. Ms. B also strongly believed technology integration would produce more discussions and influence the way she would teach and how students would learn positively. She said technology altered classroom structure and enhanced mathematics teaching. For example, she concurrently used two different types of technology tools to efficiently communicate her instructions. She began to use the overhead projector and PowerPoint at the same time because she wanted to show the instructions while she went over some examples because students were taking notes. She said that the use of two different presentation tools saved her instruction time.
She realized how much she actually integrated technology into her teaching as she watched her own recorded class sessions. When she watched multiple class sessions, she recognized that she was using more technology for teaching at the end of the research study than she used at the beginning of the study. Ms. B increased her technology integration as she learned how to use the tools.
Even though the two teachers constantly integrated technology to improve their instruction and student learning, according to the class observations, the way students learned remained the same in the three classes. For example, in the AP calculus class, instead of paying attention to and understanding the instructions, students were busy copying the presented information for future reference. Sometimes students spent most of the class time taking notes because of the amount of instructions given. 157 In AP statistics, the way students were getting instructions was not different from what I observed in AP calculus. Students still took handwritten notes, and the majority of the instructions were given orally. In other words, technology somewhat influenced the way teachers gave instructions, but the impact was not strong enough to significantly change the way they taught and the way students learned in mathematics classes.
Can technology aid this type of simple teaching process through use of the Web or email, since all students have an email account at the school? If the electronic copy of the information were available, students could pay more focused attention to the in-class instructions and download the information when they needed it.
Answers to Research Questions
I interpreted the primary data sources and identified common themes to answer my research questions. To make my answers more understandable, I organized students' responses and teachers' responses separately.
To understand what students thought about the role of technology in their mathematics learning, I placed student responses in the order of the questionnaire, student interviews, and class observations. To verifY data against the triangulation method, I wanted to discuss what I found after I first discussed what students believed and also wanted to compare what their stated beliefs about mathematics with their actions in class.
For teacher responses, I followed a similar pattern. I organized teachers' responses based on the first teacher interviews, the second teacher interviews, and class observations for the same reasons. After analyzing what teachers believed they did, I 158 discussed what they actually did in classrooms based on the class observations. While I was analyzing the two teachers, I compared Mr. A and Ms. B to understand the similarities and differences in the way they integrated technology in their instruction.
Question I,' What roles do students and teachers report for technology integration? Why do they think those are the roles?
The questiounaire responses showed that students had very strong opinions related to this question. They thought that technology integration helped them visualize the abstract concepts so that they learned better and retained the information longer. They generally thought that they would understand and learn mathematics better because they could visualize the concepts when they used the technology.
Students also believed, according to the student interviews, that technology integration improved the instruction because teachers could show what they explained while they taught. They also thought the way teachers presented the instructional materials was effective, helped them visualize the concepts, and made the instructions clear to them. They especially liked it when teachers used a video camera and a TV monitor to show how to use a graphing calculator.
Students recognized another possible role of technology in their mathematics learning. They believed that technology integration allowed them to learn mathematics in multiple ways. They thought that when they used technology, they learned mathematics by watching, listening, discussing, and manipulating objects. 159 While most of the students made positive comments about technology integration, they shared some concerns about technology integration as well. Students believed that the use of graphing calculators might make them lazy, think less, and communicate less actively with others.
I found a few common ideas between the first teacher interviews and the class observations that helped me understand what the two teachers believed the role of technology in learning mathematics should be. Both teachers thought that technology integration enabled them to clearly communicate with students.
Mr. A believed that technology helped students visualize the concepts they were learning. He thought students tended to learn better because they could understand better when he used technology for instruction. On the other hand, in the beginning of the first teacher interviews, Ms. B did not seem so confident about how technology integration influenced student learning of mathematics. She considered technology a reporting and organizing tool that students could use to learn mathematics better. However, in the latter part of the interview, she mentioned that technology would help students visually when they studied mathematical concepts. Therefore, she wanted to use more educational technology because students understood better when they visualized mathematical concepts even though she believed that technology integration would require more time and effort.
In the second teacher interviews, both teachers pointed out very similar benefits of using technology in mathematics education. Mr. A thought that technology integration was a very effective way to teach mathematics. He knew that technology integration 160 would help students learn better because technology helped students visualize what the teachers were talking about Ms. B also believed technology integration was a powerful teaching tool for her because it allowed students to visualize when they studied mathematical concepts. She considered a graphing calculator an essential mathematics learning tool for students.
Question 2: How do teachers and students use technology in mathematics class?
In the student interviews, they discussed how their graphing calculators helped them learn mathematics more effectively. They believed that the use of their graphing calculators helped them to better learn because they could perform the basic mathematical calculations faster and more accurately.
Students also used technology to approach and understand mathematical concepts in different ways. They believed that they could manipulate mathematical equations and graphs more easily when they used their graphing calculators, and that flexibility helped them understand their problems better. They also mentioned in their interviews that they learned more effectively because they did not have to remember detailed steps; they only needed to remember the correct options on their calculators to accurately perform various types of calculations.
The use of graphing calculators helped students not in only solving their problems better but also in checking and confirming their solutions quickly. Students enjoyed using their graphing calculators because they could experiment with mathematical concepts and came up with their own conclusions. Graphing calculators helped students to be active 161 learners. When they used their graphing calculators, they completed aod checked their work effectively.
From class observations, it became clear to me that students used their graphing calculators aod other technology tools for two main reasons. Technology helped students study aod understaod mathematical concepts better. Students used their graphing calculators to understaod abstract mathematics concepts by visualizing them. Students were able to take different approaches aod try different ideas fairly quickly when they used graphing calculators for problem-solving. That was why students used calculators extensively on their tests. They also seemed to understaod mathematical concepts better when they were presented visually on a TV monitor.
Students thought using graphing calculators did not help them comprehend the entire problem-solving process because the TI-S3 did not show the calculations behind the solution they had. While they could check their final answers quickly aod effectively, they could not understaod aoy unexplained, non-displayed parts of the problem-solving process when they used their graphing calculators.
Mr. A used educational technology to vary his instruction. For example, he extensively used the information on the Web to learn effective ways of varying his instruction. He frequently accessed the websites developed by NCTM aod other professional organizations. He also believed that he could vary assessment when he used computer technology. He easily made different versions of assessment tools aod instructional materials with his scanner aod a computer. 162 He also thought that technology helped students organize their work and complete their assignments better. He said that when they used computers to do their assignments, students had a stronger ownership of their own work and put more thought into their work before they finalized their conclusions.
Ms. B focused on student learning and believed that technology was an important teaching and learning tool. She believed that students completed more work when they used their graphing calculators because of the speed and accuracy of the calculations. She also thought that students would retain information longer when they used graphing calculators because they could focus on the more important processes when they did not have to worry about tedious calculations.
Ms. B believed that effective technology integration enhanced the communication between the teachers and the students because they could have instantaneous and lively communication when they used technology in their instruction. For example, while students were studying mathematics, she wanted to let them use the computers as an online search tool to enhance their learning and to generate meaningful discussion in class as well.
In the second teacher interviews, teachers shared somewhat different ideas because they watched and discussed their own classroom teachings. Both teachers believed a graphing calculator was an essential tool in mathematics classes. Mr. A thought that graphing calculators were "second nature" to students, and Ms. B strongly believed that graphing calculators improved student learning because it gave them instant feedback and let them confirm their ideas. 163 Mr. A talked about his online activities. He posted the homework assignment lists for students on the Intemet. It helped students who were absent and when they forgot their assignments. He said the lists were updated quarterly. He also offered online tutoring sessions on weekdays. He used instant-messaging software programs to tutor his students.
He thought he could improve his instruction by using a video camera and TV monitor because students who sat far away from the whiteboards could not easily see information on the boards and participate in discussions. He also used a scanner and the
OCR software to digitize the assignment answer keys. He explained that the use of a scanner and the OCR software helped students to be more self-directed learners.
Ms. B believed that students understood mathematics better when they visualized the concepts by using a graphing calculator. She also talked about how students understood the theorems they learned much better when they confirmed the theorems.
She believed that students confirmed those theorems more effectively when they could apply the theorems during problem-solving and could check whether their solutions worked on their graphing calculators.
Based on what I observed, I found that the two teachers used similar technology tools in a similar way. Mr. A used graphing calculators as a checking tool. He let students study and check their own work using their graphing calculators. He led discussions and helped students do the same problem manually before students checked their work on the calculators. When students checked their work and answers after manually solving 164 problems, they could compare the two different ways to problem-solve, and that helped
students understand clearly the role of a graphing calculator.
Students were encouraged to use their graphing calculators when they worked on
problems that were very challenging and difficult to solve manually. In AP calculus, there were certain problems that required special programming codes to perform a very
specific part of the calculation on a TI-S3. Mr. A gave the specific programming codes in
class so students could complete the challenging problems.
Using graphing calculators helped students easily visualize their problems, and
that helped student learning. Mr. A also used a video camera and a TV monitor to show
how to solve problems visually. He also used graphing calculators when he introduced
new ideas to enhance student understanding. He explained multiple complicated steps by
using a graphing calculator. He used graphing calculators to test different ideas with
students as a part of the learning process. Students used graphing calculators to test
different types of functions visually when they drew their graphs.
Mr. A used a PDA to collect assigmnents and to check attendance on a daily basis. He also used it to send emails to students and their parents as needed. He also
disseminated new grades to students and their parents by using his PDA on a regular basis.
Ms. B also used technology when she presented warm-up problems and to make
abstract concepts more easier to visualize. She used the overhead projector and
PowerPoint to present the warm-up problems and showed how to solve the problems by using a video camera and a TV monitor. 165 When instruction was presented visually, students were more attentive and, in general, they understood better. Using a TV monitor and a video camera further encouraged students to present what they understood and helped them to gain a better understanding.
Question 3: How does educational technology integration help students learn mathematics cooperatively?
