The Effectiveness of the National Board Certification as it
Relates to the Advanced Placement Calculus AB Exam
by
Fernando Antunez
A Dissertation Submitted to the Faculty of
The College of Education
In Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
Florida Atlantic University
Boca Raton, FL
December 2015 Copyright 2015 by Fernando Antunez
ii
Acknowledgements
I would like to acknowledge my dissertation chair who has worked with me relentlessly throughout the entire project. Dr. Furner’s patience, expertise, and attention to detail played a major role in the completion of my dissertation. I also want to acknowledge my other committee members, Dr. John Morris, Dr. Emery Hyslop-
Margison, and Dr. Traci Baxley for the time and incredible dedication in assisting me to
complete my dissertation.
I greatly appreciate Dr. Burnaford’s expertise and support. She has been an
inspiration and has taught me how to effectively do research. She also guided me to
complete my literature review at a very special time of my doctoral degree.
Immeasurable thanks to my friend Lawrence Moorman for helping me stay focused and
persevere and for acting as an unparalleled encourager throughout my dissertation.
I would like to thank the research office of the participating school district for
granting me permission to conduct this research. I would like to express special thanks to
Beth Tillman and Jack Ciminera for their continuous help in assisting me to properly
have all required paperwork processed and also for providing me with the data in such an
efficient manner. Finally, I would like to thank Harvard University Professor and
personal friend Dr. Natesh Pillai for his support and guidance toward the completion of
my dissertation.
iv Abstract
Author: Fernando Antunez
Title: The Effectiveness of the National Board Certification as it Relates to the Advanced Placement Calculus AB Exam
Institution: Florida Atlantic University
Dissertation Advisor: Dr. Joseph Furner
Degree: Doctor of Philosophy
Year: 2015
This study compared data related to National Board Certification (NBC) of mathematics teachers in a South Florida school district. Data included 1,162 student scores on the 2014 AP Calculus AB exam, student gender, student grade level, and eligibility for free or reduced price lunch (FRL) status. Teachers completed the
Standards’ Beliefs Instrument (SBI) (Zollman & Mason, 1992) to determine alignment of their beliefs with the National Council of Teachers of Mathematics (NCTM) standards.
Interviews were conducted with five NBC mathematics teachers to understand how they incorporate best mathematics teaching practices in their daily instruction.
A t-test analysis revealed that students taught by NBC teachers scored significantly higher (M = 3.70) on the AP Calculus AB Exam than those taught by non-
NBCTs (M = 2.74); Cohen’s d (.6429) indicated a moderately large effect size. No causation is to be implied; various confounding factors may also contribute to the variance in student scores.
v Three factorial ANOVA tests were performed to test interaction effects. Two
significant interaction effects were detected: (1) NBCT status and student grade level;
and (2) NBCT status and student FRL (free and reduced price lunch) status. No
significant interaction was found between NBCT status and student gender.
With a reliability estimate using Cronbach’s alpha, a second t-test was conducted.
A statistically significant difference was found regarding the mean scores of NBCTs and
non-NBCTs regarding their beliefs according to the vision of the NCTM. NBCTs
generally have practices that align more with the NCTM mathematics teaching standards.
Interviews with five NBC teachers of AP Calculus provided rich qualitative descriptions of their teaching philosophies, approaches, and best practices contributing to student success.
The results of this study reinforce evidence from previous research that the process of obtaining the NBC contributes to a teacher’s professional expertise and is related to student success; however, since there may be other confounding factors related to teachers, students, and their schools, the NBC cannot be considered the sole factor contributing to student success in AP coursework and exams.
vi The Effectiveness of the National Board Certification as it
Relates to the Advanced Placement Calculus AB Exam
List of Tables ...... xii
List of Figures ...... xiv
Chapter 1. Introduction ...... 1
Statement of the Problem ...... 4
Purpose of the Study ...... 7
Research Questions ...... 7
Hypotheses ...... 7
Definition of Terms ...... 8
Limitations ...... 11
Significance of the Study ...... 12
Organization of Dissertation ...... 13
Summary ...... 13
Chapter 2. Literature Review ...... 15
Introduction ...... 15
The National Board Certification ...... 16
Teacher Certification ...... 29
vii Best Practices in Teaching Secondary Mathematics ...... 39
The nature of high school math...... 39
Issues of mathematics performance...... 40
Issues with AP math ...... 42
Gender, Race, and Equity in Mathematics ...... 44
Mathematics Standards ...... 50
National Council of Teachers of Mathematics (NCTM) ...... 52
The Role of the National Board for Professional Teaching Standards (NBPTS). .... 56
Studies of Best Mathematics Practices...... 57
The Relationship Between Best Practices and AP Math Teaching ...... 69
Historical Background of AP Math Classes ...... 72
AP Calculus...... 73
AP Statistics...... 77
Relationship Between the NBC and Best Teaching Practices ...... 79
Discussion ...... 82
Conclusion ...... 87
Summary ...... 93
Chapter 3. Research Methodology ...... 96
Introduction ...... 96
Sampling Plan, Instrumentation, and Data Collection ...... 97
viii Subjects ...... 100
Research Design ...... 101
Hypothesis Test # 1...... 101
Hypothesis Test # 2...... 102
Hypothesis Test # 3...... 102
Hypothesis Test # 4...... 102
The Standards’ Beliefs Instrument (SBI) ...... 102
Hypothesis Test # 5...... 103
Interviews ...... 104
Interview Protocol/Procedures ...... 104
Interview Questions ...... 105
Quantitative Data Analysis...... 106
Qualitative Data Analysis...... 107
Summary ...... 108
Chapter 4. Analysis of Results ...... 110
Introduction ...... 110
Descriptive Statistics ...... 111
Inferential Statistics ...... 112
Sample of Interviewed NBCTs ...... 118
ix Case Studies of Insights from National Board Certified Mathematics Teachers Who
Teach AP Calculus AB ...... 118
Case Study of Ms. Johnson...... 120
Case Study of Ms. Doval...... 127
Case Study of Ms. Furgeson...... 134
Case Study of Ms. Morrison...... 143
Case Study of Mr. Glenc...... 152
Summary of the Findings from the Case Studies ...... 163
Summary ...... 171
Chapter 5. Discussion, Conclusions, and Recommendations ...... 173
Introduction ...... 173
Summary of Findings ...... 174
Discussion ...... 181
Limitations of the Study ...... 187
Conclusions ...... 189
Implications ...... 193
Recommendations for Future Research ...... 195
Summary ...... 198
Appendices ...... 200
Appendix A. Invitation and Pre-interview Questions Invitation ...... 201
x Appendix B. Initial Self-Report/Introductory Letter ...... `203
Appendix C. Adult Consent Form ...... 204
Appendix D. Face-to-face Interview Questions...... 207
Appendix E. Inside the Classroom Observation and Analytic Protocol ...... 210
Appendix F. The Standards’ Beliefs Instrument (SBI) ...... 213
References ...... 214
xi List of Tables
Table 1. Literature Comparing Studies Between NBC and Non-NBC Teachers ...... 24
Table 2. Literature Comparing the Relationship Between Teacher Certification
and Student Achievement ...... 35
Table 3. Process Standards of the National Council of Teachers of Mathematics
(NCTM) ...... 55
Table 4. NBPTS Mathematics Standards for Teachers of Students Ages 11-18+ ...... 59
Table 5. Curricular Guidelines for AP Calculus and AP Statistics ...... 74
Table 6. Characteristics of Student Subjects ...... 101
Table 7. Means and Standard Deviations for Student Test Scores of National
Board Certified and Non-National Board Certified Teachers ...... 111
Table 8. Analysis of Variance of Interaction between Certification Status and
Gender ...... 114
Table 9. Analysis of Variance of Interaction between Certification Status and
Grade Level ...... 114
Table 10. Analysis of Variance of Interaction between Certification and FRL
Status ...... 115
Table 11. Characteristics of Interviewed National Board Certified Teachers...... 119
Table 12. Summary of Ms. Johnson’s Teaching Approaches ...... 126
Table 13. Summary of Ms. Doval’s Teaching Approaches ...... 133
Table 14. Summary of Ms. Furgeson’s Teaching Approaches ...... 141
xii Table 15. Summary of Ms. Morrison’s Teaching Approaches ...... 151
Table 16. Summary of Mr. Glenc’s Teaching Approaches ...... 162
Table 17. Summary of Findings (Quantitative Portion) ...... 179
Table 18. Summary of Findings (Qualitative Portion) ...... 180
xiii List of Figures
Figure 1. Percent of teachers giving various reasons for job dissatisfaction
(Ingersoll, 2000)...... 28
Figure 2. Composition of public school teachers by route to teaching. Source:
National Center for Education Information (2011, p. 21)...... 32
Figure 3. Enrollment of boys and girls in high school math courses. Adapted
from: U.S. Department of Education, Office for Civil Rights, Civil
Rights Data Collection (2012)...... 46
Figure 4. AP mathematics enrollment by gender. Adapted from: U.S. Department
of Education, Office for Civil Rights. Civil Rights Data Collection
(2012)...... 47
Figure 5. Passing rate proportion of boys and girls on AP mathematics tests.
Adapted from: U.S. Department of Education, Office for Civil Rights,
Civil Rights Data Collection (2012)...... 48
Figure 6. Topic outline for AP Calculus AB and AP Calculus BC. Adapted from
College Board, 2015a...... 76
Figure 7. How the NBC affected teaching practices. Adapted from: National
Board for Professional Teaching Standards, 2001...... 80
Figure 8. Educational profile of NBCT survey respondents, highest educational
level. Adapted from National Board for Professional Teaching
Standards, 2001...... 81
xiv Figure 9. Technology supplies recommended by the College Board (2010)...... 84
Figure 10. Reasons why accomplished or NBCTs do not work in low-performing
schools. Adapted from Berry (2005)...... 86
Figure 11. Distribution of AP Calculus AB scores for NBCTs and non-NBCTs...... 112
Figure 12. Commonalities among National Board Certified Teachers...... 169
Figure 13. Differences and discrepancies among National Board Certified
Teachers...... 170
xv Chapter 1. Introduction
Many professions have high standards that are cultivated within their profession; the standards set for the National Board Certification (NBC) are standards that illustrate what teachers should know and what effective teaching should be (Burroughs, 2001).
These standards define what a highly qualified teacher should be able to demonstrate in
order to make instruction effective and create success for students, contributing to high
student achievement. The NBC is the source for high standards and best practices, which
leads to highly effective teaching.
In order to achieve the NBC, teachers must undergo and successfully accomplish
a two-part assessment system. As described by Gaudreault and Woods (2012), “The
process requires completion of ten assessments, four portfolios and six constructed
response exercises that evaluate content knowledge” (p. 49). The first part consists of
four components in which teachers must demonstrate that they can effectively teach and
promote students’ learning and growth. As teachers assemble their portfolios, they are
required to analyze students’ work, reflect on such work, and create videos that reflect
their classroom instruction and their interactions with students. Every single step of these
four components concentrates on student learning. Through these components, teachers
are specifically required to demonstrate pedagogical knowledge, the use of effective
teaching strategies, assessment skills, and evidence of providing effective feedback; they
must also show assessment of their own teaching strategies to demonstrate self-reflection
and the capability to improve oneself as a teacher throughout their reflection process.
1 Additionally, teachers are required to interact in professional activities and document their own professional accomplishments. The second part of the assessment requires candidate teachers to attend an assessment center where they will take a rigorous, computer-based exam and will have the opportunity to demonstrate their knowledge of subject matter in multiple topics of their certificate field. The National Board for
Professional Teaching Standards (NBPTS), a national not-for-profit organization to advance teacher competence, sets widely accepted and certifiable standards about what teachers should know and be able to do. As reported by Pyke and Lynch (2005), “The
Educational Testing Service (ETS) in collaboration with NBPTS developed the rigorous
NBPTS performance assessment to allow teachers to demonstrate their competence in meeting the high standards set forth by NBPTS” (p. 25). Teachers are not evaluated based on their students’ scores on state tests or any other student performance measures.
The NBC process for teachers is a rigorous and voluntary process that goes beyond obtaining the regular teacher state certification (National Education Association, n.d.). Becoming certified requires teachers to work on a regular basis beyond their teaching hours. It is an opportunity to develop and grow as a teacher and professional.
The NBC is both teacher and student oriented. It helps teachers become aware that success is not only concentrated in their own performance but also to focus on students’ success. Obtaining the certification is a way to validate how teachers perform in their classrooms and to test if teachers’ work stands up to the high standards of accomplished teaching; it is a reward and recognition for their hard work. Candidates who attempt to accomplish the certification are encouraged to search for guidance and assistance from other National Board Certified teachers (NBCTs).
2 McKenzie (2013) stated that “the need for quality teachers has been the focal point of education reform movements of the 20th century” (p. 154). Creating more
NBCTs and bringing them into the classrooms may represent a way to provide schools with more effective teachers all across the nation, especially for those low performing schools that are in great need of qualified teachers. McGee (2004) explained that “the achievement gap is the single most critical issue in American education” (p. 97).
Additionally, Exstrom and the National Conference of State Legislatures (2011) stated that “today we are facing deep cuts to education while struggling to close the achievement gap and turn around low performing schools” (p. 3). Furthermore, this organization pointed out that “National Board teacher certification is being used as a turn-around strategy to improve teaching and learning in low performing schools" (p. 6).
The gap in education that exists in low income and high-need schools may be reduced by having more NBCTs in the classroom (National Board for Professional Teaching
Standards, 2010b).
The Association of American Educators (2013) explained that there have been some revisions to the NBC process. These revisions have been made in order to break away from some of the barriers that teachers have encountered such as: (1) the cost;
(2) the pacing by which teachers were subjected to submit their work; and (3) the incorporation of the latest educational research. The cost has been reduced from $2,500 to $1,900. Teachers can now go through the whole process at a more leisurely pace and break the process down into different modules without being required to complete the process in less than one year.
3 Many school districts have provided a financial reward for holding the
certification. This means that becoming a NBCT represents a beneficial investment as
teachers grow professionally and get an additional bonus in their salaries. Additionally,
some universities provide graduate-level credit to teachers pursuing the certification or allow National Board work to be counted towards a master’s degree (National Board for
Professional Teaching Standards, 2014a).
Statement of the Problem
For many years, the need for effective teachers has been a highly important component to creating better schools and better students. Exposing students to effective teachers in mathematics is very important for the development of students’ mathematical thinking at their early stages. Consequently, preparing our students in mathematics has a big impact on the way that students will continue taking their math classes and how competitive they will be. If students are not well prepared, then it will be practically impossible for them to qualify for honors and Advanced Placement (AP) math classes in the future. When students have a teacher who is minimally qualified, the learning process deteriorates, and it becomes very difficult for them to catch up at later stages because they did not build the fundamentals necessary to move to a higher level of learning. This is indeed a disservice when the school’s teachers do not know how to deliver content and is further compounded when they do not possess adequate subject matter knowledge.
Understanding how students learn and being able to connect with them as well as knowing how to organize curriculum and classroom instruction ensures that children can effectively learn (Haselkorn, 2001).
4 Today, teaching is a very intense profession. Teachers are exposed to long hours, must complete extensive paperwork, and are held more accountable for students’ success.
The teaching process becomes even more diluted at times because of the numerous responsibilities that teachers are assigned. As noted by Ballard and Bates (2008),
“Accountability, high-stakes, and student achievement are popular terms among educators. Students’ performance on standardized achievement tests is used to a high extent in reflecting the quality of instruction students receive from teachers” (p. 560). It
is evident that high level education authorities, school boards, and administrators demand
an increase in the accountability levels, not only for students but also for teachers. Under
these circumstances, teaching effectively gets harder to do. Consequently, teachers leave
their professions for multiple reasons such as inadequate time, student discipline
problems, poor student motivation, poor salary, and lack of administrative support,
among others (see Figure 1). Rinke (2008) reported that about 40% of teachers leave the
profession within the first five years in the United States. This leads to the creation of
classrooms where teachers are unprepared, uncertified, and inadequately supported,
especially in science and mathematics (Sterling, 2004).
It is evident that the need for effective teachers is critical under such
circumstances. A problem that has emerged is that pursuing the NBC process is no longer
free. Federal subsidies are no longer available to candidates attempting the certification.
In order to apply, teachers must pay a total of $1,900 to cover the expense of 4
components plus a $75 processing fee (National Board for Professional Teaching
Standards, 2015). Moreover, in the past, becoming an NBCT represented a salary
increase of up to $10,000 in some states with federal and state funds. Unfortunately this
5 incentive is no longer available in many states, although some districts may provide a 5%
bonus as recognition for holding the title of NBCT. Under these circumstances, there has
been a decline in the number of teachers attempting to earn the certification in the past
few years.