The student interviews helped me understand comprehensively what students actually believed regarding cooperative learning and technology integration. They said they felt comfortable when they studied cooperatively because their school promoted project-based learning, and they had practiced working in a group setting from the beginning of their high school life. They constantly discussed their questions with peers if they did not understand what they had on their calculators. When their discussion did not answer their questions, they communicated with their teachers for further explanation.
When AP statistics classes practiced for the AP exam by discussing their ideas as a group and summarizing them on poster paper, students thought that the experience of using poster paper helped them work together and find an appropriate solution to a given problem. They also thought that the poster paper activity prepared them well for the future AP exam and that they understood better when they worked as a group.
As we discussed cooperative learning in the student interviews, the students mentioned a few detrimental aspects of technology integration as related to cooperative learning. A few students expressed some concerns about using technology and studying 166 cooperatively. Students believed that they would not work cooperatively once they had the correct answers because they worked together only to have the correct answers.
Obviously, students misunderstood the meaning of understanding a concept. They believed that having correct answers meant understanding mathematical concepts.
A few students also pointed out that teachers' technology integration usually promoted more interaction in general, but that it also limited their interactions because they did not have to interact and rely on each other when they communicated with the teachers directly. Simply speaking, teacher-student communication reduced the frequency of communication between students.
Analyzing the class observations, I identified a few conflicting ideas that were different from what I found in the questionnaire responses and the student interviews. For example, students said that they did not work cooperatively when they used their graphing calculators. However, the use of educational technology saved students presentation time. and that allowed students to have more time for discussion after their presentations.
Students discussed and learned more when they used their graphing calculators to study because the use of calculators enabled them to visualize mathematical concepts.
When they were able to see the visual representation of an equation, they understood their problems better and were able to share their thoughts and ideas. Therefore, the use of graphing calculators definitely generated more discussion between teachers and students and deepened student understanding. 167 When three AP statistics students worked on special problems on computers, I interviewed them briefly. I thought they were relatively quiet students based on the class observations. However, when I observed them using computers to complete the special assignments, they had active discussions about the assignment. They shared ideas about how to answer the question and how to use a computer effectively to complete their assignment. They were constantly interacting while they were using computers.
Teachers also expressed their ideas and opinions regarding how technology integration influences students' cooperative learning in their classes. In the first teacher interviews, there was no discussion pertaining to this topic, but in the second teacher interviews, teachers mentioned more about cooperative learning because they watched their own teaching.
Ms. B said that students would learn more when technology was available for them to explain and discuss their own thoughts with other students. She also mentioned that when students were allowed to use technology for their mathematics learning, teachers did not have to explain everything in class because students could check their own work. She also thought that students were more engaged and learned more dynamically when they were allowed to use computers and to work in groups.
While analyzing the class observations, I realized that the two teachers had similar ideas about cooperative learning. For example, Mr. A intentionally promoted cooperative learning by administering a team test. He intentionally included a question that was not familiar to students because it encouraged students to have involved interactions as they worked on their team tests. He also changed seating arrangements on 168 a regular basis to encourage students to work cooperatively. He believed that students
could have different points of view from other classmates as they sat with different
people.
Ms. B said that students were more motivated to do their work and focused on
their work when they worked collaboratively. They were especially more motivated when
they had a challenging assignment. For example, students worked on an assignment that
required using randomly-generated data. As students used their graphing calculators to
generate random numbers to solve the given problem, they discussed how they could
determine what type of random data they would need.
She also said technology-integrated instruction encouraged students to interact
and communicate more because they could easily see what they were talking about.
When students used their graphing calculators, they received instant feedback, their
discussions were more meaningful, and their calculators also clarified what they did not
understand.
Reponses of students who took two mathematics classes one from both Mr. A and
one from Ms. B were carefully analyzed because students may have compared the two
teachers' different ways of integrating technology in their instruction. In the comparison
of the two teachers, students reported no significant difference between them.
Question 4: How does educational technology influence mathematics instruction?
From the student interviews, I understood that students believed that the use of
technology influenced instruction directly and closely. They believed technology 169 integration would make teaching and learning easier because the teacher could use technology to make the concepts visual. The visualization of a concept not only allowed students to see how to solve certain problems but it also helped teachers to show their ideas effectively.
For example, students remembered how clearly they could understand how to use their graphing calculators when teachers used a video camera and a TV monitor to explain calculator procedures. Students pointed out that using a video camera and a TV monitor was an excellent way to teach because they understood better when teachers used them.
Students believed that teachers could use technology to provide students with more opportunities to practice their ideas by giving them more hands-on activities. For example, they talked about the casino day activities they experienced in AP statistics.
When they played the selected games, they learned some of the statistical concepts.
When teachers used graphing calculators effectively, instruction was more powerful. Students clearly remembered their experiences in learning with and without their graphing calculators. They solved a problem without using their graphing calculators before they learned how to solve the same problem with their graphing calculators. The difference was clear, and they leamed the effectiveness of technology integration in mathematics learning. Students were convinced that using graphing calculators in problem-solving made learning mathematics more understandable.
From the class observations, I could see the impact of technology integration on how teachers taught in the three classes. Teachers used technology every day in every 170 class for every lesson. Technology was implemented when demonstrating how to use a graphing calculator, when presenting and sharing students' and teachers' ideas, and when experimenting with different ideas.
Every day, AP statistics and AP calculus classes started with warm-up problems on TV monitors in class. To provide the warm-up problems, teachers used their computers and PowerPoint. In class, teachers also presented procedures for and explained how to use the graphing calculators by using a video camera and a TV monitor.
Students also shared their thoughts and ideas as they completed their assignments.
When teachers presented a problem, students usually explained their solutions and how they solved the problem. To present their ideas, students in the AP statistics class used a video camera and TV monitors, a whiteboard, and mini whiteboards. In the same AP statistics class, students had some assigmnents they had to complete on computer and used PowerPoint to present their work.
In the AP calculus class, however, stUdents used a video camera and TV monitor and a whiteboard. They did not have any other opportunity to use technology to do their work and present in class. However, they discussed problems as a class most of the time.
In the AP statistics class, students used their graphing calculators as suggested in the textbook. Students worked together as a group to figure out how to perform certain calculations. In the AP calculus class, however, students used calculators based on the teacher's instructions. They used special codes to perform certain types of problems. The teacher researched online and communicated with other calculus teachers across the 171 nation to learn different ideas and strategies to help students use their graphing
calculators as much as possible.
Teachers explained how they integrated technology for instructional purposes.
Mr. A used technology to pass out and collect assignments when he participated in the
Laptop Program. When he had lengthy information to pass out, for example, he simply
saved it on a disk and passed it around the class for students to copy onto their own
laptop computers.
After he started integrating technology into his iustruction, he relied on
technology so much that he simply could not effectively teach a fairly large class when
technology was not available. In the class, students could not understand the lesson when
they could not see or hear the instruction, so he used a video camera and TV monitors to
communicate with the large class. However, it was challenging for him to teach the large
class when the video camera was not working.
Ms. B talked about the benefits of technology integration in mathematics
education. She said, "Using technology would allow more visual ... it is an aid. It is not
that you cannot do it without, it but the argument is that it makes it better. It makes them
retain the information better because they are seeing it and feeling it and doing it, versus just practicing the long way on paper and they are not really getting the idea."
Analyzing the second teacher interviews, I noticed how the teachers used
technology for their instructions. Mr. A noticed that he used a video camera extensively
to clearly present instructions and to help students present their ideas. For example, he
was teaching an important theorem by using a function graph that he generated on his 172 graphing calculator. Instead of redrawing the graph on the whiteboard, he showed it on a
TV monitor and drew lines and shapes directly on the TV screen by using a dry-erase marker to explain the theorem. Students could clearly see the actual graph on the TV monitor and how to apply the theorem on a graph displayed onscreen It was successful and a great discovery for Mr. A.
Mr. A believed using technology would help students be more self-directed learners. He prepared the answer keys of the assigmnents for students. Given the keys, students could check their own work. Mr. A believed that having answer keys helped students understand the concepts better. He also used technology to organize grades and other information and to communicate with his students and their parents as well.
Another benefit of technology integration in mathematics class, according to
Mr. A, was visualizing mathematical concepts. He therefore believed that using graphing calculators and a video camera in class to show graphical representations of a function was an important part of the instruction.
As Ms. B watched her own class teaching, she recognized the way she was using an overhead projector to post instructions for students to start the class as they walked into the classroom. She thought the use of an overhead projector to present daily instructions was the most effective way of providing the instructions at the beginning of each class period.
She also believed technology integration helped students better understand and learn mathematics. Using a graphing calculator especially helped students visualize mathematical concepts and retain the information longer. 173 Ms. B also strongly believed technology integration would genemte more discussions, influence the way she would teach, and how students learn positively. She said technology altered her classroom structure and enhanced teaching mathematics.
She realized how much she actually integrated technology into her teaching as she watched her own recorded class sessions. When she watched multiple class sessions, she recognized that she was using more technology for teaching at the end of the study than she used to at the beginning of the study. Ms. B increased her technology integration as she learned how to use the tools.
Limitations of the Study
The first limitation of this study was lack of generalizability. I cannot generalize beyond the classes that I worked with, and the findings may not apply to other students and teachers. For obvious reasons, I cannot generalize from my study to all mathematics teachers and all students who learned calculus and statistics.
I cannot generalize what I found in this study beyond the classes I worked with.
However, it is my belief that teachers who read this study will recognize similarities in my descriptions of the classrooms and the study results with observations they have made in their own classrooms; thus, they may be able to take my findings and put it to good use in their classrooms.
The second limitation was the type oftechnology available. My observations and discussions were limited to what students used in their classes. For example, during class time, students had limited access to computers and to the Internet. I cannot talk about all 174 technology tools other than what participating students used. Teachers used computers, but this use was limited by available software. There are many kinds of software that could be part of a study. Potentially useful are many technologies like Skype™ and
Elluminate™--which were not available to the teachers and students in this study.