As time goes by, education is becoming more global, and students need skills now
that they did not need previously such as the proficient use of technology. Specifically for
AP math courses, students need to understand how to operate graphing calculators and
computer software among other sophisticated tech items. It has become even more
necessary to incorporate highly competent teachers who can effectively deliver good
quality instruction.
As the literature review demonstrates in the following chapter, research about the effectiveness of the NBC has produced mixed results; however, most of the studies
previously conducted have shown favorable results for NBCTs. This study explored the
effectiveness of the NBC as it relates to students’ achievement on the AP Calculus AB
Exam. This test measures students’ understanding of the concepts of calculus, their
ability to apply these concepts, and their ability to make connections among graphical,
numerical, analytical, and verbal representations of mathematics, described in the AP
Calculus AB overview (College Board, 2014) as roughly equivalent to a first semester
college calculus course.
Passing the AP exam offers the benefit of obtaining three credits at the college level for those who meet success. The requirements set by each college differ according
to their respective standards. Some colleges require a score of 5 in order to grant credit,
others require 4, but a score of 3 will satisfy most colleges’ requirements.
6 Purpose of the Study
The purpose of this study is to add knowledge about the relationship of student success on the AP Calculus AB Exam and NBC certification attainment. This study additionally explored if particular factors such as student gender, student grade level, and free and reduced price lunch (FRL) status reveal a significant interaction with teachers’
NBC status on the AP Calculus AB Exam scores. Lastly, this study attempted to reveal how best mathematics teaching practices are being utilized and incorporated in classrooms of NBC and non-NBC teachers.
Research Questions
1. Do NBC mathematics teachers produce better results than non-NBC mathematics teachers on the AP Calculus AB Exam?
2. In respect to test score performance, is there an interaction between teacher
NBC status and student gender, NBC status and student grade level, or NBC status and student FRL status?
3. What is the difference in perception of best mathematics teaching practices between NBCTs and non-NBCTs?
4. How do NBCTs incorporate best mathematics teaching practices in their daily instruction?
Hypotheses
Five hypotheses were tested in this study. These hypotheses attempted to answer three of the four research questions. The first hypothesis concentrates on comparing students’ test scores on the AP Calculus AB Exam that may reveal the effectiveness of certified and non-certified NBC teachers. The second hypothesis deals
7 with observing the interaction between teacher certification status and student gender.
Similarly, the third hypothesis explores the interaction between teacher certification
status and student grade level; the fourth hypothesis explores the interaction between
teacher certification status and student FRL status. Lastly, the fifth hypothesis focuses on
exploring the level of agreement between NBCTs and non-NBCTs in alignment with the
National Council of Teachers of Mathematics (NCTM) standards. Moreover, it is the
researcher’s belief that NBC mathematics teachers consistently utilize best teaching
practices in order to effectively prepare their students to succeed on the AP Calculus AB
Exam.
The five null hypotheses to be tested are:
Hypothesis 1. H0: u1 = u2: The mean of student test scores of NBC teachers is the same
as the mean scores of those students of non-NBC teachers.
Hypothesis 2. H 0 : There is no interaction between teacher certification status and
student gender.
Hypothesis 3. H 0 : There is no interaction between teacher certification status and student grade level.
Hypothesis 4. H 0 : There is no interaction between teacher certification status and student FRL status.
Hypothesis Test 5. H 0 :NBCTs and non-NBCTs produce equal scores on the Standards’
Beliefs Instrument (SBI).
Definition of Terms
The terms utilized in this study that require additional clarification are listed
below: 8 National Board Certification (NBC), as defined by Western Washington
University (n.d.), is an optional certification at the national level for teachers who have at
least three years of teaching experience and involves an extensive application and
portfolio process. It is an advanced teaching credential. The California Department of
Education (n.d.) website describes the NBC as the most rigorous, coveted, and respected
professional certification available in education. The term NBC is such a common term
nowadays that it can be found on multiple websites; these descriptions state that the NBC
improves teaching and student learning by enhancing overall educator effectiveness and
recognizing and rewarding highly accomplished educators who meet high and rigorous
standards.
The NBC is granted by the National Board for Professional Teaching Standards.
The Ohio Department of Education (n. d.) defined the NBPTS as follows:
National Board for Professional Teaching Standards is an independent, nonprofit
organization. It was formed in 1987 to advance the quality of teaching and
learning by developing professional standards for accomplished teaching, creating
a voluntary system to certify teachers who meet those standards and integrating
Board-certified teachers into educational reform efforts (para. 1).
The NBPTS has its own website and defines National Board Certified Teacher
(NBCT) as a teacher who has developed and demonstrated the skills of an accomplished education professional and also has demonstrated advanced teaching knowledge, skills, and practices.
Advanced Placement (AP) is described by the College Board (2015b) in its AP brochure, Why Take AP?:
9 The College Board’s Advanced Placement Program® (AP®) enables willing and
academically prepared students to pursue college-level studies — with the
opportunity to earn college credit, advanced placement or both — while still in
high school. (p. 6)
AP exams are given each year in May. A score of 3 or higher on an AP exam can typically earn students college credit and/or placement into advanced courses in college.
AP Calculus AB is a comprehensive course that encompasses the study of both differential and integral calculus. The level of this course is intended to be the equivalent of a college level Calculus I course. Students study the idea of functions, graphs, limits, continuity, derivatives, and integrals (and their applications) as outlined by the College
Board (2015a).
At-risk students is the term utilized for those who struggle or experience poor
performance at school, and according to USLegal.com (2014):
At-risk students are students who are not experiencing success in school and are
potential dropouts. Usually, they are low academic achievers who exhibit low
self-esteem. Generally they are from low socioeconomic status families. At-risk
students tend not to participate in school activities and have a minimal
identification with the school. (para. 1)
The Free and Reduced Price Lunch Program (FRL) provides cash subsidies for
free and reduced-price meals to students based on family income and size. Eligibility is
determined via an application process which parents complete and submit each year.
According to the U.S. Department of Education Office of Planning, Evaluation, and
Policy Development, Performance Information Management Service (2012), “Children in
10 households with incomes at or below 130 percent of the federal poverty guidelines are
eligible for free meals. Children in households with incomes between 130 and 185
percent of the federal poverty guidelines are eligible for reduced-price meals” (p. 2).
The Standards’ Beliefs Instrument (SBI) is a sixteen-item instrument that measures alignment of teachers’ beliefs with the NCTM standards (Zollman & Mason,
1992) (see Appendix F).
Limitations
This study is restricted to only one school district in South Florida. The researcher evaluated test scores corresponding to year 2014 only. This study is only applicable and generalizable for students taking AP Calculus AB. Most students placed in these classrooms are not at-risk students. Another limitation was the sole focus of the dependent variable AP Calculus AB scores. No assumption of causality can be made since multiple confounding variables are possible but are beyond the scope of this study.
There are many other outcomes associated with NBCTs besides student achievement as measured by test score performance on an AP exam. There are student-related influences on student achievement such as motivation, past experiences, etc., that are important to measure but were not included or captured in this study. The relationship between student success on the AP Calculus AB Exam and NBCTs found in this study may not be applicable to other subject areas or to any remedial or low level math courses.
The researcher is an NBCT in mathematics and is aware of the importance of disseminating and reporting results in a way that is accurate and unbiased. The research design allows objectivity throughout the entire study in order to avoid bias that may affect the results and/or findings.
11 Furthermore, in the United States, the National Center for Education Statistics
(2013) reported that 49.6% of public school students were eligible for FRL in 2011-2012, but in Florida, 57.6% were eligible in 2011-2012 (NCES, 2013). In Florida, 1,576,720
(58.6%) public school students were eligible for FRL in 2012-2013 (Florida Kids Count,
2012-2013); students in AP Calculus classrooms are more likely to be advantaged rather than eligible for FRL as a poverty indicator.
It is important to address that more motivated teachers may enhance students’ academic achievement. Motivation is an alternative explanation for the effect of the first t-test. The researcher interviewed only NBCTs, so this study does not provide any evidence on what strategies non-NBCTs utilize to lead students to pass the AP Calculus
AB exam. Additionally, the five NBCTs who volunteered to be interviewed were all white Americans. There was no diversity of race or color in this sample of teachers.
Foreign NBCTs or NBCTs of color were not incorporated in this study.
Significance of the Study
The number of AP students has grown in the past decades and continues to grow due to the demands of college readiness and admission requirements. In the year 2012, approximately 2 million students took 3.7 million end-of-year AP exams representing more than double the numbers from ten years earlier (Pope, 2012). The expansion of AP programs has even been reaching schools that serve the most disadvantaged students. As previously stated, many classrooms are being increasingly filled with unprepared or uncertified teachers with no pedagogical background to serve this growing mass of students.
12 The results gained in this study may assist policy makers to discuss subsequent
issues about promoting the NBC such as bringing back the monetary incentives that were
given in the past to those who hold the certification. If this were to happen, teachers may
again become interested in applying to become NBCTs. This type of research is
necessary to better inform school administrators how to identify and select quality
teachers to lead an AP math class.
Organization of Dissertation
This dissertation is composed of five chapters. Chapter 1 provides an introduction
followed by the statement of the problem, the four research questions in this study,
definition of terms, limitations of the study, and the significance of the problem. Chapter
2 deals with the review of relevant literature. Chapter 3 provides an explanation of the
research methods and statistical procedures. Chapter 4 encompasses the analysis and description of the data. Finally, Chapter 5 provides findings, conclusions, and suggestions for future research.
Summary
The NBC was designed as an efficient professional development tool for teachers.
It was created to bring more highly qualified teachers into the classrooms, especially at low income schools. Teachers need to demonstrate that they possess both pedagogical and high level content knowledge to accomplish the certification. There is evidence that participating in the NBC process strengthens teachers’ practices, even for those who tried to achieve it but did not succeed. For many years, the need for effective teachers has become paramount to prepare students not only to be successful in school and in college, but also in the real world. Teaching has become a very demanding and stressful
13 profession with a great deal of accountability for teachers, low pay, discipline problems, and lack of support from administration. As a consequence, effective teachers may leave the profession, thereby opening doors to uncertified or unqualified teachers. Another problem is that applying for the NBC process is no longer funded at the district level, and this becomes a barrier for teachers who might otherwise go through the process. This study explored the impact of the NBC as it relates to the AP Calculus AB Exam and also to teachers’ daily mathematics instruction.
14 Chapter 2. Literature Review
Introduction
The purpose of this mixed method study is to investigate the relationship between
the National Board Certification (NBC) and its effect on content knowledge and test
performance. Analyses of interaction effects were performed to explore the impact of
NBC status on student gender, student grade level, and student FRL status. An additional
focus was to explore how being a National Board Certified teacher (NBCT) relates to delivering best math teaching practices at the secondary level. Furthermore, the
researcher explored if there were significant differences between NBCTs and non-
NBCTs regarding mathematics teaching beliefs in alignments with the National Council
of Teachers of Mathematics (NCTM).There are five foci of research in this literature review. These include research on (a) the NBC, (b) teacher certification and licensing,
(c) best practices in teaching secondary mathematics, (d) mathematics standards, and e) the relationship between the NBC and AP mathematics teaching.
The researcher developed a literature review of above research topics in order to address the following research questions:
1. Do NBC mathematics teachers produce better results than non-NBC mathematics teachers on the AP Calculus AB Exam?
2. In respect to test score performance, is there an interaction between NBCT status and student gender, NBCT status and student grade level, or NBCT status and student FRL status?
15 3. What is the difference in perception of best mathematics teaching practices
between NBCTs and non-NBCTs?
4. How do NBCTs incorporate best mathematics teaching practices in their daily instruction?
The National Board Certification
The education of children in America has been the cause of much debate and conflict (Hollandsworth, 2006, p. 1). The quality of the American education system has deteriorated. Unfortunately, the national education system has been falling behind in the past decades while other countries have been making progress. According to Mahajan
(2013), the usual U.S. math curriculum is simply weak, even in well-off schools; after decades of reform in the mathematics curriculum, students in the United States are still unprepared for full participation in society. Just 6% of U.S. students performed at the advanced level on an international exam administered in 56 countries in 2006, a proportion that is lower than scores achieved by students in 30 other countries
(Huffington Post, 2012, para. 5). Some attribute these poor test scores to the poor quality of instruction that many students receive. On average, four out of 10 secondary school
teachers do not have a degree in the subject they teach (A Primer of Facts on Education
in the United States 1998, p. 1). A National Center for Education Statistics (1999) study
reported that only one in five teachers feels well prepared to teach to high academic
standards. Furthermore, some schools of education are not preparing teachers to
effectively perform with their students. According to Williams (2013):
Schools of education today operate independently, sending graduates into the
workplace without all the skills and knowledge they need to maximize the
16 achievement of all students. Some schools of education train teachers this way;
some schools train them that way. There are neither industry standards, best
practices nor transparency. That puts an enormous burden on potential employers,
and works to the detriment of students in the classroom. (para. 6)
The United States cannot ignore the quality of education of its schools because
this is already having an impact on multiple educational issues. Students are falling
behind in national standards as early as in elementary school, graduation rates have been
decreasing, and many students leave high school unprepared for college. These are just
some of the current issues that may have a great impact on the economic growth of the
nation in the future. Under these circumstances, the need for effective teachers becomes
critical. The emphasis on the need of quality teaching led to the formation of the National
Board for Professional Teaching Standards in 1987 (NBPTS, 2004).
The mission of the National Board is to advance student learning and achievement
by establishing the definitive standards and systems for certifying accomplished
educators, providing programs and advocating policies that support excellence in
teaching and leading, and engaging National Board certified teachers and leaders
in that process. (NYSUT, 2015, p. 1)
It is important to understand some of the reasons behind the creation of the
NBPTS. Going back to 1957 when the Russians launched Sputnik, the United States was highly concerned with improving the education of its citizens. “The 1957 Sputnik launch was a political issue that influenced schools through federal involvement in curriculum”
(Steves, Bernhardt, Burns, & Lombard, 2009, p. 73). In order to remain a viable player in the world market, it was thought that Americans needed to know more than the basic
17 reading, writing, and mathematics skills characteristic of an industrial era, according to
Vandevoort, Amrein-Beardsley, and Berliner (2004). Also, Harman (2001) says:
In 1983, a report called A Nation at Risk provoked a wave of reform initiatives
that engulfed the education community, and three years later the Carnegie Task
Force issued A Nation Prepared: Teachers for 21st Century (1986), which
recommended the establishment of the NBPTS. (p. 54)
In that same year, Fiske (1986) reported that the Carnegie Foundation stated that “the educational system must be rebuilt to match the drastic change needed in our economy if we are to prepare our children for productive lives in the 21st century” (p. 1). A year later, “The NBPTS was created in 1987 at the recommendation of the Task Force on
Teaching as a Profession whose goal was to identify and recognize teachers who effectively enhance student learning” (Vandevoort et al., 2004, p. 3). In 1989, the NBPTS published What Teachers Should Know and Be Able to Do (National Board for
Professional Teaching Standards, 1994, 2004). This document presented five core propositions that depicted what effective teachers do to accomplish student achievement.
These five propositions were a clear commitment to improving education (National
Board for Professional Teaching Standards, 2014c). These include:
1. Teachers are committed to students and their learning.
2. Teachers know the subjects they teach and how to teach those subjects to
students.
3. Teachers are responsible for managing and monitoring student learning.
4. Teachers think systematically about their practice and learn from experience.
5. Teachers are members of learning communities.
18 Although educational policy makers have been concentrating on standards for more than two decades, it is difficult to think of higher standards when standards are compared to those established by the NBPTS. According to Goldhaber, Perry, and
Anthony (2004):
The No Child Left Behind (NCLB) Act of 2002 focuses on the need to staff the
schools with highly qualified teachers, and designates support of the National
Board of Professional Teaching Standards (NBPTS) program as one of six
strategies for targeting teacher quality. (p. 260)
The NCLB Act came into place in 2002 and challenged America to put highly qualified teachers in every classroom by 2006 (Vandevoort et al., 2004, p. 2). It is logical to believe that the NBC served as a guideline for NCLB to create effective teachers as a factor for improving student achievement. As stated by Whitman (2002), “New teaching standards and assessment procedures promoted by the National Board . . . have been adopted as guidelines for professional development and evaluation in many school districts in the United States” (p. 1), standards based on research that recognizes the importance of education practices with the goal of improving student performance and achievement. The NBC plays a critical role for American education reform, researchers, and policy makers.
The NBPTS has 29 members, 16 of whom are classroom teachers (National
Board for Professional Teaching Standards, 2014a). In approximately 25 different areas of teaching, such as middle school language arts, upper level biology, or elementary school generalist, the Board and its consultants have specified what teachers should know and be able to do (National Board for Professional Teaching Standards, 2014a). Castor
19 (2001) stated that “National Board Certification is important to education reform because the process attests to teachers' knowledge of advanced content in the subjects they teach”
(p. 1). It is unreasonable to think about successful education reform without high quality teachers, and the NBC is improving education by producing more effective teachers. The
NBC provides an opportunity for teachers to continue with their development and to grow professionally. As the American Federation of Teachers (AFT, 2013) of
Connecticut stated, “During the 2001 Legislative Session, a law was passed that allows teachers who earn the designation of National Board Certified Teacher to utilize that experience towards their continuing education unit requirement” (para. 4).
Some universities have started to offer graduate credits for masters and even PhD programs for those who have achieved the certification. This represents another call for
NBCTs to continue to grow professionally and become even more qualified.
NBC does not focus solely on the teacher, but it is the foundation on which students can achieve more learning progress. If we want our students to be successful, we need to produce teachers who promote student academic achievement and, specifically in areas such as mathematics, develop strong mathematical problem solving thinking instead of just delivering content and having students recite information mechanically.
Research says that effective teachers are the key to the success of a school and to what the students are learning. NBC is a topic that has attracted the interest of a considerable number of researchers. According to Vandevoort et al. (2004), “the vast majority of reports on NBCTs and those that compare them with non-NBCTs are favorable” (p. 7). Goldhaber et al. (2004) performed a study in North Carolina comparing students’ end of the year test scores. They divided the sample into different groups of
20 teachers who never applied for the NBPTS certification, teachers who applied to go
through the certification process but failed, and those who applied and successfully
achieved the certification. The results revealed that students taught by NBCTs or those
teachers who would become NBCTs in the future obtained better results compared to
those taught by non-NBCTs. In most of the models in their multivariate analysis
estimating the effect of NBCTs on students’ gains, the results were in favor of NBCTs.
A study by Whitman (2002) concentrated on investigating teachers’ attitudes and
characteristics connected to effective teaching. The researcher looked at six variables
such as internal efficacy, organizational commitment, higher career commitment, job
satisfaction, cooperation, and higher profession orientation. Five of the six variables were
statistically significant in favor of the NBCTs. The study revealed that NBCTs had
completed more graduate and beyond graduate coursework.
Vandevoort et al. (2004) demonstrated that students taught by NBCTs achieve higher learning gains compared to those students taught by non-NCBTs. In their study,
they compared 35 NBCTs with non-NBCTs, and they analyzed four years of data coming
from the Stanford Achievement Test in reading, mathematics, and language arts, Grades
3 through 6 (p. 1). NBCTs surpassed non-NBCTs in almost three quarters of the 48
comparisons within the four grades, the four years of data, and three measures of
academic performance (p. 2). The authors also found that most principals, about 85%,
stated that NBCTs are the best teachers they observed.
In some states, leaders were concerned about investing money in incentives to pay
for NBCTs. They were concerned that NBCTs would not indeed produce better results
than non-NBCTs. According to Jacobson (2004):
21 Critics have questioned the expenditure of state and district money on National
Board for Professional Teaching Standards certification, but the examination of
hundreds of thousands of student test scores offers the first evidence that teachers
who undergo the process make a difference in the classroom. (p. 1)
Stephens’ (2003) study revealed that there are not significant differences between
NBCTs and their non-certified peers. Her study focused on the relationship between
NBCTs and student achievement in mathematics standardized tests in two school districts in South Carolina. In 87% of the comparisons, no significant differences were found in the test scores produced by the students in the two groups of certified and non-certified teachers. However, this study used a very small sample size and utilized less sophisticated analytical techniques. For this reason, one could criticize the validity of the results in this study. Harris and Sass (2008) criticized Stephens (2003) as follows:
Stephens (2003) studies math achievement of 154 students in classrooms taught
by NBPTS teachers in South Carolina using an ANOVA technique. It is important
to note that while the dependent variable is the change in student achievement,
equation (1) is a model of student achievement levels, not achievement growth.
(p. 9)
In addition, Vandevoort et al. (2004) questioned the validity of this study because of the poor statistical management and the difficulty of the research in running the matched pair t tests.
Cavalluzzo (2004) studied 108,000 individual student records collected from
Miami-Dade County which assessed teachers’ professional characteristics, students’ background behavior, and school environment. The results showed that with the
22 exception of teacher’s undergraduate school quality, students whose teachers were
NBCTs made larger gains compared to those students taught by teachers who did not go
through the National Board Certification process. Additionally, this study revealed that
NBCTs produced greater gains especially with minority students and those students with
special needs. Moreover, the study found that students of NBCTs performed significantly
better on the mathematics portion on the Florida Comprehensive Assessment Test
(FCAT).
The studies in Table 1 address Research Question 1 which is: Do NBC
mathematics teachers produce better results than non-NBC mathematics teachers on the
AP Calculus AB Exam? Table 1 shows five major studies about the National Board.
What these studies have in common is that all of them used inferential statistical
procedures to prove the validity of their results. Three of the four studies revealed
significant differences in favor of the NBC and showed the importance of professionalism in teaching to very powerfully reduce the achievement gap. Another commonality is that all of these studies possess the quantitative component. Two of them are mixed methods which allowed examining characteristics such as gender, race, and education level. As stated above, Stephens (2003) did not favor NBCTs, but the criticisms by Harris and Sass (2008) and Vandevoort et al. (2004) directed attention to the limitations of the validity of the results in the Stephens study.
23 Table 1
Literature Comparing Studies Between NBC and Non-NBC Teachers
Author(s) Focus Type of Study Methodology Results Goldhaber Comparison of Quantitative Conversion of test Analysis of databases of et al. students’ end-of-year into z-scores. North Carolina student (2004) test results. Independent z-tests and assessment results (NBTs applicants & reveal that students of Comparison of 5 non-applicants) NBCTs scored 7 to 15 teacher categories points higher on end-of- (applicants, non- Comparison of year exams. applicants, certified descriptive data (by applicants, non- gender, experience, African Americans were certified applicants, education level, found to have 33% less all teachers license, etc.) chance of obtaining the certification compared to Racial ethnic groups Regression analysis white applicants.
Use of marginal probabilities for District fixed effects
Linear regression model
Vandevoort Academic Mixed Use of gain scores Statistically significant et al. performance of methods adjusted for the SAT- differences in almost (2004) students in 9 test. three quarters of all elementary comparisons. In classrooms Questionnaires classrooms where students of non-NBCTs Stanford Teachers and had gained more in an Achievement Tests in principals surveys to academic year, the reading, mathematics obtain data on differences were not and language arts, in demographics. statistically significant. grades three through six Experimental design Most of the NBCTs were techniques setting female. NBCTs as the treatment group and About 85% of the the non-NBCTs as principals stated that the control group NBCTs are one of the best teachers to be Analysis of observed. descriptive data Better learning gains and Analysis of effect size student achievement for those taught by NBCTs.
Effect size results indicate that the students of NBCTs have over a one month advantage in achievement in comparison to the
24 Author(s) Focus Type of Study Methodology Results students taught by non- NBCTs.
Whitman Teaching attitude Mixed Surveys NBCTs were likely to be (2002) characteristics linked methods accomplished people, to effective teaching Logistic regression with a stronger sense of (internal efficacy, models classroom responsibility, organizational efficacy, career commitment, higher Likert-type scales to commitment, career commitment, measure attitude professionalism, and job satisfaction, and characteristics collegiality. higher profession associated by orientation. previous research Statistical significant with effective difference found in five of Education level of teaching or related the six attitude NBCTs and non- behaviors. hypotheses. NBCTs Descriptive statistics Significant differences in Teacher gender (for demographic education level in favor data). of NBCTs. NBCTs tend to have more master’s Factor analysis degrees compared to non-NBCTs. Inter-scale correlations Significant differences in favor of gender. There t-tests seems to be a correlation between Levene’s test for being NBCT and being equality of variances. female.
NBCTs are more likely to believe that each child can be taught successfully regardless of the home situation or other external factors.
Stephens Quantitative Matched pairs t-test Only one matched paired (2003) t-test was shown to be The researcher statistically significant. controlled for teachers’ years of In 87% of the matched experience and pairs there was no school poverty level. significant difference between the ANOVA on achievement on the experimental groups mathematics portion of South Carolina’s achievement test (PACT) of students of NBCTs and non-NBCTs.
Cavalluzzo Compares nearly Quantitative Comparison of Students of NBCTs 25 Author(s) Focus Type of Study Methodology Results (2004) 108,000 Miami-Dade descriptive data made larger County student achievement gains on records to assess Calculation of effect the Florida mathematics teachers’ sizes test (FCAT) than characteristics that students of non-NBCTs lead to student Multiple linear (statistically significant achievement on regression differences in most of the FCAT mathematics indicators under study) portion in the 9th and Sensitivity tests 10th grades (classification Assessed teachers’ function) characteristics were experience, certification status, degree achieved, and selectivity of the teacher’s undergraduate school Note. ANOVA stands for Analysis of Variance.
Others are critics of the NBC. The criticisms can have two focal points. One focal point
relates to the NBC and its effectiveness, and the other is about the disparity of NBCTs
and how they are located in the different schools. There is literature that suggests
limitations regarding the effectiveness of implementation of the National Board. King
(1994) stated that the National Board would create a division of classes among teachers:
“The Board’s general emphasis on professionalization, national standards, assessment, and certification is likely to result in more controls on teachers, further distance professionals from lay persons, and inhibit collaborative efforts among school people in their local contexts” (p. 98).
According to Moore (2002), five barriers are perceived by teachers in two
Tennessee counties that discourage a population of eligible teachers to apply for the
NBC. The barriers are: (a) personal obstacles; (b) teaching professionalism; (c) teacher morale; (d) evaluation process; and (e) financial considerations. Personal obstacles seemed to be the most significant of the perceived barriers. Podgursky (2001) argued that the assessments to test the content knowledge of a teacher, for example in mathematics, 26 do not really measure how well a teacher knows his or her subject area. He also stated
that “the national board certification standards are not well tailored to what states expect
their students to learn (p. 38). Darling-Hammond, in 2006, as cited in Ingvarson and
Hattie (2008), refuted Podgursky’s ideas. Darling-Hammond stated:
First generation approaches to teacher testing have primarily relied on multiple
choice tests of basic skills and subject matter. The national board assessments
seek to address teaching knowledge and skills through portfolios and
performances that included authentic, complex teaching tasks as well as
systematic, content and context based analyses on-the-job performance.
Proponents argue that such strategies not only improve the validity of teacher
assessment but also support the development of teacher education programs
organized more explicitly around the attainment of important teaching abilities.
(p. 33)
Podgursky (2001) also disagreed that teachers should be paid the bonuses offered
for NBCTs. Former President Bush did not support the National Board financially. After he took power, his administration did not recommend any federal money in support of the
NBC (Jacobson, 2004). However, former President Bill Clinton would totally reject
Podgursky’s ideas on the incentive pay matter and he took a completely different approach compared to Bush. Clinton was an advocate of the NBC, and his view on how
teachers should be rewarded was indeed very different. In a Harvard education letter he
stated that the best teachers in this country should be rewarded (Gordon, 2002), and he
also fought in Congress to keep the NBC alive at times when opponents attempted to do
away with it. Whitman (2002) stated that “critics have also demanded proof that NBCTs
27 were outstanding in ways that would effectively benefit students” (p. 21). Ingersoll
(2000) described the main reasons for teacher attrition (see Figure 1).
3.2 Class Sizes too Large 6.5
7.7 Intrusions of Teaching Time 7.4
3.9 Inadequate Time 5.5 Poor Opportunity for 2.9 PERCENT advance 6.3
21.4 Poor Student Motivation 14.6 Math/Science Teachers All Teachers 12.2 Lack of Faculty Influence 16.5
29 Student Discipline Problems 22.9
45.9 Poor Administrative Support 42.7
56.7 Poor Salary 54.3
0 20 40 60
Figure 1. Percent of teachers giving various reasons for job dissatisfaction (Ingersoll, 2000).
By looking at Figure 1, it is evident that poor salary is the number one reason for
teachers leaving their jobs, and mathematics and science teachers are at the top of the list.
Politicians and policy makers should examine issues of this kind if they want to improve
education. Paying a bonus to NBCTs could have a positive impact in teacher outcomes
and student achievement.
A problem with NCBTs is that they generally opt to work in high performing
schools. The NBC was originally designed to make better teachers, but one of the main goals was to create effective teachers who would be able to make a bigger difference in low-income schools and reach out to those students who need the most help. According
28 to Sawchuk (2011), a report from the Center on Reinventing Public Education found that
“only about 1 percent of NBCTs transferred to challenging schools since 2007-08, and
that the teachers were no more likely to stay in such schools than other teachers” (p. 1).
Berry and King (2005) concluded that “NBCTs are more likely not to teach in low
performing schools as well as in schools serving poor and minority students” (p. 2).
Greenlee and Brown (2009) observed that “school leaders face the difficult challenge of
finding teachers who are highly qualified, committed, and prepared to ensure that all
students achieve at levels mandated by NCLB” (p. 96). Nearly all teachers, especially the
accomplished ones, would prefer to work in higher performing schools. The problem is that the number of schools in need or “non-performing schools” has grown. These types of schools do not seem to be the first choice as places to work for those who hold the
NBC.
Teacher Certification
The lack of certified teachers has allowed uncertified teachers access to the classrooms due to teacher shortages. The statistics associated with out-of-field teaching are shocking. Only 41% of those who teach math have a math degree (Berliner, 2001, p. 8). Unfortunately, these people often come into the teaching profession with no teaching experience or pedagogical background. Additionally, these people lack a teaching background because they did not attend a university teacher preparation program. The lack of pedagogical content knowledge endangers the quality of education that our students receive. According to Darling-Hammond (2007):
Many math and science courses are now taught by teachers who were trained in
other fields, by emergency hires, or by teachers with background in the content
29 but inadequate teaching skills. For many new instructors, the lack of training in
content and pedagogy contribute to both high attrition rates and lower
effectiveness. (p. 20)
It is important to understand the background reasons of how non-certified teachers or other professionals switching into the teaching world are hired in the educational system.
According to Boyd, Lankford, Goldhaber, and Wyckoff (2007):
Traditionally most U.S. school districts have hired graduates of teacher
preparation programs operated by schools of education in the nation's colleges
and universities. Successful completion of such programs is by far the most
common route to teacher certification. But many difficult-to-staff urban and rural
schools, unable to hire enough teachers from traditional preparation programs,
especially in subject areas such as math, science, and special education, are forced
to hire un-certified teachers. (p. 47)
It is a reality that uncertified teachers are usually placed in minority, low performing, or rural schools. According to Berliner (2001), “out-of-field teaching is a
tremendous problem for minority school districts and rural school districts. But it's not a
problem for the wealthy school districts” (p. 8). This creates a disparity in the educational
system because all children, regardless of their race, color, or socio-economic status, should have equal opportunities to be taught by effective teachers. Even though effective teachers cannot solve the problem of socio-economic disparities, they can certainly help in producing more well-prepared students. Darling-Hammond (2006) stated that:
America’s schools are among the most unequal in the industrialized world in
terms of both inputs and outcomes. Inequalities in spending, class sizes,
30 textbooks, computers, facilities, curriculum offerings, and access to qualified
teachers contribute to disparate achievement by race and class. (p. 9)
Inequality is inevitable for the simple fact that schools are not funded in the same way.
Some districts are wealthier than others, and the local property taxes from more affluent
areas create a clear disparity in the allocation of resources. To close this achievement gap,
providing schools with quality teachers can be a better way to move forward and start
rethinking the certification process. Walsh (2001), as cited in Darling-Hammond (2002),
argued that “the least prepared teachers are at the worst city schools. One third of them
lack basic credentials for certification” (p. 1). According to Darling-Hammond (2006):
Disparities in access to well-qualified teachers are large and have worsened
throughout the 1990s. In 2001, for example, students in California’s most
segregated minority schools were more than five times as likely to have
uncertified teachers as students in predominantly White schools. (p. 12)
There are alternative certification routes that can aid in supporting professionals
who come into the teaching world by obtaining a certification that allows them to
officially remain in the system as long as they produce positive results in student success.
Darling-Hammond (2010) commented that, “Associated policy initiatives, encouraged by the federal government under No Child Left Behind, have stimulated alternative certification programs that often admit recruits before they have completed, or sometimes even begun, formal preparation for teaching” (p. 37). A point that needs to be addressed is that schools should have teachers coming from teacher training programs possessing the appropriate pedagogic and content knowledge. Ringrose (2004) argued:
31 Subject-matter expertise alone is not enough to ensure effective teaching. The
“how to teach” element includes knowledge of child and adolescent development,
instructional strategies for various types of learners, assessment and evaluation
strategies, classroom management, and strategies for teaching those of differing
abilities. (p. 4)
It is evident that there are a number of people entering today’s classrooms with either no certification whatsoever or only a temporary certification. According to the
National Center for Education Information (2011), “there are 3.2 million public school
teachers teaching the nation’s 49.4 million children attending PK-12 schools (p. 8). Of
those 3.2 million teachers, 17% reported entering the teaching career through various
alternative ways (see Figure 2).