Conclusions and Recommendations
Concluding this study, I would like to further discuss some of the findings and make some recommendations. First, when students and teachers were asked what they thought the role oftechnology in mathematics learning was, they strongly believed that technology helped students visualize mathematical concepts and helped student learning.
Students said technology would help them learn mathematics in different ways. When they used technology, they believed that they could be more active learners because they controlled what and how they learn. Teachers pointed out that they could communicate with students more effectively when they integrated technology.
Second, students and teachers discussed how they used technology for mathematics education. Students believed that they learned mathematics better when they used their graphing calculators because of calculator speed and accuracy. They also said using their graphing calculators aided their learning because they concentrated on the procedures and logic instead of tedious calculations. Teachers pointed out that students understood various theorems better because they confirmed their solutions and felt comfortable when they used theorems. 175 Even though there were many positive impacts of technology integration in mathematics education, students pointed out that they could not understand the entire problem-solving process because their graphing calculators skipped many steps.
Teachers mentioned how students changed their attitude toward their assignments when they used technology tools. Students showed stronger ownership when they used computer technology to do their assignments.
Teachers used PowerPoint to present instructions and coursework (such as warm up problems) and to review concepts. They also used online tools such as the Internet to study instructional strategies and to communicate with other teachers, parents, and students. Teachers and students believed using a video camera and a TV monitor to give instructions was effective.
Third, students and teachers explained how technology integration encouraged cooperative learning. Students believed that when they used their graphing calculators, they had more discussions with their peers about what they studied. From the class observations, I saw many students actually communicated while they were working on a computer to complete a special assignment.
Some students believed that technology integration would not promote cooperative learning in mathematics learning. They thought that they would not discuss the problem with their peers when they had the correct answers to the problems they worked on. They said that they would collaborate with other students to get the correct answers. They believed that once they had the correct answers, they therefore would not collaborate any further with their peers. However, my observation of the classes and my 176 data analyses suggested that this perception was not supported. Contrary to this expressed
belief by students, I found almost all students acted cooperatively when they used technology.
Fourth, how does technology influence mathematics instruction? Students said
technology made instruction clear because it allowed them to visualize the concepts.
Teachers believed technology enhanced their instruction by allowing them to project
what was on a graphing calculator and what was drawn on the whiteboard. Technology
integration enabled teachers to present and teach mathematical concepts differently by
allowing them to show an accurate graph of a function on a TV monitor and to
manipulate graphs as discussion evolved.
There were three main findings common to the two teachers. They continuously
used their video camera to give instructions and to get students' attention. They practiced
more student-oriented teaching when they integrated technology by letting students
present their ideas freely. Lastly, they had more interactions with students as they
instructed while using integrated technology.
Recommendations for Teacher Education
From my study, I found that teaching mathematics with technology was more
effective when teachers clearly understood how to use technology tools and when
teachers showed students how to use these tools for very specific purposes. Therefore,
training teachers in how to use specific technology for their own teaching content would
be more effective instead of teaching how to use technology in general. Students strongly 177 indicated that when teachers used technology in their instruction, teachers' demonstrations of specific steps in graphic calculator functions and uses helped students learn better and reduced instructional time.
Providing teachers with numerous opportunities to observe how other teachers actuaIly use technology in classroom settings would be one of the most effective ways to prepare teachers to use specific technology tools for teaching particular concepts.
Offering mini hands-on workshops would be another way to prepare teachers. However, developing and offering different workshops according to what each teacher might need more specifically would be very challenging.
To accommodate what teachers need without putting in too much time, effort, and resources, on the part of both teachers and their trainers I would like to recommend that a collection of various video sessions be developed for training in each technology as an ideal and practical solution (Lu & Rose, 2003). One of the participating teachers suggested that teachers visit each other on a regular basis for a similar purpose. However, requiring teachers to visit each other could increase expenses and cause other unforeseen challenges.
If pre-service teachers can have opportunities to learn and use a database of
, video-recorded technology sessions, they can watch the sessions when they have time and as needed. If they reflect on some of the possible implications of what they watched and on how they would integrate what they learned into their future curriculum, the effect on their learning would be amplified. 178 From responses from and observations of teachers in my study, I found that there are some implications of my research for teacher education. Teachers needed a lot of good practical examples of how to integrate technology into their specific curriculum and instruction. They also wanted more hands-on, activity-based learning instead ofiong, traditional lectures. Teachers also needed to learn about different software that allows users to generate and manipulate visual representations of mathematical concepts software such as Cabri™, Fathom™, and the Geometers Sketchpad™.
Good teacher education programs for secondary mathematics teachers should include several areas of training: technology tools, learning labs, and opportunities for observation and reflection. First, technology tools are necessary; while they study in the colleges of education, all pre-service teachers should own or have access to software they would need to use when they teach in their own classrooms. Second, pre-service teachers in colleges of education would benefit from learning labs. In such labs they should have opportunities to practice and discuss what they found or what they did not understand regarding the use of certain technology tools. All pre-service teachers should have many examples of how to use technology and time to observe and reflect on using that technology. When they want to see how technology integration can be done effectively, they should be able to access the examples and compare them with what they are actually doing in practice. As they reflect upon what they can improve for their own technology integration, pre-service teachers would prepare themselves more effectively.
I believe that pre-service teachers would not consider integrating technology an additional job for them to perform if, as Nolan (2004) pointed out, they understand the 179 role of technology in education and have meaningful experiences in ways to integrate technology into their teaching in a well-planned teacher education program while they are taking college courses.
Recommendations for Professional Development ofExperienced Teachers
I think my study has some implications for professional development for veteran teachers. It was very clear that teachers understood the effectiveness of technology in their teaching and that they knew students would learn better and their teaching would be more effective with appropriate technology integration.
Teachers wanted to have more technology training and workshops to learn and understand how to implement different technologies into their own teaching. They were motivated to integrate different types of technology to improve their performance and to teach more effectively. As I conducted my study, participating teachers implemented more technology tools as they learned how to use different types of technology for their instruction.
However, if it takes too long for teachers to learn how to integrate technology, teachers would not be willing to spend the time. Teachers generally have a lot of work to complete in a very limited time, so they tend to hesitate to participate in any activity that would take their time away from completing their required job. There are currently too many meetings that teachers are mandated to attend, and those meeting times take up much valuable work time for teachers. One of the interviewed teachers pointed out that 180 using part of the meeting times for technology training would be a very effective way to provide essential technology training.
In general, most learning of how to use technology does not take too long or require a high level of technological skill or knowledge, according to my secondary and
College of Education teaching experiences. Therefore, developing short, practical, and effective professional development sessions would be the key to success.
Many times, teachers worried about dealing with technology because they did not have much experience in using it. Also, if they do not understand how to use it, they tend not to use it. In general, when teachers understand how to use a type of software, they tend to use more of the other types of technology tools as well. I believe helping teachers feel comfortable with using technology would be the best way to prepare and encourage them to use more technology in their classrooms.
I strongly believe that training teachers to use technology they already have in their daily lives would help them feel comfortable about integrating technology in their instruction. After my study was completed, I looked back and thought about how the findings of the study would help professional development. It was very clear that students and teachers comfortably used their graphing calculators but not all them knew how to use all the features of their graphing calculators.
While I was teaching in the Educational Technology Department, I taught teachers how to incorporate Microsoft Office™ into their teaching for various teaching purposes. Even though Microsoft PowerPoint,TM Excel,TM and Access™ are commonly included in most computers, not many mathematics teachers clearly know all the 181 available features of each software application and which of those features can be used for teaching mathematics. Therefore, they hardly incorporate them into their teaching. As
I taught this course, I noticed that teachers loved the content they were learning and were eager to implement the content into their lessons.
Another area in which we need to prepare teachers for effective technology integration would be in the use of online sources. There are a lot of specific content-based instructional materials available for teachers online. Web developing and searching skills would be very valuable and effective teaching tools for teachers when they make connections between their lessons to real life situations. For example, if teachers could pullout a set of data from a Website of an organization students want to work for and have them discuss the data as part of class learning, that would definitely motivate students to learn more.
Due to the vast amount of information on the Internet, it is very time consuming for each teacher to evaluate all of the information efficiently and effectively. Therefore, development of a meaningful online resource database should be accomplished through departmental collaborative efforts. Teachers can collect and use real data from numerous organizations that would help students apply their mathematical knowledge and understand the concepts in practical settings.
When teachers feel comfortable using in class the available basic software, technology tools, and online searches and resources, they can start learning how to use video sources in their instruction. Teaching teachers how to use video files and teaching them the associated editing skills would be a good direction for professional 182 development. When teachers learn how to use video files effectively, they can create their own instructional materials to meet their needs for their own classes. Teaching them how to edit video files would be a very powerful tool for classroom teachers.
Recommendations for Technology Integrated Mathematics Classrooms
I am going to present a picture that I believe will allow teachers to take full advantage of technology integration. This is what I believe should be happening in technology-integrated mathematics classrooms. Technology-integrated mathematics education should allow students to work on more challenging problems without allowing them to worry about how to use their technology tools. Students should focus on learning the mathematics contents rather than on how to use tools. With technology integration, students should be able to communicate and exchange their ideas and thoughts with other students and their teacher as easily as they use their cell phone to call their friends.
In this study, when students walked into the classroom, they began to work on their warm-up problem by watching the TV monitor, because teachers had already used their computers and PowerPoint to post the problem on TV. Students used their graphing calculators as they worked on the problems and discussed the problems with their peers.
When they could not understand or could not solve any part of the problem, they looked up information in their textbook or waited for their teacher to begin the discussion.
When they understood a particular problem and wanted to share their solution, ideas, and suggestions, they put their calculators under the video camera to show the class. The camera transmitted their image or solution onto the TV monitor. After 183 presenting or sharing, students discussed their work with their peers and teacher, who usually gave feedback or made comments about the presentations.