Pathway to Teacher Certification
Traditional college campus- 5% 1% 3% based undergraduate teacher education program 8% Traditional College campus- based graduate teacher education program Alternative school-based program administered by a college 18% Alternative school-based program administered by a school district 65% Alternative program but don’t know who it was administered by Not Sure
Figure 2. Composition of public school teachers by route to teaching. Source: National Center for
Education Information (2011, p. 21).
32 Figure 2 displays four different ways in which people access alternative teacher
certification in this country. Unfortunately, 17% of the existing 3.2 million results in an
estimated total of 544,000 teachers, or more than half a million teachers, who may not
possess the appropriate qualities necessary to teach. This may result in classrooms staffed
with teachers who do not have the pedagogical and/or content knowledge and the ability
to properly manage a classroom. Darling-Hammond (2002) stated that “teachers who hold full certification in their major are more powerful predictors of student achievement compared to the teacher’s level of education, for example, having a master’s” (p. 32). To achieve alternative certification, states may require taking college courses to fill in gaps in content knowledge, educational psychology, and pedagogy, and passing teacher certification examinations. According to Kane, Rockoff, and Staiger (2008):
• Traditionally, federal and state governments have regulated teacher quality
with ex ante certification requirements. To gain legal permission to teach,
prospective teachers have been required to study full-time for one or two
years in an approved education program. However, recruiting difficulties
have forced many districts to hire large numbers of uncertified or
alternatively certified teachers. (p. 1)
Boyd et al. (2007) stated that there is a positive and a negative aspect in obtaining
certification. The positive aspect is that obtaining certification keeps ineffective teachers
from the profession. They also stated that the drawback is that the certification procedure
may keep potential good teachers away because it can become an arduous process to some. The Abell Foundation proposed elimination of the coursework requirements for teacher certification and required only a bachelor's degree and a passing score on an
33 appropriate teacher's examination in Maryland (Darling-Hammond, 2002, p. 2). Similar
changes in other states were implemented to comply with the NCLB requirements for
highly-qualified teachers. This underscores why it is important to study if there is a
positive association between teacher certification and student achievement.
Understanding the provisions of traditional teacher certification and subject area
examinations can help in better understanding the value and validity of the NBC.
The research in regards to teacher certification and student achievement is limited
(see Table 2). Table 2 provides a snapshot of some of the important studies that have been performed. Throughout these four studies there is a comprehensive focus on the different kinds of certification status ranging from being non-certified, temporarily certified, to certified and non-certified international teachers. The selection of studies in
Table 2 presents quantitative and qualitative methods. Even though there are mixed results in these studies, most of them find in favor of certification. It is evident that certification is an issue that matters in education in this country, and at times it is a predictor of teaching effectiveness.
As can be observed from Table 2, mixed results have been found across the different research studies (Ringrose, 2004). However, several studies found positive results in favor of certified teachers compared to those who do not have certification or find themselves in the processes of obtaining it. Boyd et al. (2007) researched how teacher preparation and certification can affect student achievement. They found out that teachers who score well on the certification exam can improve students’ outcomes.
34 Table 2
Literature Comparing the Relationship Between Teacher Certification and Student Achievement
Author(s) Focus Type of Methodology Results Study Boyd et al. How preparation Qualitative Surveys Highly selective alternative route (2007) and certification programs can produce effective requirements Analysis of teachers who perform about the affect student descriptive same as teachers from traditional achievement. data (tables) routes after 2 years
Traditional teacher Teachers who score well on preparation certification exams can improve programs student outcomes
Alternative routes Research evidence is simply too of certification thin to have serious implications for policy Teacher supply
Clotfelter, NBC Quantitative Analysis of Students in classes with non- Ladd, & tests scores certified teachers scored eight Vigdor Quality of points lower on the Algebra I EOC. (2007) undergraduate Comparison students of effect There was no practical significance sizes in a student’s odds of passing the Teacher examination between students in characteristics certified teachers’ classrooms and those in uncertified teachers’ Student classrooms. characteristics Female students scored slightly Teacher level of higher than males. education Students eligible for FRL scored Teacher lower than non-FRL students. experience Asian students obtained the highest Institution from scores compared to other races. where the teacher obtained his or her Students from certified teachers’ degree scores were higher and statistically significant compared to those
students coming from the non- certified group.
Effect size is small; this weakens the power of statistically significant differences.
Ringrose Effects of teacher Quantitative Three Students taught by uncertified (2004) certification on statistical teachers produced significantly freshman high tests were poorer results in 3 out of 6 tests school students’ used for the scores. mathematics analysis of achievement the data: Most of the teachers in the district (Algebra I EOC) covariance possessed less than two years of
35 Author(s) Focus Type of Methodology Results Study as measured by analysis, experience (national average is 15 the State of Texas hierarchical years). multiple regression, One third to one half of the and logistic teachers in grade levels 3 and regression higher were without standard certification (striking finding).
TFA teachers produced positive results on student achievement.
Certified teachers are more effective compared to those uncertified but certified teachers were not found to perform better than TFA teachers.
Darling- Effects of teacher Quantitative Multiple No differences were found in math Hammond, certification Linear and reading achievement between Holtzman, Regression certified and non-certified teachers. Gatlin, and Comparison of Heilig certified and Analysis of Teaching experience revealed (2005) uncertified effect sizes statistically significant differences. teachers Analysis of Statistically significant differences Comparison of descriptive in math student achievement comparing lower and third quartiles TFA teachers and data of teachers holding initial non-certified certification. teachers Least square Uncertified teachers, international Teacher regression teachers, and teachers with experience estimate alternative certification seem to work for different kind of schools Teacher where certified teachers tend not to background and be hired. level of education International recruits and Students grades 3 uncertified teachers scored lower and higher than certified teachers in student through 1995- achievement in math and reading. 1996 to 2001- 2002 school years
Kane et al. Effectiveness of Quantitative Not clearly Initial certification status of a (2008) recently hired stated but teacher has small impact on teachers in New one may students’ test performance. York City assume independent On math achievement, no Math achievement t-tests for difference found between teaching comparison fellows and certified teachers or Reading of two between uncertified and certified achievement groups were teachers. Teaching utilized to perform the Teachers recruited through TFA statistical are significantly more effective than 36 Author(s) Focus Type of Methodology Results Study experience tests. both uncertified and certified teachers at math instruction and Teacher Analysis of statistically indistinguishable in effectiveness by descriptive reading instruction. initial certification statistics In math, students assigned to Uncertified Multiple teaching fellows performed no teachers regression differently than similar students analysis assigned to traditionally certified International teachers. teachers Calculation of fixed Teaching fellows have very similar Alternative effects retention rates to regular certified certification teachers (with teaching fellows teachers Logistic having slightly higher retention regression rates in the first two years. TFA of hazard corps members would be roughly rates twice as likely to be in their first year of teaching (25.6%) than Calculation certified teachers (11.1%). of interaction effects There are few or no differences in average value-added by initial certification status.
Little or no difference in the average teacher effectiveness of certified, uncertified, and AC teachers does not imply that selection of teachers is unimportant.
Note. EOC stands for End of Course, TFA stands for Teach for America.
Ringrose (2004) performed a study comparing teachers who were certified and those who were not. She concentrated on test scores of Algebra I End of Course (EOC)
Exams. Her study found that students taught by teachers who hold certification produced better results on the Texas Algebra I EOC exam compared to those students taught by non-certified peers. Darling-Hammond et al. (2005) presented a case study in which they attempted to answer the question of whether certified teachers are more effective compared to those who did not meet the testing or training requirements. This study incorporates the effectiveness of teacher certification as a variable linked to student 37 achievement. The study utilized a large set of individual student records published by
Houston Independent School District in Texas. The study also examined teacher
backgrounds and experience. The Teach for America (TFA) candidate recruits were
compared in effectiveness to experienced certified teachers. The methods utilized were
multiple regression, measuring effect sizes, and descriptive statistics. The study was
reinforced by incorporating variables such as students’ ethnicity and socioeconomic
level. The main findings were that the students taught by non-certified teachers made less progress on three out of six tests and students of certified TFA teachers performed comparably to students of other certified teachers. The findings showed no statistically significant differences on all six tests.
A study by Kane et al. (2008) produced mixed results. Their study researched teachers with various certification statuses. No differences were found in student achievement in the areas of math and reading when comparing certified and non-certified
teachers. They found significant differences on teaching experience between those two
groups. In addition, significant differences occurred in favor of teachers in the top
quartile holding an initial certification compared to those located in the lower quartile.
The statistically significant differences in math and reading were found among
international recruits and non-certified teachers when compared to certified teachers.
Results showed higher z-scores for certified teachers in this study:
Prior research has found a relationship between teacher effectiveness and the
selectivity of the college a teacher attended (Summers & Wolfe, 1977), tests of
teachers’ verbal ability (Hanushek, 1971), or a teacher’s own ACT (American
38 College Testing program) scores when applying to college (Ferguson, Ladd, H.F.
& Ladd, H. 1996). (Kane et al., 2008, p. 622)
Best Practices in Teaching Secondary Mathematics
It is important to consider that effective learning cannot be accomplished without effective teaching. It does not matter how well a lesson is planned or designed, or how much teachers know about mathematics, if the elements of effective teaching are not in place. Effective teaching needs to be present in order to develop students’ mathematical understanding.
The nature of high school math. Math is the subject that helps us understand the world around us and is a form of reasoning. Math is part of many of humanity’s progressions. The Education Alliance, a West Virginia not-for-profit public education advocacy organization, stated that “thinking mathematically consists of thinking in a logical manner, formulating and testing conjectures, making sense of things, and forming and justifying judgments, inferences, and conclusions” (2006, p. 1). Battista (1999) argued:
Mathematics is at first and foremost a form of reasoning. In the context of
reasoning analytically about particular types of quantitative and spatial
phenomena, mathematics consists of thinking in a logical manner, formulating
and testing conjectures, making sense of things and forming and justifying
judgments, inferences and conclusions. (p. 428)
Mathematics teaching requires an understanding not only of the essence of mathematics but also of current research about how students learn (Battista, 1999, p. 433). According to Daro (2006), the focus is on finding the right answer but not on
39 building the foundations necessary for understanding higher level math. These foundations can only be built with a mathematics program that teaches concepts, skills, and problem-solving (p. 35). Daro’s statement relates to one major problem in American education, which is how mathematics is taught at the early stages. There are teachers who teach mathematics in a way that does not allow students to think critically. They make their students solely perform systematically and often times solve math problems through repetitive procedures without context. By the time these students get to high school, they have a mindset that leads to an incorrect conception about mathematics. When students are taught in a repetitive manner, they do not see how mathematics is related to real- world situations or how it is meaningfully applied to the world that surrounds them.
Issues of mathematics performance. When analyzing the results of international mathematics test competitions such as the Program for International Student Assessment
(PISA) or the Third International Mathematics and Science Study (TIMSS), it is no surprise that the United States falls behind other countries that outperform our students.
“United States students are continuing to trail behind their peers in a pack of higher performing nations” (USA Today, 2010, p. 1). In those higher performing countries, students are taught to learn by constructing their knowledge through higher order thinking processes. The TIMSS (1996) reveals that U.S. eighth graders scored significantly below the international average in math and only outperformed seven nations (Bernstein, 1997). U.S. students scored below the international average of 41 nations in mathematics (NCES, 1999, p. 4). According to Vigdor (2013), “The performance of American 15-year-olds on the PISA in 2000 and 2009 confirms the lack
40 of progress among secondary students. The United States is among those countries whose
math performance worsened over this time period” (p. 45). According to the National
Center for Education Statistics (NCES, 2005), regarding results for the PISA in 2003:
In 2003, U.S. performance in mathematics literacy and problem solving was
lower than the average performance for most of the OECD countries. The United
States also performed below the OECD average on each mathematics literacy
subscale representing a specific content area (space and shape, change and
relationships, quantity, and uncertainty). (p. iii)
The PISA has a scale that measures student proficiency levels. Those levels range from level 1 through 6, 6 being the highest level of proficiency. “In mathematics literacy, the United States had greater percentages of students below level 1 and at levels 1 and 2 than the OECD average percentages.” The report went on to say, “The United States also had a lower percentage of students at level 4, 5, and 6 than the OECD average percentages” (NCES, 2005, p. iii). The results from the PISA 2000 were very comparable to the results in 2003.
The results of the 2009 PISA revealed that the United States ranked 31st in
mathematics achievement among 65 participating countries (Organization for Economic
Co-operation and Development, 2013). A TIMSS publication, A Splintered Vision, had
suggested that “the unfocused state of the U.S. mathematics and science curricula and the
poor student performance is likely a result of unfocused curricula” (Masini & Taylor,
2000, p. 4). The percentage of students in the U.S. Class of 2009 who were highly
accomplished in math is well below that of most countries with which the United States
generally compares itself (Hanushek, Peterson, & Woessmann, 2010).
41 A large issue that also emerges with the lack of students’ mathematical
preparation and skills is their poor performance when they enter college. “In the past 25
years, nationwide, students are coming to college unprepared for math” said Clukey
(2013, p. 1). “As many as 40% of the nation’s high school graduates say they are
inadequately prepared to deal with the demands of employment and post-secondary
education” (Achieve Inc., 2005). According to Carmichael (2012), even in
Massachusetts, one of the top states in secondary education, “just 46 percent of public
high school students in Massachusetts read at a 12th-grade level, and only 36 percent can handle the equivalent level of math, other states do worse” (p. 1). Curtis (2012) detailed student data in Maine:
According to data from the state’s community college system, 65 to 70 percent of
students planning to attend the state’s community colleges are unprepared to
enroll in college level math courses, while about 40 percent of students entering
four-year colleges are also unprepared. A recent study by the National Center for
Education Statistics showed that . . . nearly two-thirds, or 63 percent of students
who require remedial math courses in a college or university fail to earn degrees.
In contrast, nearly two-thirds, or 65 percent, of students who do not require
remediation complete associate’s or bachelor’s degrees. (p. 1)
In addition, Doubleday (2013) argued that “less than half of the students who took the
SAT in 2013 are ready to succeed at the post-secondary education according to a College
Board report released on September 26” (p. 1).
Issues with AP math. AP math classes are designed to prepare students to be
college ready. If students do not learn mathematics properly in their early years, they face
42 a gap in achieving success; these students are less likely to be ready for advanced
mathematics classes when they are in high school. The need for effective math teaching
practices and qualified teachers is critical to help students become successful. For
students to be able to handle AP mathematics classes, they need to have a solid
foundation of mathematical concepts and skills that have prepared them to be ready to
handle a high level of mathematic challenges. There are no requirements set to take an
AP math class, but the reality is that most of the students who sign up for those classes are students who previously followed the “honors” mathematics pathway by taking classes such as Algebra I Honors, Geometry Honors, Algebra II Honors, and Pre-
Calculus. It is evident that Pre-Calculus is highly recommended, yet it is not a mandatory
pre-requisite. On many occasions students sign up for AP Statistics after either Pre-
Calculus or AP Calculus.
It is undeniable that effective teachers using best mathematics practices can make
a difference in providing marked improvement of mathematics education. We need to
have a system that encourages teachers to be curious and adventurous trying new
technologies and methods, thereby helping students to think critically and creatively and
resulting in effective math thinkers and problem solvers. We need teachers who
understand that teaching the curriculum is not just teaching procedures without also
giving meaning to mathematics. To achieve this goal, the need for such highly effective
math teachers and best practices becomes critical. The NBC plays an important role in
attempting to enhance American education improvement, its reform, researchers, and
policy makers.
43 Gender, Race, and Equity in Mathematics
One fact about gender in mathematics is that historically, most famous mathematicians were men, and mathematical discoveries were developed by men. As cited in Simon (2000), “Wertheim (1995) traces the exclusion of women from mathematics to around 600 BCE” (p. 782), while Simon (2000) stated that even in the
1950s, women were not permitted into physics buildings at Harvard and Princeton
(p.784). According to the European Commission (2009), “for much of human history, women were officially excluded from the scientific realm. Many women throughout the centuries have managed to overcome their marginalization and excel in their chosen field” (p.3). The cited statements above indicate that women were banned from mathematics participation in society in the past. This opens the question if women tend to develop less interest in mathematics participation in today’s education at the higher levels of mathematics.
Often times teachers who teach upper level mathematics courses in a high school make the observation that these types of classrooms are filled mostly with male students.