Unfortunately, many times students had to wait for clarification from their teacher to confirm what they did before they proceeded to the next step. If the teacher was busy or engaged in discussions with another student, students could not make a decision on their own, in many cases. Students were generally passive learners for both mathematics and in the use of technology tools. Until the teacher explained how to use certain features of the device, they were not comfortable trying to use it on their own. The teacher centered instruction was strong, even though technology was used by both teachers and students.
However, in ideal technology-integrated mathematics classrooms, students should be able to experiment with their own ideas without waiting for a teacher's permission. To turn students into self-motivated and active learners, as Schwartz (1993) described, students should have access to technology outside of classroom and they should use it actively. This can be done by giving students more of the technology tools they need to use for studying mathematics or allowing them to use technology tools they already possess. These tools include gaming, online learning, and mobile devices. When students use their technology devices without worrying about how to use them, their learning would be improved.
Students voiced their opinions about the role technology had in their education and the influence of emerging educational technology such as gaming, online learning, and mobile devices. They use technology on a daily basis and, in many ways, are far 184 ahead of their teachers and parents not only in the sophistication of their technology use, but in the adaption of emerging technologies for learning purposes.
Are mathematics classes currently taking full advantage of the available technology in their classrooms? I do not believe so. For example, instead of using a video camera to show the whole class how to use the buttons on a calculator, a teacher could create a mini instructional video clip in the form of a Podcast so students could watch it whenever they need the instruction. Perhaps students could create their own version of instructional video clips to share with other students.
What type of technology usage would make mathematics education even better?
Some educational software programs promote mathematical learning-programs such as the Geometers Sketchpad, Cabri, and Fathom. When students work with these types of software, they can manipulate objects on computer screens, and that feature would help students understand mathematical concepts. However, there is much open-source content on the Internet that has already been collected and organized by teachers and that can help students study mathematics. For example, the hugely popular, user-generated online encyclopedia Wikipedia (www.wikipedia.org) would be an excellent resource that students can take advantage of when they study mathematics.
Future mathematics education would be heavily based on integrated technology.
The success of mathematics education may strongly depend on how well students use technology tools. Students willieam the same content but in a very different context.
Therefore, preparing teachers and students for efficient technology usage will be one of the keys to successful mathematics education in the classroom of the future. 185
APPENDICES
Appendix A: Questionnaire
Thank you for taking the time to answer this briefsurvey. Your honest answers will contribute to expanding what is known about mathematics edueation. Your participation is strictly voluntary and should you choose to complete this survey, I assure you that responses will never be identified in any way with you as an individual Therefore, please do not place your name, SS#, DOB, etc. on this form. I will be the only person who sees these forms-they will never be seen by your teacher, TA, or anyone associated with your grade. Mahalo for your Participation! Tae Ha. taeha@,hawaiLedu. (808) 277-4176.
1. Gender: Male Female
2. Grade Level: Freshman Sophomore Junior Senior
3. Class: AP Statistics AP Calculus
4. What is your anticipated career field (e.g. Engineer)?
5. Why did you choose an AP course (as opposed to "regular" course)?
6. Is this your first time to take AP course? Yes No
7. Please list all math classes taken so far in high school?
For the following statements, please indicate your level of agreement with these statements by circling the appropriate number from 1 = Strongly Disagree (SD) to 4 = Strongly Agree (SA). The term "educational technolo2Y" denotes any technology you use for learning such as JUllphing calculator, TV, or computer.
SD D A SA 8. I have experience with educational technology in different class 1 2 3 4 before. 9. Educational technology should be available to all students who 1 2 3 4 enroll in math and science. 10. Educational technology is primarily designed for students who will enter the work force immediately after graduating from high 1 2 3 4 school. 11. Educational technology is designed for students who will 1 2 3 4 pursue a college degree after graduation from high school. 186 12. Educational technology is most appropriate for students 1 2 3 4 enrolled in special education. 13. Using educational technology will help students develop a skill I 2 3 4 by making projects. 14. The idea of using educational technology in a curriculum 1 2 3 4 reflects industry. 15. Educational technology does not scare me. 1 2 3 4 SD D A SA 16. The effective usage of educational technology is guided by the 1 4 technological literacy needs of students. 2 3 17. Educational technology is for only applied science. 1 2 3 4 18. My experience with educational technology was positive. 1 2 3 4 19. I am able to get the help that I need in this class. 1 2 3 4 20. I feel comfortable working in groups in this class. 1 2 3 4 21. I use computer to complete work outside of class time. 1 2 3 4 22. I prefer to work by myself. 1 2 3 4 23. I prefer to work in a group setting. 1 2 3 4 24. I have to do more self-motivation in this class than in other 1 2 3 4 math classes I take. 25. People who like computers like mathematics. 1 2 3 4 26. I am a more active learner in this class versus more traditional 1 4 math classes since more educational technology is available. 2 3 27. Teacher doesn't use educational technology in an adequate 1 4 way in this class. 2 3 28. Using computer will be necessary in my anticipated career 1 field. 2 3 4 29. Technology could be helpful in my anticipated career field. 1 2 3 4 30. Using educational technology in a class is mainly to help 1 2 3 4 students learn distinct machine skills. 31. I learn less when I have to use educational technology than I 1 2 3 4 would without educational technology. 32. Using educational technology helps me understanding 1 2 3 4 mathematical concepts better. 33. When I use educational technology I am motivated to learn 1 2 3 4 mathematics. 34. When my teacher uses educational technology to explain a 1 2 3 4 mathematics concept, I can understand better. 35. Educational Technology helps me learning in general. 1 2 3 4 36. Educational Technology helps me learning mathematics. 1 2 3 4 37. If I could drop this course, I would. 1 2 3 4 38. I think that I am doing well in this class. 1 2 3 4 39. Grades are my principle way to judge how well I am doing in 1 2 3 4 this class. 40. This mathematics is important 1 2 3 4 187 41. I feel comfortable making mistakes in this class. 1 2 3 4 42. The mathematics that I learn in this class applies to the real 1 2 3 4 world. 43. Women are better than men at subject like English and 1 2 3 4 History. 44. Making a mistake is part of the learning process in this class. 1 2 3 4 45. This class is different than other mathematics classes that I 1 2 3 4 have taken. 46. I have used calculator in my previous math classes. 1 2 3 4 SD D A SA 47. I prefer calculators to computers in mathematics class. 1 2 3 4 48. Memorizing is important in mathematics class. 1 2 3 4 49. Mathematics is a very structured branch of knowledge. 1 2 3 4 50. Men are better at mathematics than women. 1 2 3 4 51. Time is highly organized and structured in this class. 1 2 3 4 52. I wish that time was more structured in this class. 1 2 3 4 53. My grade in this class seems to be a fair reflection of my 1 2 3 4 understanding of the material. 54. I had wrong answer due to technological error. 1 2 3 4 55. Mathematics frustrates me. 1 2 3 4 56. Everyone can learn in this class. 1 2 3 4 57. It is easier to learn the content of this class on the computer 1 2 3 4 than it would be to learn it in a lecture/recitation format. 58. It is easier for me to learn from the computer. 1 2 3 4 59. Using educational technology frustrates me. 1 2 3 4 60. I learn a lot from the teacher in this course. 1 2 3 4 61. I am motivated to use more educational technology for 1 2 3 4 learning new ideas in this class. 62. Men do better than women in mathematics classes. 1 2 3 4 63. I am a visual learner. 1 2 3 4 64. Mathematics and computers go together well. 1 2 3 4 65. I learn a lot from myself in this course. 1 2 3 4 66. This class would be very different with a different teacher. 1 2 3 4 67. People who like mathematics like computers. 1 2 3 4 68. Mathematics makes me feel uneasy and confused. 1 2 3 4 69. Mathematics is a very worthwhile subject for everyone. 1 2 3 4 70. Mathematics is enjoyable and stimulating to me. 1 2 3 4 71. Mathematics has been my worst subject. 1 2 3 4 72. Mathematics helps develop a person's mind and teaches 1 2 3 4 him/her to think logically. 73. Mathematics problems can be solved in various ways by using 1 2 3 4 different representations such as tables, graphs, and equations. 74. I like using more than one representation such as graphs, 1 2 3 4 tables, and equations to solve mathematics problems. 188 75. Given a mathematics problem, I find it easier to focus on one 1 3 4 representation than to deal with many representations. 2 76. When a mathematics problem is presented with more than one representation, it means that there are as many questions as 1 2 3 4 representations. 77. Solving a mathematics problem with different representations such as graphs, tables, and equations results in totally different 1 2 3 4 answers.
After this point, please choose one of the options that indicates yonr opinion the most and explain brief] • Mabalo a in! 78. I like using D Equations, D Tables, D Graphs the most when solving mathematics problems. Because
79. D Equations, D Tables, D Graphs make mathematical topics the easiest for me to understand. Because
80. IfD Equations, D Tables, D Graphs were the only option I had to solve mathematics problems then I would have the most difficult time doing the problem. Because
81. I frod the hardest to construct D Equations, D Tables, D Graphs when solving problems using paper and pencil. Because
82. I will usually start solving mathematics problems with D Equations, D Tables, D Graphs. Because
83. I find D Equations, D Tables, D Graphs confusing when working on a mathematics problem. Because
84. Using educational technology in our class was D Confusing, D Helpful in understanding mathematical representations such as graphs, tables, and equations. Because 189
85. Educational technology is valuable when I solve mathematics problems using equations 0 Yes D No. Because
86. Educational technology is valuable when I solve mathematics problems using tables o Yes DNo. Because
87. Educational technology is valuable when I solve mathematics problems using graphs DYes 0 No. Because 190 Appendix B: Questionnaire Results
Thank you for taking the time to answer this brief survey. Your honest answers will contribute to expanding what is known about mathematies education. Your participation is strictly voluntary and should you choose to complete this survey, I assure you that responses will never be identified in any way with you as an individuaL Therefore, please do not place your name, SS#, DOB, etc. on this form. I will be the only person who sees these forms---they will never be seen by your teacher, TA, or anyone associated with your grade. Mahalo for your participation! Too Ha. taehalalhawaiLedu. (808) 277-4176.