This raises the question of whether math ability is related to gender. Wade (2013) stated:
Math ability, in some societies, is gendered. That is, many people believe that
boys and men are better at math than girls and women and, further, that this
difference is biological (hormonal, neurological, or somehow encoded on the Y
chromosome. (para. 1)
She also stated that there is no difference in math aptitude before the age of 7, but at later stages the difference is observed in tests like SAT in which boys score 30 to 35 points higher compared to girls (para. 3). The gap between boys and girls seems to increase in
44 high school, where by the 12th grade, males show very significant advantages over
females of the same age in mathematical achievement tests. According to Rosselli,
Inozemtseva, Matute, and Ardila (2009), “gender differences in the level of achievement
in mathematics among students in middle school and high school compared to elementary
school have been reported by Mullis et al. (2000) and Willingham and Cole (1997)”
(p. 217).
Former president of Harvard University Laurence H. Summers, as cited in Angier
and Chang (2005), stated that “one factor in women’s lagging progress in science and
mathematics might be innate differences between sexes” (para. 1). According to a U.S.
Department of Education, Office for Civil Rights, Civil Rights Data Collection report
(2012), “girls are equitably represented in rigorous high math courses such as geometry,
algebra II, and calculus” (p. 19). Figure 3 depicts the enrollment of both genders in math
courses.
By looking at Figure 3, it can be observed that gender enrollment presents no
disparities in regards to mathematics courses that are non-AP; however, the standards and
mathematical content level involved in these types of mathematics classes are not as
rigorous and demanding as compared to AP math courses. AP courses have been
designed for those students proficient in abstract thinking and who can meet the college
mathematical demands while they are in high school. In many schools, students need to meet certain criteria before they can be accepted into an AP math course. Among those criteria are getting good grades in previous honors classes, and traditionally AP math classes are to be taken after successfully passing and completing Pre-Calculus. It is indeed a different type of commitment.
45 Enrollment in Math Courses 100% 90% 51% 50% 49% 51% 80%
70% 60% 50% 51% Boys 40% 49% 50% 49% Girls % Students of % 30% 20% 10% 0% Enrollment Geometry Algebra II Calculus Courses
Figure 3. Enrollment of boys and girls in high school math courses. Adapted from: U.S.
Department of Education, Office for Civil Rights, Civil Rights Data Collection (2012).
A U.S. Department of Education, Office of Civil Rights, Civil Rights Data
Collection report (2012) stated that “girls outnumber boys in enrollment of AP science,
AP foreign languages, and several other AP classes; however, in AP mathematics
(calculus and statistics) boys have consistently outnumbered girls” (p. 3). Figure 4
demonstrates the difference in enrollment of AP courses regarding boys and girls.
Figure 4 depicts the consistent differences in AP calculus and AP statistics
enrollment in favor of male students every year for an entire decade. It is interesting to
note that girls have outnumbered boys when registering for AP classes such as science,
foreign languages, and several other AP classes. However, boys have consistently
outnumbered girls on AP math class enrollment by up to 10,000 (Perry, 2013). According
to the National Science Board (2014):
46 AP Mathematics: Trends in Course Taking 230,000
220,000 210,000 200,000 190,000 Girls 180,000 170,000 Boys
Number Students of Number 160,000 150,000 2000 2002 2004 2006 2009 Year
Figure 4. AP mathematics enrollment by gender. Adapted from: U.S. Department of Education,
Office for Civil Rights. Civil Rights Data Collection (2012).
The proportion of male and female students in the class of 2012 taking
mathematics and science exams varied by subject. Male students were more likely
than female students to take advanced AP courses, including calculus BC (59%
versus 41%), physics B (65% versus 35%), and both physics C courses (about
75% versus 25%). (p. 5)
An additional component that deserves to be considered is the passing rate on the
AP mathematics exams. Differences in test results have been observed and such
difference was found in favor of male students (see Figure 5).
As can be observed, the boys’ higher passing rate of 60% was compared to the girls’ passing rate of 55%. In both groups of girls (African American and white), girls passed the AP test at a lower rate. The College Board (2013) stated that:
47 AP Test-Passing 70%
60% 64% 60% 60% 50% 55% 40% 30% Girls 30% 32% Boys
% Students of % 20% 10% 0% All Black White Race
Figure 5. Passing rate proportion of boys and girls on AP mathematics tests. Adapted from: U.S.
Department of Education, Office for Civil Rights, Civil Rights Data Collection (2012).
In fall 2013, the College Board implemented the AP Science Technology
Engineering and Mathematics (STEM) Access program in 335 public high
schools across the country. With the support of a $5 million Google Global
Impact Award to DonorsChoose.org, these schools started offering new AP math
and science courses with the goal of enabling underrepresented minority and
female students who have demonstrated strong academic potential to enroll in and
explore these areas of study and related careers. (para. 15)
It is interesting to note that many jobs that require the understanding of
mathematics such as engineers, mathematicians, astronomers, and others are traditionally
performed by males. Researchers at Yale University published a study proving that
physicists, chemists, and biologists are likely to view a young male scientist more
favorably than a woman with the same qualifications (Pollack, 2013). He also noted that
only one fifth of PhDs in this country are awarded to women. AP Calculus and AP
48 Statistics are classes frequently taken by students whose future careers will require the application of mathematics.
According to Oxford Dictionaries, an online service, stereotyping is defined as “a widely held but fixed and oversimplified image or idea of a particular type of person or thing” (Oxforddictionaries.com, 2014). Stereotypes usually focus on groups of people involving gender, race, or nationality. Spencer, Steele, and Quinn (1999) explained that
“when a stereotype about one’s group indicts an important ability, one’s performance in situations where that ability can be judged comes under an extra pressure, . . . and this extra pressure may interfere with performance” (p. 6). The researchers also found that stereotyping, when it comes to women’s math performance, depresses their performance and participation in math related areas. As observed, stereotyping may become a threat as it may give women a poor self-image of how they perform mathematically in comparison to men.
Beilock, Rydell, and McConnell (2007), reported on a study that the University of
Chicago conducted to find out how a group of women did on a math test after being told that women do poorly in math compared to men. A control group was not reminded that women do worse than men. She explained that women who were casually reminded that there are gender differences in math and that they may not perform as well indeed performed significantly worse compared to the control group (Beilock et al., 2007). This outcome provides evidence that the presence of stereotyping may have an effect on math performance. FARS News Agency (2010) reported that:
. . . teachers and parents can play a negative role by guiding girls away from
math-heavy sciences and engineering; and they continue to hold stereotypes that
49 boys are better in math, and that can have a tremendous impact on individual girls
who are told to stay away from engineering or the physical sciences because girls
can’t do the math. (p. 1)
With specific regard to AP mathematics classes, stereotyping may induce girls to stay
away from taking such demanding classes.
It is evident that the perception of mathematics ability has brought a great deal of
discussion and has awakened researchers’ interest to explore if it is true that there are
inherent gender differences. Based on the gender differences found in research related to
mathematical performance between men and women, the researcher’s interest has been to
investigate if there is an interaction effect between teacher certification status and student
gender on the AP Calculus AB Exam.
Mathematics Standards
Educational standards are accepted measurements of comparison for quantitative
and/or qualitative value (Calison, 1999, p. 38). According to the National Board for
Professional Teaching Standards (2011b), “mathematics standards describe what accomplished teachers should know and be able to do” (p. 16). Policy makers have been
trying for years to institute stricter standards in the American educational system. Starting
with former President Bill Clinton by bringing Goals 2000, later on with former President
George W. Bush with the implementation of the No Child Left Behind policy, to
President Barack Obama and his Race to the Top, education in the United States has been
changing drastically thanks to the existence and revisions of standards. The standards
have often been accompanied by high-stakes testing. Both are often used as determining
factors for students’ graduation and entrance to college. Finally, the latest standards that
50 have come to the educational world to improve American education are the Common
Core State Standards (CCSS), a project that President Obama supported emphatically.
CCSS are being implemented from Kindergarten through 12th grade, and they constitute a
method of measuring student achievement and teacher performance. It is clear that there
are many components of the CCSS that are important to educators and policy makers.
Among those components are historical foundations, purpose, funding and support, diversity, and the impact on mathematics including Algebra 1 and AP mathematics courses.
The federal government has offered $350 million in order to help the states to establish this brand new evaluation system that will measure student proficiency
(Rothman, 2011).The incentives also serve the purpose of helping teachers to strive to improve their teaching and student achievement.
Mathematics standards are not just a matter of measuring students and teachers.
They have been created to ensure student success, ensure that students are well prepared to enter a 4-year college and also to ensure that some students are able to perform accordingly in more advanced mathematics classes after they leave high school.
Standards define what students should understand at their respective grade levels and
what they should be able to do as they learn mathematics. Ideally, there should be a good match with what the standards propose and the reality of what the students are mastering.
Especially in mathematics, it has been observed that there is a big disconnection between what students know coming out of their respective high schools and the expectations that
colleges and universities have when they open their doors to those students for the simple
fact that many students go to college unprepared. This disconnection can also be
51 observed as there are mathematics middle schools teachers who complain that students come unprepared from elementary school; in the same fashion, there are mathematics high school teachers who blame the middle school teachers for leaving their students mathematically unprepared. By observing such behavior, it is evident that standards should be coherent in the sense that students should continuously progress from one grade to the next.
Hart, Carman, Luisier, and Vasabada (2011) stated: “The College Board recognizes that defining clear, consistent, and rigorous standards is just the beginning of ensuring educational improvement” (p. 2). If standards are developed in ways that are clear, consistent, rigorous, and relevant, it is undeniable that they can be very important to meet the demands of student success, college readiness, and also readiness to take more AP classes. It is important to note that teachers instruct their AP Calculus AB students according to the standards set by the College Board.
National Council of Teachers of Mathematics (NCTM). The NCTM is a non- profit, professional, educational association. It is the largest organization dedicated to the improvement of mathematics education and to meeting the needs of teachers of mathematics (Russell, 2006). In 1989, the NCTM published the Curriculum and
Evaluation Standards for School Mathematics (Reed & Oppong, 2005, p. 2). The goal of this document was to improve mathematics teaching in all schools in America. Since
1989, the NCTM has been releasing documents updating and evaluating curriculum standards in math. The NCTM updated their standards in the 2000 publication, Principles and Standards for School Mathematics (Reed & Oppong, 2005, p. 2). According to
Burris (2014):
52 In April 2000, the National Council of Teachers of Mathematics released its
Principles and Standards for School Mathematics. This document updates the
1989 Curriculum and Evaluation Standards and includes some components of
both Professional Standards for Teaching Mathematics and Assessment Standards
for Teaching Mathematics as well. (p. 2)
According to the NCTM (2000), “Content Standards explicitly describe the five strands
of content that students should learn, whereas the five Process Standards highlight ways
of acquiring and applying content knowledge.” The Process Standards are directly related
to methods and best teaching practices to be implemented in the classroom (Burris,
2014). Table 3 depicts the NCTM Process Standards (2013).
Problem solving serves as a foundation for any strong math curriculum. This is
the element in the mathematics curriculum that exposes student to reasoning, logic to
apply mathematical concepts and apply the procedures that the students previously
learned from the teacher. One example of problem solving can be illustrated when a student needs to use logical reasoning and decide which process he or she needs to develop in a situation where an algorithm or a formula cannot be directly applied.
Reasoning and proof is an effective math strategy that helps the students justify
their results. “By exploring phenomena, justifying results, and using mathematical
conjectures in all content areas and—with different expectations of sophistication—at all
grade levels, students should see and expect that mathematics makes sense” (NCTM,
2000, p. 4; also see Principles and Standards for School Mathematics, 2000).
Communication requires that students express their thinking orally or verbally. It
also allows for discussion to exchange ideas and perspectives, see different points of
53 view, agree and disagree with others and in such way to gain mathematical understanding.
Connections is a strategy that focuses on having students relate what they study in the classroom with real-world situations and apply math to other subjects or sciences.
Students need to understand their mathematical ideas through different representations.
Students can express mathematical knowledge in diverse format, for example, an equation with words or a sentence, or representing the number 25 as a fraction 100/4 among many examples.
One of the main goals of the NCTM and the standards was mathematics literacy that included applying “knowledge actively in order to learn it well” (Ravitch, 1995, p. 127). There are several approaches on how students should develop their mindset and see how mathematics is related to the world in which they live. Most of the approaches overlap in the way they define the best mathematics teaching practices. Nevertheless, the standards set by the NCTM are research based and these are standards that have been re-evaluated and updated. According to Zemelman, Daniels, and Hyde (2005):
Taken collectively, the NCTM standards and their related materials offer a
significantly broadened view of the nature of mathematics, what it means to know
mathematics, how students can learn mathematics, and what kinds of teaching
practices best foster this learning. (p. 110)
54 Table 3
Process Standards of the National Council of Teachers of Mathematics (NCTM)
Process Skills attained Problem Solving Instructional programs from prekindergarten trough grade 12should enable all students to: • Build new mathematical knowledge though problem solving • Solve problems that arise in mathematics and in order contexts • Apply and adapt a variety of appropriate strategies to solve problems • Monitor and reflect on the process of mathematical problem solving Reasoning and Instructional programs from kindergarten through grade 12 should Proof enable all students to: • Recognize reasoning and proof as fundamental aspects of mathematics • Make and investigate mathematical conjectures • Develop and evaluate mathematical arguments and proofs • Select and use various types of reasoning and methods of proof Communication Instructional programs from kindergarten through grade 12 should enable all students to: • Organize and consolidate their mathematical thinking through communication • Communicate their mathematical thinking coherently and clearly to peers, teachers, and others • Analyze and evaluate the mathematical thinking and strategies of others • Use the language of mathematics to express mathematical ideas precisely Connections Instructional programs from kindergarten through grade 12 should enable all students to: • Recognize and use connections among mathematical ideas • Understand how mathematical ideas interconnect and build on one another to produce a coherent whole • Recognize and apply mathematics in contexts outside of mathematics Representation Instructional programs from kindergarten through grade 12 should enable all students to: • Create and use representations to organize, record, and communicate mathematical ideas • Select, apply, and translate among mathematical representations to solve problems • Use representations to model and interpret physical, social, and mathematical phenomena Note. Adapted from National Council of Teachers of Mathematics, 2015, retrieved from http://www.nctm.org/Standards-and-Positions/Principles-and-Standards/Process/.
55 Hollandsworth (2006) stated that, “what the NCTM emphasized with these standards is that learning the basics is important but memorization without understanding is useless” (p. 78). The standards created by the NCTM are research based, they project high expectations, and they require teachers to be more aware of their practices to develop understanding among students.
The Role of the National Board for Professional Teaching Standards
(NBPTS). Similar to the NCTM, the NBPTS created a comparable set of standards
defining those teacher qualities and qualities of teaching that define accomplished
mathematics teaching (Reed & Oppong, 2005). Hollandsworth (2006) stated that “from the five core propositions,” the “NBPTS developed standards that describe the highest level of teaching in different disciplines” (p. 1). According to Reed and Oppong (2005),
“NBPTS went further than NCTM’s suggestion that teachers be aware of issues of equity.
Through their certification process, NBPTS asks teachers to reflect on equity-focus and
what it implies for teachers” (p. 3).
The NBPTS recognizes best practices and acknowledges that NBCTs should
model these practices (Hamsa, 1998). The standards of the NBPTS, even though they are
expressed with a different word format, present similar characteristics to the NCTM
standards. Both standards concentrate on mathematical reasoning, problem solving,
forming and testing conjectures, justification, and proof on how to communicate results
(National Board for Professional Teaching Standards, 2010a, p. 13). Standards that
encourage teachers to make students use their mathematical thinking knowledge
recognize situations in which mathematics can be useful in real life. The synthesis of
information is a significant and invigorating enhancement, setting the foundation for
56 future students’ success. Table 4 describes the NBPTS mathematics standards for teachers of students aged 11 to18+, which explain the important facets of accomplished teaching.
The standards promoted by the NCTM and the NBPTS have served as a foundation for different studies on best practices. There are many standards that teachers need to align in their daily instruction in order to deliver best math teaching practices. It is important that teachers align classroom strategies using proven teaching methods.
Looking at the NCTM and the NBPTS standards at the national level should be the first step taken. Then, teachers can focus on their state and district standards. Standards at the national, state, and district level in addition with best classroom practices need to be cohesive so that student learning is at its best.
Studies of Best Mathematics Practices
Johnson (2000) suggested that the use of manipulatives improves mathematics achievement if they are used long term. Johnson (2000) analyzed a literature review of previous researchers who researched the use of manipulatives. Through his analysis from previous research, he emphasized that manipulatives make a connection with concrete examples and real world situations that promote a better understanding of math processes. Some studies revealed that manipulatives improve thinking processes and reduce math anxiety. It is important that students think analytically in mathematics which means making reasoned decisions or judgments about what to do and think and that they consider the criteria or grounds for a thoughtful decision and do not simply guess or apply a rule without assessing its relevance (The Critical Thinking Consortium, 2013).
57 Students who are able to make connections and think critically have a better
attitude toward learning mathematics, but it is important that the mathematics teachers
intervene effectively to help students focus on the underlying mathematical ideas that
build bridges between students’ work and their understanding of math. Within this
literature review, English and Halford (1995) did not agree that the use of manipulatives
promotes understanding of algebraic thinking.
Brophy (2000) presented a summary of best teaching practices based on research in classrooms. In his study, he focused on curriculum and instruction, assessment, classroom management and organization, and practices that support effective instruction.
He stated that there is no single strategy in particular that works on every occasion. A good teacher has to be aware of how to manipulate methods that will maximize student learning. He suggested a model featuring principles of effective teaching practices: (a) a supportive classroom climate; (b) opportunity to learn; (c) curricular alignment; (d) establishing learning orientations; (e) coherent content; (f) thoughtful discourse; (g) practice and application; (h) scaffolding students’ task engagement; (i) strategy teaching;
(j) cooperative learning; (k) goal oriented assessment; and (l) achievement expectations.
These 12 principles are not mutually exclusive. They must interact with one another to achieve effective teaching and help students achieve their goals. The methodology in this study included performing classroom observations and analyzing students’ outcomes on standardized tests (Brophy, 2000, pp. 9-33).
58 Table 4
NBPTS Mathematics Standards for Teachers of Students Ages 11-18+
Standard Result of teaching the standard Standard I: Accomplished mathematics teachers acknowledge and value the individuality and Commitment to worth of each student, believe that every student can learn and use mathematics, Mathematics and are dedicated to their success. Accomplished mathematics teachers are Learning of All committed to the fair and equitable treatment of all students, especially in their Students learning of mathematics. Knowledge of Mathematics, Students, and Teaching Standard II: Accomplished mathematics teachers have a deep and broad knowledge of the Knowledge of concepts, principles, techniques, and reasoning methods of mathematics, and Mathematics they use this knowledge to inform curricular goals and shape their instruction and assessment. They understand significant connections among mathematical ideas and the applications of these ideas to problem solving in mathematics, in other disciplines, and in the world outside of school. Standard III: Accomplished teachers use their knowledge of human development and individual Knowledge of students to guide their planning and instructional decisions. They understand the Students impact of prior mathematical knowledge, home life, cultural background, individual learning differences, student attitudes and aspirations, and community expectations and values on students and their mathematics learning. Standard IV: Accomplished mathematics teachers use their knowledge of pedagogy along with Knowledge of their knowledge of mathematics and student learning to inform curricular the Practice of decisions; select, design, and develop instructional strategies and assessment Teaching plans; and choose materials and resources for mathematics instruction. Accomplished mathematics teachers stimulate and facilitate student learning by using a wide range of practices. The Teaching of Mathematics Standard V: Accomplished mathematics teachers create environments in which students are Learning active learners, show willingness to take intellectual risks, develop self-confidence, Environment and value mathematics. This environment fosters student learning of mathematics. Standard VI: Ways Accomplished mathematics teachers develop their own and their students’ abilities of to reason and think mathematically— to investigate and explore patterns, to Thinking discover structures and establish mathematical relationships, to formulate and Mathematically solve problems, to justify and communicate conclusions, and to question and extend those conclusions. Standard VII: Accomplished mathematics teachers integrate a range of assessment methods Assessment into their instruction to promote the learning of all students by designing, selecting, and ethically employing assessments that align with educational goals. They provide opportunities for students to reflect on their strengths and weaknesses in order to revise, support, and extend their individual performance. Professional Development and Outreach Standard VIII: To improve practice, accomplished mathematics teachers regularly reflect on what Reflection they teach, how they teach, and how their teaching impacts student learning. They and Growth keep abreast of changes and learn new mathematics and mathematical pedagogy, continually improving their knowledge and practice. Standard IX: Accomplished mathematics teachers collaborate with families and communities to Families and support student engagement in learning mathematics. They help various Communities communities, within and outside the school building; understand the role of mathematics and mathematics instruction in today’s world. Standard X: Accomplished mathematics teachers continually collaborate with other teachers Professional and education professionals to strengthen the school’s mathematics program, Communities promote program quality and continuity across grade levels and courses, and improve knowledge and practice in the field of mathematics education. Note. Adapted from National Board for Professional Teaching Standards, 2010a, pp. 19-21.
59 Marzano, Gaddy, and Dean (2000) identified nine instructional practices that
work well with all types of all knowledge and subjects in improving student achievement.
In their report, they stated that these nine practices are the most likely to produce beneficial student learning. The identified practices are: (a) identifying similarities and differences; (b) summarizing and note taking; (c) reinforcing and providing recognition;
(d) homework and practice; (e) nonlinguistic representations; (f) cooperative learning;
(g) setting goals and providing feedback; (h) generating and testing hypotheses; and (i) activating prior knowledge. This study was a meta-analysis in which the researchers considered results with thousands of educators over a period of 30 years. They considered multiple strategies that could influence improvement in students, but they presented a table which depicts the nine strategies that produced the highest effect sizes through their statistical analysis. That is how the instructional practices were considered to be the most effective. The researchers concluded that “no strategy works equally well in all situations” (p. 5).
Walberg and Paik (2000), similar to Marzano et al. (2000), also identified nine best instructional practices that show consistent academic improvement and can be applied in all K–12 levels. These strategies are research based on practices over half a century, and they have shown large positive effects in student learning in varying conditions. The research methods consisted of analyzing 170 books, compiling 91 research syntheses, and surveying 61 educational researchers. Parent involvement has a great impact on student achievement, and interaction of parents with the student affects the child’s behavior inside and outside the school environment. Graded homework allows opportunity for feedback, discussion, and reteaching concepts that have not been grasped;
60 these are key factors to maximize the impact on homework. The quality of homework and relevance is as important as the amount. Aligned time on task refers to the alignment of learning activities and educational goals. Students who are actively focused on the learning process make greater progress, and it is the teacher’s task to deliver effective and meaningful activities. Direct instruction emphasizes systematic sequences of lessons, presentation of new content, student-guided practice, and feedback. It is highly important that teachers effectively deliver clear instruction and organize their presentations.
Teaching learning strategies is an approach that focuses on transferring part of the direct teaching functions of planning, allocating time, and reviewing to learners. It is important to allow students to monitor their own progress. Advance organizers help students to understand new concepts by making connections with prior learning. It is an opportunity for students to connect the old with the new and in that way have a better understanding of the new lesson. Tutoring is a great strategy, especially in mathematics. Peer tutoring can be remarkably effective and promotes learning both in the tutored and those who tutor. Mastery learning deals with the sequencing and monitoring process of learning. It allows time for remediation for those who need it, and frequent assessment allows the teacher to correct learning gaps when needed. Cooperative learning refers to when students work in pairs or groups and teach themselves. Cooperative learning increases student learning; students can learn teamwork, take leadership tasks, and give and receive criticism. Adaptive learning is necessary when there is the need to focus on students that present barriers to learning such as students with special needs. This requires the teacher to expand horizons to a broad spectrum of teaching approaches. All the previous teaching practices can be combined in order to tailor instruction to individual or group needs.
61 The NCTM is an advocate of the use of technology and they recognize it as a tool to better assist problem solving and math application skills. According to The Education
Alliance (2006):
NCTM recommends that students and teachers have access to a variety of
instructional technology tools, teachers be provided with appropriate professional
development, the use of instructional technology be integrated across all curricula
and courses, and that teachers make informed decisions about the use of
technology in mathematics instruction. (p. 9)
In addition, as described and cited by The Education Alliance (2006), Sabean and
Bavaria (2005) defined a list of the most important principles to teach mathematics.
Among those principles are: (a) teachers know what the students need to learn, (b) teachers utilize heuristic methods focusing on developing conceptual understanding, experiences, and prior knowledge of mathematics; (c) students provide written justification for problem solving strategies, (d) problem based activities focus on concepts and skills. and (e) a mathematics curriculum that emphasizes conceptual understanding. The Education Alliance (2006) further cited Sabean and Bavaria (2005) as supporting the use of concrete materials and the role of discovery to improve mathematics achievement.
Busatto (2004) presented findings from a large project which was designed to investigate the educational practices that make a difference in numeracy outcomes and to what extent those practices can be applied to other school contexts. In this research study,
45 public and non-public schools were studied as case studies including students from different socio-economic levels, languages, and backgrounds. The methodology used to
62 conduct the case studies was by performing interviews with teachers, parents, students, and community members.
Classroom observations were also performed and the researchers analyzed many school documents. The researchers found that the most important factors making a difference across three different contexts are: within the classroom, throughout the school, and beyond the school. These factors and strategies provide guidelines for successful numeracy in teaching. Within the classroom factors are: language as focus of learning, assessment to identify and accommodate, and purposeful pedagogy. This context deals with effective strategies such as peer work in pairs or groups, scaffolding, oral and written language to explain strategies and justify solutions, and the incorporation of resources that allow students to manipulate concrete materials to build knowledge. The researcher also argues that strong classroom management with a strong teacher presence enhances these strategies. Throughout the school factors making a difference are: a school of commitment to numeracy, school policies that support numeracy, and specialized programs to support numeracy. This context relates to leadership to unify and direct, collaboration between staff to facilitate communication, identification of areas that need remediation in mathematical teaching and learning, and special school wide programs that support learning in mathematics. Beyond the school factors strategies includes a shared vision, communicating about learning, and mathematics at home. This context relates to parent and community involvement, programs to support parents’ understanding of mathematics policies and practices, providing parents with math understanding so that they can help their children at home, and homework that invites the participation of family members.
63 Busatto (2004) found that “the schools which demonstrated greater than expected growth in numeracy achievement were those that focused on either the language of mathematics, or the engagement of practical resources to support concept development in numeracy” (p. 26) and suggested that discussion between teachers and students concerning the effective use of resources is paramount for improving numeracy learning.
Furner, Yahya, and Duffy (2005) supported the idea that to succeed in today’s world, the knowledge of mathematics has become critical. Even though students are not conscious that they use math knowledge in everyday situations, math is present in their lives whether they count, put gas in the cars to calculate how far they will go, receive change when they buy something, or many other facets of everyday life. Furner et al.
(2005) stated “all students deserve equal access to learning math, and teachers must make an effort to ensure this” (p. 16). The researchers proposed 20 effective strategies to teach mathematics, and they suggested that these strategies can be effective to reach all students, including English Language Learners (ELL) and special education students.
This literature review was based on a call for papers on a section called 20 Ways so the researchers investigated 20 ways to teach math based on best practices (personal communication). The strategies are:
• Teach vocabulary
• Relate math problems and vocabulary to prior knowledge and background
• Apply problems to daily life situations
• Encourage drawings to translate and visualize word problems
• Have ELL/special education students pair with typical students for
computer/cooperative activities
64 • Encourage children to think aloud when solving word problems, and have students
give oral explanations of their thinking, leading to solutions
• Have students write original word problems to exchange with classmates
• Explain directions clearly and repeat key terms
• Encourage students to follow the four-step-problem-solving process
• Realize that not all mathematic notations are necessarily universal
• Group students heterogeneously during cooperative learning
• Make interdisciplinary connections to what students are learning in math
• Make cultural connections for students when teaching math
• Rewrite word problems in simple terms
• Concretize math concepts with Total Physical Response (TPR)
• Create word bank charts and hang them in the classroom for viewing
• Take Internet field trips and use mathematics software
• Use children’s literature to teach mathematics and develop the language
• Using auditory, visual, and kinesthetic teaching approaches for different learning
styles enables teachers to reach more students than the traditional direct instruction
• or paper and pencil drill and practice forms of instruction
These strategies, all research based, can be applicable to Grades K–12. The researchers suggest that the strategies could help teachers to more effectively deliver instruction and adapt the strategies to each individual’s classroom needs.
Willig, Bresser, Melanese, Sphar, and Felux (2009), five experts in the content area, summarized 10 best math teaching practices to help ELL students to succeed in math. These experts argued that ELLs have a double learning challenge as they must
65 learn the English language and mathematics at the same time. For that reason, it is the teacher’s responsibility to make lessons clearly understood while ensuring that the students receive the proper language instruction so that they can grasp math content as well. The best 10 teaching practices are (pp. 27-29):
1) Create vocabulary banks: make charts and phrases containing words,
expressions, & illustrations
2) Use manipulatives: make math comprehensible; students construct physical
models of abstract ideas, and they gain confidence at the same time
3) Modify teacher talk and practice wait time: give students time to properly
process information and formulate responses. Reduce the amount of talk
4) Elicit non-verbal responses: like thumbs up or down to agree or disagree to
an idea without the need of talking
5) Use sentence frames: to contextualize and bring meaning to vocabulary
6) Design questions and prompts for different proficiency levels: beginner,
intermediate, and advanced.
7) Use prompts to support student responses: starting with sentences like “it’s
a polygon because….” to encourage them to give an answer
8) Consider language and math skills when grouping students: students benefit
from working in groups and they have the opportunity to engage in math
conversations.
9) Utilize partner talk: encourage participation and remove the pressure to
speak alone.
66 10) Ask for choral responses from students: expose them to new vocabulary and
help with pronunciation, syntax, and grammar.
The Education Alliance (2006) identified essential characteristics of an effective
standards classroom to close the achievement gap. Those characteristics that make an
effective classroom are: (a) lessons address specific standards-based concept skills; (b)
there are student-centered learning activities; (c) lessons are focused on inquiry and
problem solving; (d) critical thinking and knowledge application skills are emphasized;
(e) students use adequate time, space, and materials to complete tasks; and (f) various and
continuous assessments are designed to evaluate both student progress and teacher
effectiveness.
Steedly, Dragoo, Arefeh, and Luke (2008) analyzed multiple studies focusing on
effective math instruction for students with disabilities. The methodology utilized in this
study was to review a total of 183 studies combined in five different meta-analyses. The researchers found four strong methods of instruction that show the most promise and that can also be applied when teaching both students with disabilities and mainstream students as well (p. 4). The four approaches are:
• Systematic and explicit instruction, a detailed instructional approach in which
teachers guide students through a defined instructional sequence. Within
systematic and explicit instruction students learn to regularly apply strategies that
effective learners use as a fundamental part of mastering concepts.
• Self-instruction, through which students learn to manage their own learning with
specific prompting or solution-oriented questions.
67 • Peer tutoring, an approach that involves pairing students together to learn or
practice an academic task.
• Visual representation, which uses manipulatives, pictures, number lines, and
graphs of functions and relationships to teach mathematical concepts.
The researchers highlighted the importance of teachers determining which instructional interventions best support the educational goals based on students’ ages, needs, and abilities.
Gasser (2011) wrote, “this article draws on the 21st Century Skills Movement and
the successful teaching practices of Asian schools in order to provide five suggestions
that secondary math teachers can incorporate into their classrooms” (p. 108). The
researcher stated that American students are always compared to their international peers
and tests such as the TIMSS and PISA and that such tests demonstrate how our students
fall behind in comparison to those in Asian countries. The researcher’s method of
research consisted of examining teaching practices in countries like Japan, China,
Taiwan, Korea, Singapore, and India and determining five instructional practices for
secondary educators to incorporate in our country with our students. These strategies can
provide students with the necessary skills to be successful in future jobs. Problem-based instruction is a strategy that deals with working with word-problems and problems that deal with real life situations. The students can see how math is utilized outside the classroom. Problem solving also encourages active thinking. Student-led solutions allow students to think through problems and create their own solutions. Gasser (2011) argued that teachers should not give definitions at the beginning of each lesson but rather start with problems that lead to discover rules and better understand definitions. Risk taking
68 encourages students to try to perform tasks and not be ashamed in case of failure. The students need to see failure as a growing experience. The teacher plays a critical role in setting the tone to create such classroom atmosphere and learning environment. Having fun in confronting students with real world problems, for example with games, computer software, or surfing the web could be a great strategy to enhance math learning, especially in settings where students need to be prepared to face standardized tests.
Collaboration time is a strategy that suggests that the teacher should not be in total control of the learning because this reduces the child’s ability to learn. The researcher stated that “students should develop their abilities to listen to others and to defend their stances” (p. 114). By using all the above five strategies students will be prepared for the
21st century; “…instruction can still be fast paced, and standards do not have to be neglected in order to place a greater focus on our students' critical thinking, problem solving, communication, and collaboration skills” (p. 115).
The Relationship Between Best Practices and AP Math Teaching
Good and effective teaching practices are needed to lead students to success. It is imperative that best teaching practices are also present at the AP level. According to
Klopfenstein (2003):
Many AP teachers do not have a college major or minor in the AP subject in
which they teach. Given the rigor of the material taught in AP classes, however, it
is critical that AP teachers have a deep conceptual understanding of their subject.