1. Gender: Male (53%) Female (47%)
2. Grade Level: Freshman (0%) Sophomore (0%) Junior (12%) Senior (88%)
3. Class: AP Statistics (68%) AP Calculus (26%) Both (6%)
4. What is your anticipated career field (e.g. Engineer)? (Most common response to this item was "Engineer.") See Appendix C.
5. Why did you choose an AP course (as opposed to "regular" course)? (Most common response to this item was "College Preparation.") See Appendix D.
6. Is this your first time to take AP course? Yes (97%) No (3%)
7. Please list all math classes taken so far in high school? (Most common response to this item was "IMP 1,2,3, & Trigonometry.'') See Appendix E.
For the following statements, please indicate your level of agreement with these statements by circling the appropriate number from 1 = Strongly Disagree (SD) to 4 = Strongly Agree (SA). The term "educational technology" denotes any technolol!Y you use for leaminll; such as waphinll; calculator, TV, or computer.
SD D A SA 8. I have experience with educational technology in different class 0% 3% 35% 62% before. 9. Educational technology should be available to all students who 0% 3% 21% 76% euroll in math and science. 10. Educational technology is primarily designed for students who will enter the work force immediately after graduating from high 18% 47% 21% 15% school. 11. Educational technology is designed for students who will 9% 21% 53% 15% pursue a college degree after graduation from high school. 12. Educational technology is most appropriate for students 32% 35% 21% 9% eurolled in special education. 13. Using educational technology will help students develop a skill 0% 12% 59% 26% 191 by making projects. 14. The idea of using educational technology in a curriculum 0% 6% 76% 18% reflects industry. 15. Educational technology does not scare me. 3% 6% 32% 59% SD D A SA 16. The effective usage of educational technology is guided by the 3% 15% 64% 18% technological literacy needs of students. 17. Educational technology is for only applied science. 20% 68% 6% 6% 18. My experience with educational technology was positive. 3% 3% 53% 41% 19. I am able to get the help that I need in this class. 0% 6% 41% 53% 20. I feel comfortable working in groups in this class. 0% 3% 44% 53% 21. I use computer to complete work outside of class time. 3% 27% 38% 32% 22. I prefer to work by myself. 3% 35% 44% 15% 23. I prefer to work in a group setting. 0% 18% 71% 9% 24. I have to do more self-motivation in this class than in other 6% 24% 29% 41% math classes I take. 25. People who like computers like mathematics. 6% 61% 24% 9% 26. I am a more active learner in this class versus more traditional 0% 35% 47% 15% math classes since more educational technology is available. 27. Teacher doesn't use educational technology in an adequate 44% 50% 3% 3% way in this class. 28. Using computer will be necessary in my anticipated career 0% 3% 53% 44% field. 29. Technology could be helpful in my anticipated career field. 0% 0% 32% 68% 30. Using educational technology in a class is mainly to help 0% 56% 38% 6% students learn distinct machine skills. 31. I learn less when I have to use educational technology than I 20% 65% 15% 0% would without educational technology. 32. Using educational technology helps me understanding 0% 18% 64% 18% mathematical concepts better. 33. When I use educational technology I am motivated to learn 3% 38% 50% 6% mathematics. 34. When my teacher uses educational technology to explain a 3% 15% 70% 12% mathematics concept, I can understand better. 35. Educational Technology helps me leaming in general. 3% 15% 70% 12% 36. Educational Technology helps me leaming mathematics. 6% 9% 73% 12% 37. If! could drop this course, I would. 50% 29% 15% 3% 38. I think that I am doing well in this class. 15% 12% 59% 15% 39. Grades are my principle way to judge how well I am doing in 6% 32% 38% 24% this class. 40. This mathematics is important 3% 3% 59% 35% 41. I feel comfortable making mistakes in this class. 12% 35% 32% 21% 42. The mathematics that I learn in this class applies to the real 3% 12% 53% 32% world. 192 43. Women are better than men at subject like English and 32% 53% 9% 6% History. 44. Making a mistake is part of the learning process in this class. 0% 3% 59% 38% 45. This class is different than other mathematics classes that I 0% 3% 41% 56% have taken. 46. I have used calculator in my previous math classes. 0% 0% 38% 62% SD D A SA 47. I prefer calculators to computers in mathematics class. 0% 9% 53% 38% 48. Memorizing is important in mathematics class. 3% 21% 52% 24% 49. Mathematics is a very structured branch of knowledge. 0% 3% 62% 35% 50. Men are better at mathematics than women. 35% 38% 21% 6% 51. Time is highly organized and structured in this class. 0% 12% 59% 29% 52. I wish that time was more structured in this class. 12% 56% 21% 12% 53. My grade in this class seems to be a fair reflection of my 9% 21% 47% 21% understanding of the material. 54. I had wrong answer due to technological error. 24% 44% 29% 3% 55. Mathematics frustrates me. 24% 29% 38% 9% 56. Everyone can learn in this class. 0% 6% 62% 32% 57. It is easier to learn the content of this class on the computer 21% 58% 18% 3% than it would be to learn it in a lecture/recitation format. 58. It is easier for me to learn from the computer. 9% 62% 26% 3% 59. Using educational technology frustrates me. 26% 56% 18% 0% 60. I learn a lot from the teacher in this course. 0% 6% 29% 65% 61. I am motivated to use more educational technology for 0% 21% 68% 9% learning new ideas in this class. 62. Men do better than women in mathematics classes. 38% 35% 18% 3% 63. I am a visua1learner. 0% 0% 62% 35% 64. Mathematics and computers go together well. 0% 9% 73% 18% 65. I learn a lot from myself in this course. 6% 24% 59% 12% 66. This class would be very different with a different teacher. 3% 12% 18% 68% 67. People who like mathematics like computers. 9% 50% 26% 12% 68. Mathematics makes me feel uneasy and confused. 24% 46% 24% 6% 69. Mathematics is a very worthwhile subject for everyone. 3% 21% 56% 21% 70. Mathematics is enjoyable and stimulating to me. 6% 21% 47% 26% 71. Mathematics has been my worst subject. 41% 38% 12% 9% 72. Mathematics helps develop a person's mind and teaches 3% 3% 56% 35% himlher to think logically. 73. Mathematics problems can be solved in various ways by using 0% 0% 44% 56% different representations such as tables, graphs, and equations. 74. I like using more than one representation such as graphs, 9% 18% 52% 21% tables, and equations to solve mathematics problems. 75. Given a mathematics problem, I find it easier to focus on one 0% 18% 56% 24% representation than to deal with many representations. 76. When a mathematics problem is presented with more than one 0% 50% 38% 6% 193 representation, it means that there are as many questions as representations. 77. Solving a mathematics problem with different representations such as graphs, tables, and equations results in totally different 18% 62% 18% 0% answers.
Aft.r this point, pleas. choose one of the options that indicates yonr opinion the most and explain brietl • Mahalo a ainl 78. I like using D Equations, D Tables, D Graphs the most when solving mathematics problems. Because (See Appendix F) Itern# E T G Multi NR 22 6 5 1 78 o 65% 18% 15% 3% 0%
79. D Equations, D Tables, D Graphs make mathematical topics the easiest for me to understand. Because (See Appendix G) Itern# E T G Multi NR 9 6 17 1 1 79 26% 18% 50% 3% 3%
80. IfD Equations, D Tables, D Graphs were the only option I had to solve mathematics problems then I would have the most difficult time doing the problem. Because (See Appendix H) Item # E T G Multi NR 10 11 12 1 80 o 29% 32% 35% 0% 3%
81. I fmd the hardest to construct D Equations, D Tables, D Graphs when solving problems using paper and pencil. Because (See Appendix I) Itern# E T G Multi NR 7 5 20 2 81 o 21% 15% 59% 0% 5%
82. I will usually start solving mathematics problems with D Equations, D Tables, D Graphs. Because (See Appendix J) Item # E T G Multi NR 22 4 5 2 1 82 65% 12% 15% 6% 3% 194
83. I find 0 Equations, 0 Tables, 0 Graphs confusing when working on a mathematics problem. Because (See AppendixK) Item # E T G Multi NR 13 3 11 7 83 o 38% 9% 32% 0% 21%
84. Using educational technology in our class was 0 Confusing, 0 Helpful in understanding mathematical representations such as graphs, tables, and equations. Because (See Appendix L) Item # C H CIH NR I 32 I 84 o 3% 94% 3% 0%
85. Educational technology is valuable when I solve mathematics problems using equations 0 Yes 0 No. Because (See Appendix M) Item # Y N YIN NR 28 5 I 85 o 82% 15% 0% 3%
86. Educational technology is valuable when I solve mathematics problems using tables DYes 0 No. Because (See Appendix N) Item # Y N YIN NR 27 6 I 86 o 79% 18% 0% 3%
87. Educational technology is valuable when I solve mathematics problems using graphs o Yes o No. Because (See Appendix 0) Item # Y N YIN NR 30 3 1 87 o 88% 9% 0% 3% 195 Appendix C: Student Responses to Item #4
Item # nses 1. Business 2. Military 3. Travel Agent 4. Pharmacist 5. Education 6. Nursing 7. Corporate Lawyer 8. Business 9. Business Admin. 10. Education 11. Graphic Artist 12. International Relations 4. What is 13. Mechanical Engineer your 14. Computer EngineerlProgrammer anticipated 15. Mechanical Engineering career 16. Engineer field (e.g. 17. Computer Security Engineer Engineer)? 18. Pharmacist 19. Optometrist 20. Aviation/Astronomer 21. Forensics 22. ArtIdesign 23. Engineer 24. Engineer 25. Medical Doctor 26. Nursing 27. Medical 28. Science Engineer 29. Graphics 30. Mathematics Related 196 Appendix D: Student Responses to Item #5
Item # Responses 1. Challenge 2. Teacher Recommendation 3. Less interest in other courses 4. College prep. & Challenge 5. Future Work & College 6. Challenge 7. College prep. 8. College prep. 9. Challenge 10. To take higher math course 11. Liked the teacher 12. College prep. 13. College prep. 5. Why 14. Want to learn calculus did you 15. College prep. choose an 16. Like challenges APcourse 17. Like challenges (as 18. Like challenges opposed 19. College prep. to 20. College prep. ''regular'' 21. Requirement course)? 22. College prep. 23. College prep. 24. College prep. 25. Availability 26. Like challenge 27. Teacher recommendation 28. College prep. 29. Like challenge 30. College prep. 31. College prep. 32. My choice 33. College prep. 34. College oreo. 197 Appendix E: Student Responses to Item #7
Item # Responses 1. IMP 1,2,3 2. IMP 1,2,3, Pre-Calc. 3. IMP 1,2,3 4. IMP 1,2,3, Trigonometry 5. IMP 1,2,3, Trigonometry 6. IMP 1,2,3, Trigonometry & Pre-Calc. 7. IMP 1,2,3 & Trigonometry 8. IMP 1,2,3 9. Algebra I, IMP 1,2,3 10. IMP 1,2,3 l1.IMP 1,2,3 12. IMP 1,2,3 13. IMP 1,2,3, Trigonometry, & Pre-Calc. 7. Please 14. Algebra 1,2, Geometry, IMP 3, Trigonometry, & Pre-Calc. list all 15. IMP 1,2,3, Trigonometry, & Pre-Calc. math 16. IMP 1,2,3, Trigonometry, & Pre-Calc. classes 17. IMP 1,2,3, Trigonometry, & Pre-Calc. taken so 18. IMP 1,2,3, Trigonometry, & Pre-Calc. far in high 19. IMP 1,2,3, Trigonometry, & Pre-Calc. school? 20. Geometry, Algebra 2, Trigonometry, & Pre-Calc. 21. IMP 1,2,3, Geometry, & Trigonometry 22. IMP 1,2,3 23. IMP 1,2,3 24. Algebra 1, Geometry, Trigonometry 25. IMP 1,2,3 26. IMP 2,3 27. IMP 1,2,3 28. Pre-Algebra, IMP 2,3 29. IMP 1,2,3 30. IMP 1,2,3 31. IMP 1,3 32. IMP 1,2,3 33. IMP 1,2,3, Trigonometry, & Statistics 198
Appendix F: Student Responses to Item #78
ltem# Choice Responses 1. I can easily break it down. (easy to understand) 2. They are faster and less time consuming to solve with. (easy to understand) 3. They are easy to interpret (easy to understand) 4. I like the algebraic part (personal preference) 5. It is straight up in front of me and it's easier with definite answers. (clarity) 6. I am not very good at drawing or making tables and graphs. (less time consuming) 7. They are easier to solve. (easy to understand) 78 8. There is a set formula. (easy to understand) 9. I can see the numbers and work the equation using algebra. "I likeo using (easy to understand) Equations, 10. I can easily calculate using the equation. (clarity) 11. I find it less work. (less time consuming) Tables, o 12. It is more understandable. Equations are exact and easy to o Graphs E the most spot errors. (clarity) when 13. By remembering and practicing math formulas I understand solving better. (easy to understand) mathematics 14. Equations are what stick in my head most of the time and I problems. can see visually what my mistakes are. (personal preference) Because" 15. They are straightforward. (clarity) 16. I can easily solve the problems by using algebra. (personal preference) 17. It helps me think logically as opposed to visual representations. (clarity) 18. It seems easiest for me. (easy to understand) 19. I like working with numbers rather than with bars and lines and drawings. (personal preference) 20. It's easier. (easy to understand) 21. It is systematical and requires less analysis. (clarity) 22. Easiest to work with .. ~ to understand)
T 1. It spreads out your data easily . (easy to understand) 2. All the data is easy to find and the information is organized. (easy to understand) " " •. 1.. .11 Lt. "1.. .1..1. /.' ' ..' J. ,~~~ " 4. In this subject tables simplify what you have for your data. (clarity) 5. They are easy to read and understand the data that was given. 199 (easy to understand) 6. It is easy out of the 3 for me to remember and the table is the most accurate out the 3. (easy to understand) 1. You can visually see the data. (clarity) 2. They are easier to understand as a visual learner. (easy to understand) G 3. A visual always helps to better understand the problem. (easy to understand) 4. It's easy to understand. (easy to understand) Multi 1. (T/G) I am a visual learner. (easy to understand) 200 Appendix G: Student Responses to Item #79
Item# Choice Responses 1. Teacher's explanation helps me better to understand the 79 math. (teacher's help) 2. They summarize the data that is generally used. (clarity) "[] 3. I keep in mind that both sides have to balance each other in Equations, order for my answer to be correct. (clarity) DTables, 4. The equation ties in with the topic. (personal preference) DGraphs 5. You could see the different steps to solve the equation. make E (clarity) mathematical 6. I have to understand what I am doing in order to solve it. topics the (clarity) easiest for 7. They are easier to understand and all I have to do is solve the me to equation. (simplicity) understand. 8. With equations you get to practice and understand more. Because" (clarity) 9. I understand them better. (personal preference) 1. The data is right in front of you. (clarity) 2. All the data is easy to find and the information is organized. (helps organizing data) 3. You can see what happens. (visnalize) T 4. All your data is nice and neat. (helps organizing data) 5. It shows you the data, which is precise rather than graphs that you mostly guess at the data (clarity) 6. They are easier to understand and the data is accurate. (clariJy) 1. I am a visual learner. (visnalize) 2. They show the problem in a visnal way. (visnalize) 3. There are visnals. (visnalize) 4. It's a visnal representation of the equations and tables. (visualize) 5. You can visnally see the data (visualize) 6. I am a visnallearner. (visnalize) G 7. They are visually assisting. (visnalize) 8. Visnal representation is good for me. (visualize) 9. I graph the numbers. (visualize) 10. I can visually see what is occurring. (visualize) 11. It gives me a visnal aide to help me understand a mathematical concepts what teacher is trying to explain. (visnalize )
G 12. They are visnal and I can "see" the problem/situation. (visua '". ". •. . ,'.' . ."., 14. It is a visual representation. (visualize) 201
15. Easy to reacl. (easy to understand) Multi 1. (BIG) You find patterns/trends in the data (visualize) 202 Appendix H: Student Responses to Item #SO
Item# Choice Responses 1. Gives me a headache. 2. Too many numbers. 3. I don't think letters should be in math problems since they make the equations more confusing. 4. I could not view it in any other form. 5. They can also be confusing. 6. Sometimes you don't know when to plug in where to use E certain functions. 7. Although I like using equations, they get confusing sometimes. S. Too much steps might forget steps if! haven't practice them SO in a long time. 9. There are times when the equation may go wrong. If 0 10. I do not know many equations. Equations, 1. You would need to find the equation and have to make the Tables, o graph anyway, so it's sort ofa step backwards. Graphs o 2. You have to figure out the equation yourself and you don't were the have the data drawn out on a graph. only option 3. I rely more on equations and graphs. I had to I would need to find the pattern myself. A table would be solve 4. confusing because it would be a group of numbers with an mathematics problems idea of how they were created. I can't process math patterns in my head, I'd rather write out then I T 5. calculations. would have the most 6. Tables just give out straight numbers where it is harder to see the relationship. difficult It is not as easy to "view" data in just numbers as opposed to time doing 7. seeing a sloped line on a graph. the S. I find tables too boring. problem. Because" 9. I don't know what equations to use for. 10. Sometimes confusing. 1. Too much info at once. 2. Creating graphs can sometimes be confusing and unclear. Graphs take a lot of time to make. 3. There is no equations or exact #'s. All of them are at the same level to me. G 4. I sometimes get confused with the points and their location. 5. I would have to look for the data. 6. I need more than just graphs to solve a problem. 7. You can't get the numbers first. S. You cannot see the numbers. 203 9. Depending on the problem it's not easy to calculate exact numbers. 10. I have difficulty seeing the answer in a graph. 11. All of your data will show but you are uncertain of the actual # is. 12. It is hard to read graphs that are not clear. 204 Appendix I: Student Responses to Item #81
Item# Choice Resoonses 1. Some equations are confusing. 2. You need all the components in order to make the correct equation. E 3. I think more, but it's good! 4. It must work for every data you received. 5. Too much steps, might forget them ifI don't practice it often. 6. I cannot solve it in mv head. 1. I am a perfectionist and it takes too much precision. 2. It takes longer. T 81 3. Tables make it a little bit harder to find the relationship. 4. There is a lot of writing and I don't have the patience. "I find the 1. It's not neat. hardest to 2. I write sloppy. construct 3. Graphs take a lot of time just to solve one question. 0 4. They are not exact. Equations, 5. You have to draw the graph and eyeball the points and it o Tables, wouldn't be accurate. o Graphs 6. It does not come out accurate. when 7. My lines are never straight and sometimes they are incorrect. solving 8. I have to figure out how to label the area. problems 9. I can't draw straight lines. using 10. You need to draw it fairly accurately. paper and G 11. I didn't memorize the specific graphs. pencil. 12. I cannot see the graph too well. Because" 13. I like using the calculator. 14. When I draw graphs they are not accurate. 15. Graphs might not be drawn accurately causing inaccurate answers. 16. Graphs are still a mystery to me, so I don't remember all the rules. 17. Takes a while to set up. 18. It is hard to get the graph right. 19. Proportions are usually inaccurate. Labeluig/numbering axis can throw off graphing. 205 Appendix J: Student Responses to Item #82
Item# Choice Responses 1. I am used to it. I was taught that way. 2. They take the least amount of time to solve, as long as you understand it. 3. I like algebra. 4. It gives me exact answers and I can calcu1ate other answers. 5. They are easy to set up and I understand them very well. 6. I like using equations the most. 7. I can further elaborate with a graph after figuring the equation. 8. It is what we mostly work with. 9. It's a logical process. 10. That's what's used mostly. E 11. I am able to accurately solve the problem. 82 12. I am most comfortable with them. 13. It is the most accurate. "I will 14. Equations are easy for me to comprehend and use. usually start 15. It's easiest form if! know what I am doing. solving 16. They are straightforward and can solve using algebra easily. mathematics 17. That's what math is all about. problems 18. They are simple most of the time. with 0 19. It is easiest to use for me. Equations, 20. I don't like drawing out tables or graphs. I just go straight to o Tables, answering the problems. o Graphs. 21. Everything is in front of me and I will understand it as I go Because" thromm it. 1. It spreads out your data easily. 2. Tables are easy to read & understand. 3. The data is organized. T 4. These are easy to make and the numbers are precise. 5. You can make a graph out of a table and you can make an eQuation with a table data. 1. Plot things out. 2. You can visually see data. G 3. My mind has a visual concept. 4. I can recognize and understand them. 1. (Etr/G) All-which ever is comfortable. Mu1ti 2. CEff/G) If denends on the nroblem that is askine: vou. 206 Appendix K: Student Responses to Item #83