(p. 4).
The recommendation of the College Board (2010) is that teachers who teach an
AP course should have considerable experience or have an advanced degree in the
69 discipline that they teach. Teachers of AP Calculus courses need to be well prepared to teach the concepts and skills of these specific courses.
In a study of teaching practices in AP classrooms, Henderson (1996) found that the most effective teachers delivered more questions, had higher engagement rates, provided more feedback to students, and had greater participation rates. Simonsen (1995) conducted a multiple case-study in order to investigate high school AP calculus teachers’ subject matter and pedagogical perceptions. The researcher collected data by utilizing questionnaires, interviews, observational field notes, videotapes of classroom instruction, journals, and written instructional documents. The sample consisted of six AP calculus teachers. In regards to instructional practices, the researcher argued that there are differences between teachers who were part of the calculus reform and those who were not. The differences were found in factors such as: (a) commitment to the textbook;
(b) planning; (c) use of multiple representations; (d) attitude toward graphing technology;
(e) classroom atmosphere; (f) examinations; and (g) appropriate level of rigor to teach AP calculus.
Flores and Gomez (2011) conducted a case study during the years 2008-2010 in an urban high school in California that serves a population of approximately 4,100 students, mostly classified as disadvantaged and Latinos. They noted that nearly all the teachers want to teach the top students (p. 71). However, the administration in that school opted to select those most highly credentialed teachers in the school to teach the AP classes. The researchers found that the most effective practices to teach AP classes are:
(a) scaffolding, which means to provide support structures to prepare students for the next level; (b) teachers’ collaboration to discuss instruction practices; (c) evaluating the AP
70 program to look at data to see if students are performing or dropping and to compare such data with previous years; (d) organizing parent meetings to have parents involved in their children’s education; (e) having AP teachers visit other schools to refine and develop best practices; and (f) providing tutoring.
Timmons (2009) reviewed literature from scholarly journals, the Delaware
Department of Education website; performed interviews with Algebra I students, math teachers, and guidance counselors; and reviewed students’ SAT scores. The best practices that the researcher found to promote student retention in challenging AP courses are: educating parents, creating a college culture, tutoring, summer school, broader honors enrollment, identifying gifted underachievers, and offering online AP courses.
Imig (2008) examined teaching strategies that are correlated with teaching effectiveness on AP classes. The researcher examined teaching practices in AP calculus, biology, English, and United States government. This mixed method study incorporated observations, surveys, analyses of gradebooks, and standardized students’ AP scores.
Statistical methods included linear regression for correlation analyses and analysis of descriptive data. The findings were that high pass rate teachers used more class time than low pass rate teachers. High pass rate teachers also provided more assignments compared to low pass rate teachers. Finally it was also found that there is a slight negative correlation between the teacher pass rate and Bloom’s taxonomy types of questions utilized in the classroom.
The College Board (2010) has developed curricular guidelines for each AP course. Table 5 displays best teaching practices recommended for AP Calculus and AP
Statistics. Many of the best teaching practices presented in Table 5 are strategies that
71 research has suggested are best teaching math practices at the secondary level although
some can be additionally applied at lower levels. Some of the strategies presented in
Table 5 are more sophisticated because at the AP level some strategies require the use of
more advanced technology and software. However, other strategies overlap and are often
applicable not only at the secondary level, but some at the elementary and middle school
levels. Strategies such as the use of manipulatives, problem solving, problems related to
real-world situations, graphic organizers, collaboration, and use of technology seem to be
common denominators of teaching practices applicable at different levels of mathematics
instruction.
Historical Background of AP Math Classes
Since its inception in 1957, the AP Program was designed to allow high school
students to earn college credit (Klopfenstein & Thomas, 2009, p. 873). The program primarily served elite students in private high schools. Even though the program has not changed much in the past 50 years, its scope has broadened dramatically (Klopfenstein &
Thomas, 2009, p. 873). Nowadays, the AP Program reaches high schools all over the
country, and access for any student to take an AP course regardless of race, color, or
socio-economic status is significantly easier.
An AP math course is a course that presents a more advanced curriculum than
regular or honors high school math courses. The two AP math courses that the College
Board (2010) offers are AP Calculus and AP Statistics. AP classes were originally
designed to provide students the opportunity to face college level classes while they are
in high school. Klopfenstein and Thomas (2009) argued that taking AP classes is a
predictor of early college grades and retention (p. 873). Students who take advanced level
72 math classes in high school are more likely to get a college degree. Students are also given the opportunity to receive college credit if they can pass an AP exam. Particularly in mathematics, students who pass the AP Calculus or AP Statistics exams may not need to take those classes when they become college students.
The AP program has grown dramatically in the past 50 years (Klopfenstein &
Thomas 2009, p. 873), and the number of students taking AP exams has risen substantially since then. Banchero and Napolitano (2006) stated that “Ex-president Bush planned to triple the number of advanced high school math and science teachers nationwide because of the AP Program” (p. 4). The number of schools in the United
States offering AP classes is as follows: (a) 13,353 Calculus AB; (b) 6,023 Calculus BC; and (c) 6,953 Statistics (College Board, 2012).
AP Calculus. Looking back through the years, a fact that brought concern in the educational system was the way in which calculus was taught. Too many students in the
1980s were failing as undergraduate students. For this reason a calculus reform was born and “the reform movement got under way after a meeting in Washington in 1987 at which mathematicians announced a crisis in calculus education: As many as 40 percent of undergraduates were failing” (Wilson 1997, p. 12). As Kasten et al. (1998) were cited in Biggers (2006):
Data available from the late 1980s showed an annual enrollment of about 300,000
students in mainstream calculus at the four-year college level. Less than 50% of
the students enrolled at the four-year college level were successful in completing
the course. (p. 2)
73 Table 5
Curricular Guidelines for AP Calculus and AP Statistics
AP Calculus Best Practices AP Statistics Best Practices • Discovery lessons • The use of real data from primary and secondary • Online applets and demonstrations sources • Projects and Learning Labs • Application of formulas • Flexible Skills Grouping • Authentic problems to support • Appropriate Pacing statistical literacy • Depth and Complexity • Use of graphing calculators and statistical applications to • Application of formulas to meaningful, manipulate data and produce real-world situations reports • Use of spatial sense in two or three • Online applets and dimensions through explorations with demonstrations real objects • Math journaling that focuses • Math manipulatives on explanation of concepts • Authentic problems to support and problem solving mathematical literacy • Statistical news articles • Use of graphing calculators and • Videos and data-gathering applications to develop designs, collect activities data, and produce reports • Projects and other hands-on • Opportunities for problem and solution experiences formation • Exploration of data sources • Math journaling that focuses on including univariate, bivariate, explanation of concepts and problem categorical solving • Word Wall of Academic • Graphic organizers and Foldables that Vocabulary help students make sense of mathematics concepts and promote and • Flexible Skills Grouping, reflect higher order thinking skills Pacing, and Depth and Complexity • Socratic Seminars
• Word Wall of Academic Vocabulary Note. Adapted from: Austin Independent School District, Advanced Academic Services, pp. 6-7; retrieved from http://curriculum.austinisd.org/adv_ac/preAP/documents/ VerticalTeamManualMathematics12092013.pdf.
Hoyt and Sorenson (2001) stated that “those concerned with the cost for remediation have suggested that public funds should not be used to pay a second time for
74 that which students should have learned in high school” (as cited in Biggers, 2006, p. 2).
Judson and Nishimory (2005) stated:
During the past several decades, many mathematicians and mathematics educators
felt that there were problems with the ways in which calculus was taught. A
common complaint was that students merely learned “recipes” for solving
calculus problems and did not gain a sound conceptual understanding of calculus.
(p. 24)
Calculus reform had a big impact on the calculus curriculum in universities, high schools, and in their respective books incorporated into the curriculum. The federal government funded numerous projects (Judson & Nishimory, 2005). Kennedy (1997) explained that attached to this reform came the incorporation of technology, especially with the introduction of graphing calculators (as cited in Judson & Nishimory, 2005, p.
25). Wilson (1997) stated that “the reform courses encourage students to rely on computers and graphing calculators to get past paper-and-pencil computation” (p. 12). It is evident that no one wants to see students fall into the same problem in today’s calculus education.
Calculus AB is a class designed to be a calculus level college class. This class is for those students who have a genuine interest in mathematics and are committed to earn college credit while they are still in high school. Students need to obtain at least a 3 to pass this AP exam on a scale 1 to 5; however, some universities require a 4 or a 5 to grant college credit, and such requirement varies from university to university. This exam is given by the College Board in May and consists of two sections: (1) a multiple choice question containing a portion of 28 non-calculator question to be completed in 55
75 minutes in addition to a 17 calculator question portion to be completed in 50 minutes; and
(2) a free response question that requires the student to demonstrate the ability to solve
problems showing step-by-step procedures on how to get to the final solution (a two 30
minute calculator portion and four 60 minute no calculator portion). The pre-requites for
Calculus AB depend heavily on the skills and knowledge previously learned in Algebra I,
Geometry, Algebra II, Trigonometry, and Pre-Calculus. Most of the students who take
Calculus AB are students who mostly took honors mathematics classes throughout high school. The College Board provided clarification about the AP Calculus AB Exam
(College Board, 2015a), “the difference between Calculus AB and Calculus BC is one of scope. AP Calculus BC includes all topics in AP Calculus AB, as well as additional topics such as differential and integral calculus (including parametric, polar, and vector equations).” Figure 6 depicts the topic outline for Calculus AB and Calculus BC.
Topic AP Calculus AB AP Calculus BC
Functions, Graphs, and Limits X X
Continuity and Limits X X
The Derivative X X
Applications of the Derivative X X
The Integral X X
Applications of the Integral X X
Differential Equations X X
Polar and Parametric Functions X
Sequences and Series X
Figure 6. Topic outline for AP Calculus AB and AP Calculus BC. Adapted from College Board,
2015a.
76 It can be observed that there are two additional chapters in Calculus BC in comparison with Calculus AB, which requires students to go at a faster pace. In many schools, students take both Calculus AB and the following year they continue to take
Calculus BC. Only a minority of students take Calculus BC without previously taking
Calculus AB. Because the data from Calculus BC can sometimes be polluted data, for this study the researcher has opted to concentrate on student test performance for
Calculus AB only.
AP Statistics. The AP Statistics test was first offered in 1997 (Roberts, Scheaffer,
& Watkins, 1999). The enrollment of students in AP Calculus was a clear manifestation of the advances made in teaching calculus. As reported by Roberts, Scheaffer, and
Watkins (1999), in 1979, the College Board appointed a Task Force for Advanced
Placement Mathematics in order to offer possible additional courses in mathematics
(p. 308). Statistics was becoming important in different areas such as social sciences and the future curriculum; for those reasons the College Board decided to incorporate this AP class as an offering. According to the College Board (2010):
Every semester about 236,000 college and university students enroll in an
introductory statistics course offered by a mathematics or statistics department. In
addition, a large number of students enroll in an introductory statistics course
offered by other departments. Science, engineering and mathematics majors
usually take an upper-level calculus-based course in statistics, for which the AP
Statistics course is effective preparation. (p. 4).
77 It is evident that the AP Statistics course became a popular course, but teaching statistics is not a job that any teacher can perform effectively. According to Garfield
(2013):
There are many challenges that are present in the process of teaching statistics
effectively: (1) many statistical ideas and rules are complex, difficult, and/or
counterintuitive. It is often difficult to motivate students to engage in the hard
work of learning statistics; (2) many students have difficulty with the underlying
mathematics (such as fractions, decimals, and algebraic formulas), and that
difficulty interferes with learning the related statistical content; (3) the context in
many statistical problems may mislead the students, causing them to rely on their
experiences and often faulty intuitions to produce an answer, rather than select an
appropriate statistical procedure; (4) students equate statistics with mathematics
and expect the focus to be on numbers, computations, and one right answer. They
are uncomfortable with the messiness of data, the different possible
interpretations based on different assumptions, and the extensive use of writing
and communication skills; and (5) The language used in statistics includes many
familiar words but with different or more precise meanings than students use. For
example: normal, random, sample, average, variable, and distribution are
introduced to students as new vocabulary words with statistical definitions, yet
students often resort to their own familiar interpretations of these terms. (para. 2)
Garfield (2013) also argued that a teacher needs to have pedagogical content knowledge besides just knowledge of statistics. Shulman (1986) agreed with Garfield on this point that pedagogical and content knowledge need to merge. Rumsey (2002) stated that “we
78 must be sure to promote the use of the scientific method in all of our students: the ability
to identify questions, collect evidence (data), discover and apply tools to interpret the
data, and communicate and exchange” (p. 1). This sounds very complex to achieve if the
teacher does not incorporate the correct pedagogy necessary to accomplish such a broad
goal. It is unclear how well high school mathematics teachers are able to teach
AP Statistics, but without content knowledge in this particular subject, even certified
teachers are not able to teach AP Statistics effectively. For that reason, NBCTs are more
suitable to teach classes of this caliber.
Relationship Between the NBC and Best Teaching Practices
What teachers know and are able to do in the classroom is very important. As
education in the United States shifted from educating the elite during colonial times to education for all, the quality of the education provided has come into question (Imig,
2008). If mathematics teachers do not possess the appropriate content knowledge for the subject area they teach, it then becomes difficult to incorporate the use of best teaching practices. In order to become a NBCT, a teacher needs to complete a high level assessment in which he or she has to demonstrate content knowledge in the following areas: algebra and functions, calculus, discrete mathematics, geometry, statistics and data analysis, and families of functions (National Board for Professional Teaching Standards,
2013a, p. 1). These test topics represent advanced knowledge of mathematics for which
teachers need to be well prepared. In order to prepare for this assessment, teachers need
to study, review concepts, and in some cases look for additional help from peers who
teach advanced mathematics courses. This shows that there is a clear connection between
the NBC and the content knowledge a teacher needs to possess. Without content
79 knowledge, a teacher cannot utilize effective teaching practices and deliver high quality instruction.
The National Board for Professional Teaching Standards (2001) conducted two comprehensive research surveys. The studies had the purpose of evaluating the impact of the NBC on teachers. A sample of 600 NBCTs constituted respondents to the surveys.
Among some of the findings in these two surveys, it was found that 91% of the respondents believed that the NBC positively influences their teaching practices (see
Figure 7).
Reinforces current teaching practices ...... 37% Plan lessons more thoughtfully ...... 33% Use more student-focused instruction...... 31% Apply performance standards to teaching ...... 25% Try new teaching tools and/or techniques ...... 20% Dropped ineffective teaching practices ...... 13% Use rubrics in student assessments ...... 11% Other (e.g., engage in continual self-assessment) ...... 10% No effect on teaching practices = 1% No response = 8%
*Percentages reflect multiple answers given by respondents, so do not total 100%.
Figure 7. How the NBC affected teaching practices. Adapted from: National Board for
Professional Teaching Standards, 2001.
Of the respondents, 83% confirmed they have become more reflective about their
teaching; respondents also reported that they have transferred their experience into
teaching practices. Another interesting factor to observe in this study is that out of the
600 participants, more than 74% of those NBCTs have either master’s degree credits or a
master’s or doctoral degree. Figure 8 illustrates the educational level of the NBCTs in
this study.
80 No Response Educational Profile of NBCT Survey 4% Respondents. Highest Educational Doctoral Degree 5% Level. Bachelor's 5%
Master's Degree Bachelor's + 37% Credits 17%
Master's + Credits 32%
Figure 8. Educational profile of NBCT survey respondents, highest educational level. Adapted from National Board for Professional Teaching Standards, 2001.
Figure 8’s graphical representation depicts that the NBCTs for this study have
impressive educational credentials. As can be observed, 37% of the sample of NBCTs
hold a master’s degree; 32% hold a master’s and graduate credits above their master’s;
and 5% have a doctoral degree. This adds up to 74%, which clearly depicts that most of
the NBCTs hold advanced degrees. This information provides relevant support for
Research Question 1: Do NBC mathematics teachers produce better results than non-
NBC mathematics teachers on the AP Calculus AB Exam?
81 Discussion
“The number one factor in enhancing student learning is the capability of the
teacher” (Wise, 1998, p. 1). At this high level of mathematics, students need to be
exposed to quality teaching. If teachers are not prepared to teach advanced level courses
such as AP mathematics classes, then the ones who will be negatively impacted are the
students. It is important to understand that at the high school level, there are many teachers who teach mathematics but they are not certified to teach students above ninth grade. From a project focused on promoting rigorous outcomes in math and science education (Project PROM/SE), which included sampling 4,000 teachers, Kuchment
(2012) stated, “there is an increasingly strong push for the inclusion of probability and statistics in high school, as is found in the Common Core State Standards, yet less than half of the surveyed mathematics teachers felt well prepared to teach it” (para. 20).