ltem# Choice Responses 1. Confusing. 2. Too many numbers. 3. You need to save for something to set the right answer. 4. Some are really big and you cannot solve quickly. 5. I don't know. 6. I like using equations the most. 7. One mistake can affect the overall outcome. 8. Sometimes I find it hard to come up with one if a problem E requests it. 9. Sometimes, I can't remember the equation. 10. Most equations have many variables. 83 11. Sometimes I don't understand how the equations fit in with the problem. "I find 0 12. You tend to forget stuff sometimes. and I tend to forget some Equations, of the steps sometimes. DTables, 13. I don't know manv eauations. DGraphs 1. It is not directly clear how the numbers were formed. confusing T 2. It does not say or you don't see a relationship. when 3. It's all numbers with no nictures. working on 1. Sometimes. not aJJ parts of graphs are clear to understand and a will therefore take longer to solve. mathematics 2. Sometimes I don't know how to read them. problem. 3. You have to look for the data. Because" 4. It is harder to get numbers. S. I usuaJly don't understand how equation matches the curve. G 6. I cannot see the numbers. 7. Graphs aren't that easy to read when you don't understand. 8. Sometimes you have to look a certain interval or change the range. 9. You need to guess on aJJ of the data. 10. It is so large. At times graphs can be misleading and throws me off. 1. (None) None of them is confusing to me. Multi 2. (None) None-Understand all. 2Cr7 Appendix L: Student Responses to Item #84
Item# Choice Responses 1. It visualizes the concepts. 2. It broke it down for me. 3. Using computers to interpret tables and graphs are a lot faster and easier to show when there are a lot of values to show. 4. I don't want to do all the work by hand. 5. It made visualization possible. 6. You can see the problem from all angles. 7. It's easier than doing it by hand. 8. They make it easier and faster to solve a problem. 9. It was easy to create, graphs, tables and equations. 10. I can't solve anything without a TI-83. 11. It does work faster. 84 12. It helped me understand their counection. 13. I can see it. "Using 14. I was able to understand the concepts better. educational 15. I am a visual learner. technology in 16. On the calculator they are more accurate than ifhand our class was written and also much faster. o Confusing, H 17. We don't use a lot of technology in the class anyway but I o Helpful in find calculators helpful. understanding 18. We could present the problems to help our peers mathematical understand. representations 19. It helped us to understand relationships. such as 20. It helps aid me through the problem solving process. graphs, tables, 21. It explains a lot and easy to understand. and equations. 22. Helps me to grasp the concept much better. Because" 23. The Technology made it easier to view and making it easier to understand. 24. You learn from it and that it's a really helpful to use. 25. We were shown how to do it and shown how it relates in answering the problem. And it also made it a lot easier to answer the problems. 26. That is what math is all about graphs, tables, and equations. 27. Technology is becoming a big part oftoday's life. 28. It is easier to understand and graphical displays are better than hand drawn. 29. It gives us a visual way to comprehend. C 1. Understanding some concepts is difficult. Multi 1. (HIC) There are so many different functions you have to 208 remember but at the same time it's very helpful. 209 Appendix M: Student Responses to Item #85
Item# Choice Responses 1. Easier. (Easy) 2. Without calculators it would take a long time for one problem. (Fast) 3. Educational technology helps us to solve and plug-in values into equations, rather than having to solve by hand. (Fast) 4. Without it we would get low grades. (Grade) 5. I would go crazy without calculators. (Easy) 6. It's faster than doing it by hand. (Fast) 7. It's easier to solve problems. (Easy) 8. All I have to do to solve the equation is type it into the calculator. (Easy) 9. I could not view it in any other form.. (Unclear) 10. It really helps solving things. (Easy) 11. It shows the processes. (Visual) 85 12. The calculator is a useful tool. (Unclear) 13. It is more accurate. (Easy) "Educational 14. It visualizes the problem. (Visual) technology is Y 15. Even if! hand write and solve the equation, I can still valuable when check accuracy fast. (Fast) I solve 16. We needed the calculators to solve hard equations. (Easy) mathematics 17. It helps to solve easier and faster. (EasylFast) problems 18. I can verify my work. (Easy) using 19. The calculator has build in equations. (Easy) equationsD 20. Visuals are better. (Visual) Yes D No. 21. Calculators do computations easily. (Easy) Because" 22. It's easier to find out the answer and to plug in numbers for the equation. (Easy) 23. I can't add, subtract, multiply, and divide too fast by hand or in my head. (Fast) 24. I need a calculator to help me calculator high numbers that I cannot figure out. (Easy) 25. It is applied to most projects. It is easier to express the problem with educational technology. (Easy) 26. It advances our learning. (Grade) 27. Equations are easy to work with. (Easy) 1. It only helps on certain subjects and it's not too direct to the point. N 2. All you need is a calculator unless calculators are educational technology. 3. I can solve most equation by hand unless the problems are 210 too hard. 4. Teacher can also write it out 5. I will still need to do it algebraically unless it requires calculations to be done on a calculator. 211 Appendix N: Student Responses to Item #86
Item# Choice Responses 1. Visible. (Visual) 2. Without calculators it would take a long time for one problem. (Fast) 3. Tables are easier to format/input values on a device that can automatically do it for you. (Easy) 4. They are easy to read. (Easy) 5. You can automatically tum it into a graph. (Visual) 6. This makes it easier. (Easy) 7. I can plug in data and see patterns. (Visual) 8. You can create tables on the calculator. (Easy) 9. There is a table option on the TI-83. (Easy) 10. It gives you the whole table of numbers. (Easy) 11. The tables are easier to view. (Visual) 12. I can graph those numbers. (Visual) 13. I can get more accurate answers. (Accurate) 86 Y 14. It makes things accurate. (Accurate) 15. Making tables can be done fast and accurately on "Educational calculators. Also many can be made. (Fast) technology is 16. Helps me to organize my information/data. (Easy) valuable when 17. The calculators could easily create tables for us to use. I solve (Easy) mathematics 18. Makes data more organized. (Visual) problems 19. I can store my work and verify it (Easy) using tables 0 20. Calculators have special functions. (Easy) Yes No. 0 21. It is easy to make tables or calculator/computer. (Easy) Because" 22. You are able to find sums and even filling the blank. (Fast) 23. The calculator automatically does the work for me. (Easy) 24. I need a calculator to help me calculator high numbers that I cannot figure out (Easy) 25. You can make nicer and comDlete tables. (Visual) 1. It only helps on certain subjects and it's not too direct to the point. 2. All you need is a calculator unless calculators are educational technology. 3. It's easier when it's on paper. N 4. Teacher can also write it out 5. I would be able to graph in on the computer. 6. Though educational technology is easier to apply with tables there are more different ways to solve a problem with or without educational technology. 212 Appendix 0: Student Responses to Item #87
Item# Choice Responses I. Visible. (Visual) 2. Without calculators it would take a long time for one problem. (Fast) 3. Computers and calculators have the ability to create graphs quickly and with less error than creating graphs by hand. (Fast) 4. They are easy to read & make the problem easier. (Easy) 5. It graphs the data accurately. (Accurate) 6. The TI83 can provide graphs for me. Easier! (Easy) 7. I can plug in data and run tests fairy easy. (Easy) 8. You can create tables on the calculator. (Easy) 9. It is exact and could only be thrown offby a user error. (Accurate) 10. It graphs out the equations. (Visual) II. It connects the graph to the equation. (Visual) 87 12. It makes the graphs faster and more accurate. (Fast) 13. I can easily graph using calculators. (Easy) "Educational 14. It can graph more accurately. (Accurate) Technology is 15. It makes things accurate. (Accurate) valuable when 16. Graphs on calculators can be accurate and altered fast like I solve Y the window range the plotting and finding point value. mathematics (Accurate) problems 17. I can use my calculator to see what the graph look like. using graphs (Visual) Yes DNo. o 18. The calculator easily creates graphs and we could use other Because" technology to show others in the class our graphs. (Easy) 19. Easier to view changes in data. (Easy) 20. I can store my work and verify it. (Easy) 21. Easy to use and understand. (Easy) 22. Calculator and computers easily make and interpret graphs. (Easy) 23. You are able to look at a picture of your data. (Visual) 24. It gives me a visual of what I am trying to figure out with the push of a few buttons rather than doing it on my own. (Visual) 25. I need a calculator to help me calculator high numbers that I cannot figure out. (Easy) 26. In order to create precise material in graphs and charts, there is a need to use computer (educational technology). (Accurate) . _. '\~JI 213 28. Graphing calculators present all the data on a traceable and easily seen graph. (Visual) 1. It only helps on certain subjects and it's not too direct to the point. N 2. All you need is a calculator unless calculators are educational technology. 3. A calculator can tell you each point on the graJ.lh. 214 Appendix P: Teacher Interview Questions
Teacher Interview Questions
The following topics will be used recollections of computer technology as part of our discussion. Thank you for your very generous cooperation.
1. What are your earliest recollections of educational technology as part of instruction in your entire teaching career?
2. Could you describe your professional experiences using educational technology as part of your research and teaching?
3. Can you trace the history of your personal experiences as educational technology was integrated into the curriculum? How well did you think it fit? What would have some of the benchmarks, milestones, and memorable events been? Why?
4. What are some of your memories of how other person (to be discussed anonymously, of course) at any school you have taught were affected by the introduction of educational technology into their research and curricula? How well did they think it fit?
5. What is your personal vision for the future of educational technology at Kapolei High SchoollLeeward DistrictlHawaii State DOE?
6. How does your technology background influence on your perception of using educational technology while you teach mathematics?
7. How does your technology background influence on students' learning mathematics?
8. What do you believe motivates or would motivate mathematics teachers to use more educational technology for their instructional purpose?
9. Would you be willing to share your course syllabus with this researcher? 215 Appendix Q: Student Interview Questions
Student Interview Questions
1. What is involved in learning mathematics?
2. How do you learn mathematics?
3. What do you think are some benefits oflearning mathematics?
4. Which of the following aspects of mathematics have you learned in your high
school mathematics classes? (Basic mathematical skills, Series of mathematical
concepts, Social skills such as how to communicate, and How to apply
mathematical skills in life)
5. How do you know when you have learned something?
6. How do teachers help you learn mathematics?
7. What is the student's role in the mathematics classroom?
8. What is the role of technology in learning mathematics?
9. What has been your biggest challenge or obstacle in learning mathematics?
10. What is your experience of cooperative learning in mathematics class?
11. What is your experience of using technology in mathematics class?
12. Did using technology help you understand and learn mathematical concepts
better? 216 Appendix R: Hmnan Subjects Approval Letter
UN.VER& .... ~ OF HAWAI'. CommiI:I8B on Human 8tmDes
MEMORAND'DM
May 23,2007
TO:
FROM:
SUBJECT: CHS #15317- "EducotlOlllal RI>..pfoIt·"
Your project !dentI1led above .... reviJ>wed and -46.IOI(b)(I).SoMces Your_ (DHHS)7;E,:4~S :j~l! rov!ew oflbis stady and will be
AnyCHS 1DIlII1IiclpaIe through Ibis _ problems TbIs Sllbj_asmaybe...... ".re'~Jated;to=~=':a m. __ _ 8erYi UDiversity policy requIres)lOU to _ as an essential part ofyour project records, any ~ pertaining to tho use ofbumans as Sllbjecliln your researcb. TbIs incbukis any information or materials COIlV8)'ad to, and received from, tho subj-. as wen as any execuIad CODSODI1'om!s, da1a and analysis results. These records must be majntainad fur at 1eastthree years o1ler project complellon orterm1natioo. Jflbis is._ project, )'1J1Ishould be aware that these recorda .... Sllbject to inspac:tIon and review by aoIhorlzed ,epxesen1all_ ofthe UnlvorsIty, State and Fedetal govetmmml8. . Please no!ItY Ibis om.. when your proJect is ggrnn!eted. We may ask that)lOU provhle infutmalion tegard!ng your oxpetleJwes with bumao Sllbj_ and with tho CHS review process. Upon not1ficat!on, we will close our fi1es pertaining to)lOUt project. Any subsequent _ of tho project will requite .new CHS application. Please do not hesitate to contact me if)'1J1l have any quastloas 01 requite assis1anoo. 1 will be happy to assist you in any wey 1 con. Thank)lOU fur your cooperalioo and efforts tbtaugboutthis review process. I wish)'1J1l sm:cess in this endeavor. Enclosme 2540 MaDe Way, spalding 263. ~. Hawa1'1881122-2303 TeI8phona:AnEqual __ (808) 96&5007. FacsfmDe: (008)_ 9f58..88S3. ~ www.hnall.sdtilrb 217 Appendix S: IRB CertificationIDeclaration of Exemption OMS No. 0990-0263 Protect/on of Human Subjacta Assurance IdentlflcatlonJIRS CertlflcetlonJDeclaration of exemption (Common Rule) PdJq: ___-IOhjacI!...,aotta-Ol_hl"" __IIm.,_"_""'_tn""_tnta_." _."AgooIm_""CoomoIRuJo_ .... ' •• '891hol",,,,, ___"IRS"""'" """"with"" appI!ca1Imor_""'" _ ... _frcmOl ...... "o11a' ..... "'''''_Rule...... 1011hl __hl""_01 AuoloT. ___"_---IIRBI..,,,..,,""""tn """-"""'""_"AuoloThI_wllb""CoomoIRaie...... ---"proposeIs"'- I._Typo 2. Typoof_ 3. Name ofFederal_orAgencyand. Wknown. II ORIGINAL [J GRANT II CONTRACT [I FELlOWSHIP AppIIoa6on or ProposaI_ No. II CONTINUATION II COOPERATIVE AGREEMENT pq EXEMPTION [J OTHER: 4.11110 of App!Ica1Ion or Ae6vIty 6. Name of PrInoIpaIlnws1Iga1or. PIogmm 01_. Fellow. or ~_. 0Ih0r T.. VounaHa a. Assurance_ of this Project (RBspondto one of""'-.vIng) [Xl1l!lsAssurance,"" fliewlih _of Heal til and Human ServIces. covers this ~ Assurancelden1fftcaftonNo. f..3528 theexplJatlondate Sentmnber23 2QQ8 IRBRegIstratIonNo. IQRW9P1§9 [I No assunmco has ..... flied tor this _on. Thlslns1ltu1kmdedareslha' Hwill provide an Assuranco and ~1Ion oflRB review and approval upon request. [Xl ExBmpIion staIuS: Human _ are _. butthls~qualHIes tor _lion under SecIIon 101(b). paragmph~ 7. CeI!IfI1:a!Ion of IRB RevIew (Respond ID one oftllafollawlng IF)'OII hava an Assurancoon Il10) [J 1I!Is~"'''''' _ andappmvad bylha IRS In __!ba Common RuJeand anyotilargovemJng _OIlS. by. [IFWlIRB-...""(..... ofIRB_&l __ or [J EIIpocIIIedReYlewon(cIa!e) [I RIess_ one_approval. provideoxp1mtiCn cIa!e_-:----:-c::- [I 1I!Is ~ _ muIIIple proj_. some of which hava not_ re~_. The IRB has granled a"""""" on _ that au _ _ by!ba Common _will be _ and approved _!bayare InIIiaIed and that __!ur!berCOl1llloatlon will be submilled. e.CcmmenIs CHS#16317 9. The official sIgnIne beIcw _ that!ba _on proy!dsd above Is corractand that. as mqutred, future reviews will be pe:rfonned unto study cIosutB and G&Jllfttallon will be proyfded. UnIWIsIIy of HawalI at Manoa 11. Phone No. (wIIh area """"J (908) 958-5IlO7 2444 Cole SIreeI, BacluT 14. Nama of DIIIcIaI 15._ Compllanoa 0fII00l WtnJam H. 0en To: Parent/Guardians From: Tae Young Ha Re: Permission to participate with Educational Research Date: Wednesday, October 13,2004 Aloha! I am a mathematics teacher (for IMP 3) at Kapolei High School and a Ph.D. candidate of the Curriculum Studies program at University of Hawaii. The purpose of this letter is to inform the parents of the students in class and obtain their permission for their children's participation with an educational research. This research will be conducted during the second quarter of school year 2004 - 2005 as part of the completion of my dissertation research project. FYI, this research will focus on "the technology integration in secondary mathematics curriculum." The outcome of this research will serve Kapolei High School, Leeward District, and the State of Hawaii school system as a valuable resource to understand iftechnology integration influence on the quality of mathematics education. This research project will include extensive videotaping twice a week to understand the actual classroom learning and videotaped small group interview sessions to follow-up the classroom observations. The classroom observations and small group interviews will be conducted during the second quarter ONLY. I would like to sincerely and humbly ask for your support for this research project. Please give your child your permission to be part of this meaningful research project by signing the form included in this letter and returning by next class time. If you have any question regarding this letter or this research project, please feel free to contact me either by calling at (808) 277-4176 or by sending an email to [email protected]. I would like to express my sincere appreciation for your support and constant interest in public education in advance. Looking forward to working with your child! Sincerely, T. Ha Administration Approval: ------3 Parents' Signature: Date: ______219 Appendix U: Teacher Consent Form To: Teacher Name From: Tae Young Ha Re: Agreement to participate with Educational Research Date: Wednesday, October 20,2004 Aloha! As you know I am currently working on my Ph.D. in Curriculum Studies program at University of Hawaii. The purpose of this memo is to inform you about the research project that I will conduct in your AP Statistics classes and obtain your consent to participate with this educational research project. This research will be conducted as part of the completion of my dissertation research project. FYI, this research will focus on "the technology integration in secondary mathematics curriculum." The outcome of this research will serve Kapolei High School, Leeward District, and the State of Hawaii school system as a valuable resource to understand if technology integration influence on the quality of mathematics education. This research project will include extensive videotaping twice a week to understand the actual classroom learning and videotaped small group interview sessions to follow-up the classroom observations. The classroom observations and small group interviews will be conducted during the second quarter ONLY. I would like to sincerely and humbly ask for your support for this research project. Please be part of this meaningful research project and see how technology can make change in your teaching and your students' learning mathematics. If you have any question regarding this research project, please feel free to contact me either by calling at (808) 277-4176 or by sending an email to taeha@hawaiLedu. I would like to express a sincere appreciation for your support and constant interest in public education in advance. Looking forward to working with you and your students! Sincerely, T. Ha By signing, I, , voluntarily give my consent to participate with the educational research project that Tae Young Ha will conduct in my classes while I am teaching. I fully understand all the conditions of this research and what to expect while the research is conducted in my classroom. Teacher's Signature Date 220 REFERENCES Baptist, P. (2005, November 25th - December 1st). 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