Additionally, there is a fact that cannot be ignored; even though some mathematics teachers majored in mathematics, still some of them do not possess strong academic preparation.
There are two types of certification for high school teachers. Teachers can be certified to teach math in Grade 9 up through Grade 12. Many math teachers in today’s high schools are not certified to teach math in all the high school grade levels. Clearly those teachers do not meet the necessary requirements to teach AP mathematics because, in the first place, most students taking AP math courses are juniors or seniors; their teachers need to be certified through the 12th Grade. Whitman (2002) observed that
NBCTs are more likely to teach in higher grades (p. 116).
82 In most colleges, students are required to take a mathematics course. Depending on the college major that a student plans to pursue, some colleges require either a calculus course or a statistics course. Taking an AP math class at the high school level may represent a gateway for acceptance. At the same time, these AP math courses require students to be challenged so that they can reach their fullest potential. A great deal of critical and higher order thinking needs to be part of the daily instruction. By considering how an AP math course needs to be taught, it is reasonable to believe that teachers who are knowledgeable in the subject matter, possess pedagogical content knowledge, and are effective in using technology become essential in teaching these classes.
Calculus reform implemented the use of graphing calculators, and today that is a mandate for both AP Calculus and AP Statistics exams. The AP Statistics course adheres to the philosophy and methods of modern data analysis (College Board, 2010). The use of statistical software packages is necessary to complement instruction; therefore, the teacher needs to explain how to update calculator software and keep up with emerging technology. The use of technology is quite comprehensive when needed to effectively teach AP math courses. The following are technology supplies that the College Board
(2010) suggests teachers utilize on daily instruction (see Figure 9).
Figure 9 depicts a very comprehensive scope of technology tools that must be understood and well presented to students to guide them to success. It can be observed that to be able to manage and understand these tools, teachers need to know more than basic technology. The mandatory certification does not require teachers to be knowledgeable in technology. On the other hand, the NBPTS requires that most
83 candidates include information in their portfolios on how they use technology for
instruction (National Board for Professional Teaching Standards, 2010a, 2013b).
Figure 9. Technology supplies recommended by the College Board (2010).
NBC renewal requires The Profile of Professional Growth (PPG), drawn from the
five core propositions of the NBPTS (National Board for Professional Teaching
Standards, 2014c). According to A Guide to Understanding National Board Certification, a joint project of the American Federation of Teachers (AFT) and the National Education
Association (NEA) (American Federation of Teachers, 2008-2009):
Although the PPG is separated into three components, the parts are integrated and
complimentarily designed so that NBCTs can demonstrate the connections they 84 make between their continued professional development and student learning. The
professional growth experiences must include effective and appropriate use of
technology and current knowledge of the field that ultimately impacts student
learning. (p. 61)
Since the NBC concentrates on the importance of the use of technology related to student achievement and growth, it may be reasonable to assume that NBCTs could be better prepared instructors to teach an AP math course compared to those who do not hold the certification.
Principals, school administrators, and department heads in high schools need to have good judgment in selecting the right teachers to teach AP math courses. The impact that AP courses have on students’ college careers and lives is of great magnitude. Taking
AP math is a pathway to success in math college courses. Having highly qualified and well prepared teachers to teach AP math courses is critical so that students can be prepared for success in future mathematics classes in high school and in college.
It is necessary to address the importance of having these highly qualified teachers reaching all kinds of student populations to enable every student to be successful and make education more equitable. Indeed, equity becomes a very important issue when looking at the impact of these highly qualified teachers on AP courses. There is evidence that keeping effective (or accomplished) teachers or NBCTs in low performing schools becomes a problem. Berry (2005) explained that there are many reasons why NBCTs do not stay or do not consider working in these types of schools. Figure 10 itemizes reasons
Berry identified that keep accomplished or NBC teachers from working or staying at low performing schools.
85 • Accomplished teachers do not work for weak principals. • Accomplished teachers do not want to work in a school where they cannot use their teaching expertise (and are forced to use highly scripted materials). • Accomplished teachers need adequate resources (e.g., classroom libraries, science equipment, current technology) to do their best teaching. • Accomplished teachers would more readily move to low-performing schools if they “could do so with kindred spirits" -- that is, similarly skilled and valued colleagues who have the time to learn from and support one another. • Accomplished teachers would expect -- and labor market forces would require -- salary incentives to teach in hard-to-staff schools. • Board-certified teachers need administrators who know and embrace the NBPTS process and cultivate teacher leadership. • Board-certified teachers will need to have additional preparation in the area of leadership if they are expected to promote school change. • Board-certified teachers and other teachers already teaching in the school will need professional development in collaboration, team building, and cultural competence.
Figure 10. Reasons why accomplished or NBCTs do not work in low-performing schools. Adapted
from Berry (2005).
Additionally, Berry, Raspberry, and Williams (2007) observed,
Poor children and those of color are far less likely to be taught by qualified
teachers—no matter how the term “qualified teacher” is defined. Studies
consistently show that teachers who are better trained, more experienced, and
licensed in the subjects they teach are more likely to be teaching in more affluent
schools, serving more academically advantaged students. (p. 1)
Even though classes like AP Calculus AB are strongly heterogeneous in the ability levels of the students who are capable of taking such a class, it is difficult to ignore the facts of funding, resource distribution, and quality of teachers among other influences on the quality and expertise of those who are teaching the students.
86 Conclusion
The NBC is one of the existing tools to measure teaching accomplishment, to help
teachers improve themselves as professionals, and is seen as a way to prove that a teacher
understands how students can effectively learn and that the teacher knows the subject he
or she teaches well. Some criticize the NBC by stating that the certification does not make any difference in producing more effective teachers and eventually better student achievement. Even though there have been mixed results about the effectiveness of the certification, most of the existing research has shown that the majority of the studies favor the positive impact of being an NBCT. Among those studies are Vandevoort et al.
(2004) and Whitman (2002). As reported by National Board for Professional Teaching
Standards (2014b), multiple studies revealed NBCTs outperforming non-NBCTs on student achievement tests (e.g., Cavalluzzo, 2004; Clotfelter et al., 2007; and Goldhaber,
Perry, & Anthony, 2004). All of these studies revealed statistically significant differences in favor of NBCTs. As Sawchuk (2015) stated, “The evidence continues to mount that teachers who earn national-board certification are more effective than other teachers, both at the high school and elementary levels.” It is important to take into consideration that if more researchers keep on producing studies on the NBC, mixed results may be found.
Politicians, policy makers, and higher education authorities need to reconsider how to encourage and attract NBCTs to work in schools that could benefit from good teachers to make a difference and improve them. Those who believe that paying bonuses to NBCTs is a mistake obviously do not understand that this could be a very effective way of attracting and retaining highly-qualified teachers to low performing and low socio-economic schools.
87 It takes more than observational studies to probe whether certification makes a difference in student achievement. The NBC needs to have support from the politicians, districts, superintendents of schools, and people who are in charge of deciding how to allocate the money in meaningful ways in order to retain the NBC as a strategy for educational reform. It is also clear that the American educational system needs to make changes. Producing more NBCTs could be an effective way to make positive changes and make American schools institutions that provide competitive opportunities for a brighter future.
As stated in Boyd et al. (2007), “states set their certification requirements independently, subject only to the NCLB requirements for highly qualified teachers”
(p. 48). If a “mandatory” certification can make a difference in teacher effectiveness and student achievement, then the NBC should make a more significant increase in both variables. State certification is mandatory for every single person who wants to be a teacher and stay in the educational system in this country; however, the NBC is optional.
Additionally, uncertified teachers and people coming from alternative routes are being placed in our classrooms. That is not a guarantee for good and effective quality of instruction for our children. However, the NBC guarantees a nationally recognized level of teaching effectiveness.
The NBC allows teachers to work and be certified in 49 states in this country
(Boyd et al., 2007), but no state requires the NBC. The NBC is voluntary; applicants are teachers who want to grow professionally and accomplish a certification that goes above and beyond a regular and mandatory certification. The standards of the NBPTS are much more demanding compared to the standards of state teacher certification. There is no
88 comparison in the amount and quality of work that needs to be produced to achieve the
NBC compared to the time and effort required to obtain the mandatory certification. In
addition, the time and money that a teacher needs to spend to achieve the NBC cannot be compared to routes to achieve minimum certifications. Moreover, passing the NBC exam is a much more difficult process and a greater challenge compared to passing a basic knowledge exam such as the General Knowledge test or the Subject Area test (tests’
names vary according to each state). As can be observed in this study, being an NBCT is
associated with possessing solid content knowledge.
Teachers who are qualified and know what and how to teach will help students be
successful but, according to the Center for the Future of Teaching and Learning (2000, p.
19), as cited in Roth and Swail (2000), “teachers who are under qualified or ill-equipped
will not produce successful students.” It is important to ensure that we have classrooms with teachers who can help students meet academic standards that are appropriate to their grade level so that they can be prepared to meet the next challenge.
Math is a science that needs to be well taught. The history of mathematics
education and the results on international exams have demonstrated that our system is not
performing at the level that is expected, and unfortunately American students have been
outperformed by many other countries. However, it is important to examine the level of
fairness while comparing American students with students in high performing countries.
Equity is not proportional since in some of the better performing countries like Finland or
Singapore, where the funding factor may be less complicated due to the much smaller
number of students and it is easier to provide and reach out to students. As Strauss (2012)
explained, “some of the schools in those high performing countries are funded in a way
89 there is a guarantee of equal allocation of resources to each school regardless of location or wealth of its community” (p. 2). That is something that has not yet been accomplished in the United States, but the inequality issue keeps producing disparities in the educational system, and there may be a possible correlation between equity and educational outcome. As explained by Tirri and Kuusisto (2013),
One reason for Finland’s success in international comparison studies is the
Finnish government’s “equal opportunity and high-quality education for all”
principle. The first practical implementation of this principle is that education is
free at all levels. The second is the government’s strong financial support of
public sector educational institutions. This has led to a situation where no
significant differences exist. (p. 84)
Additionally, the diversity of students and language barriers that the student population in the United States faces is indeed not comparable to other countries. The diversity component is clearly visible across the US, in contrast with countries where the student population is significantly more homogeneous, and in that way, most of the student population learns mathematics without these barriers. These international exams like PISA or TIMSS do not take into account the social, economic, or cultural factors.
The Organization for Economic Co-operation and Development (2012) reported that
“Socio-economic disadvantage has a notable impact on student performance in the
United States: 15% of the variation in student performance in the United States is explained by students’ socio-economic status” (p. 4). They also remark that “This contrasts with less than 10% in a number of countries/economies, including Finland,
Hong Kong-China, Japan and Norway” (p. 4). These numbers clearly depict that students
90 in the United States vary much more in regards to socio-economic status. Carnoy and
Rothstein (2015) indicated that conclusions drawn from international test comparisons
can be misleading. They stated,
Our examination of international test results finds that U.S. relative performance
would seem to be better if U.S. social class composition were similar to that of
comparison nations; that U.S. performance on international tests that are aligned
with the U.S. curriculum is superior to performance on unaligned tests; and that
disadvantaged students in the U.S. have been making more rapid gains than
disadvantaged students in nations with higher score levels. All this should make
policy makers cautious in jumping to conclusions about the shortcomings of U.S.
schools and how to improve them. (p. 122)
It is important that we understand that education in the United States faces issues that
other high performing countries, and international test results should be carefully
analyzed before making erroneous conclusions.
Additionally, many schools are so poorly equipped that from the earliest stages of learning, these schools do not provide the necessary foundations and skills to make our schools successful and competitive; a critical shortcoming is that these schools do not possess highly qualified teachers (Burris, 2014). Gaps of math teaching and learning are created beginning at the elementary school level. The need for highly qualified teachers, teachers who understand how to deliver instruction, and teachers who understand what our students need, becomes paramount for our educational system. Having NBCTs in elementary and middle schools could be a contributing factor to improving education.
91 Having NBCTs at the high school level contributes even more to make our students successful towards graduation and college readiness.
NBCTs are generally experienced teachers. Research shows that they possess content knowledge and are teachers who in general hold advanced degrees in the subjects they teach. The College Board recommends that teachers who teach AP math classes should hold an advanced degree. Clearly the College Board and the NBPTS are linked to one another in a way that both these institutions concentrate on building the connection between effective teachers and producing effective quality of math teaching practices at the AP level.
The NCTM and the NBPTS established grounds for math standards and math teaching practices, and much of the research on best teaching practices has been based on such standards. Research on best teaching practices suggests different ways of instruction to maximize student learning of mathematics. Regardless of the strategies suggested by the different researchers, it is important that classroom instruction and best teaching practices are aligned and properly connected to deliver effective math instruction so that our students can be properly prepared for the next mathematics level, graduation, and for their future. The NBC, as demonstrated in this and other research, positively affects that relationship between classroom instruction and best math teaching practices. Moreover, this relationship makes a transition at the AP level, where we can observe a direct relationship between the NBC and AP math instruction. NBCTs are more suitable candidates to deliver effective teaching and lead students to success because of the high demands in content knowledge, knowledge of technology, and understanding of students required to teach these types of classes.
92 Summary
Chapter 2 shares the relevant literature as it relates to the NBC, teacher
certification and licensing, best practices in teaching secondary mathematics,
mathematics standards, and the relationship between the NBC and AP mathematics
teaching. There are various studies comparing student achievement using test scores from
students of NBCTs and non-NBCTs. With some exceptions of some studies that showed
mixed results, most of the studies depicted significantly higher achievement favoring
students of NBCTs.
A problem that schools face in America is teacher attrition. Many teachers leave
the profession for multiple reasons, and this has become a crisis for our educational
system. Moreover, the number of low performing schools has increased and classrooms
are being filled up with people who come into the teaching profession without the proper
certification requirements or without adequate content or pedagogical knowledge. This is
a disservice to the students and also creates a gap of inequality, especially in low
performing schools, and the level of education provided by today’s low performing
schools is often times unacceptable. This chapter presented different studies that
compared certified and uncertified teachers. In a similar fashion as NBCTs and non-
NBCTs were compared, in regards to the traditional teacher certification mixed results
have been found; however, most of those results were in favor of certified teachers
revealing that they are more effective than their non-certified peers. It is important to note that bringing qualified teachers, especially more NBCTs, contributes to breaking away with the disparities found between low and high performing schools. Even though low performing schools are not the first choice of NBCTs, promoting such certification along
93 with providing financial support and reward may be a way to bring effective teachers into classrooms in need of good teachers.
From very early on, multiple issues have been found with the way in which mathematics are taught in American classrooms. The issue of not having students prepared to think critically or see how mathematics is connected with the real world becomes a problem as students are often times taught to think mechanically and with no reasoning whatsoever. These are some of the reasons why policymakers have been trying to institute standards in the American education system multiple times to ensure students success. Especially in mathematics, there has always been a disconnection between what the students should know and what they really know coming out of high schools and going into college; and this is a problem that begins right from elementary school.
Through the years, the standards of the NCTM have heavily supported mathematics learning and described the knowledge and skills that students should learn from prekindergarten throughout grade 12.
This chapter depicted literature related to best mathematics teaching practices.
Incorporating some of these practices has a positive impact on student learning. At the same time, student learning becomes a paramount component toward accomplishing the goal of passing the AP exams. When students pass an AP exam, this becomes an indication that the student went through rigor and hard work; and that they have demonstrated they possess the ability to face college work prior to leaving high school.
This fact also brings students the financial benefit of obtaining college credit for free and it is needless to say how important this is for low-income students. Increasing the number
94 of students taking AP exams is a way of improving students’ chances to be admitted to colleges of their choice.
The need to have good educators who can effectively prepare students from early stages up to the AP calculus level is highly important, but it is paramount to place these teachers who can teach math effectively in classrooms with all kinds of students regardless of the students’ mathematical abilities. Higher level educational authorities should realize how important keeping these teachers in the profession has become, and for that reason, they should indeed re-evaluate the provision of incentives that recognize teachers’ accomplishments and extra efforts. It is necessary to expose AP students to teachers who know their subject well and possess adequate pedagogical knowledge in order to teach them to think critically, analytically, and to use logical reasoning from what they learn. Apart from comparing student test score performance of NBCTs and non-NBCTs, it was the researcher’s goal in this study to investigate if NBCTs possess the right tools and qualities to equip students to be successful on the AP Calculus AB Exam, if they incorporate best mathematics teaching practices emphasized in the literature, and finally if they align their practices with the standards of the NCTM and state standards.
95 Chapter 3. Research Methodology
Introduction
This chapter describes the methods of research employed in this study. In order
to research the impact of the NBC, this mixed method study explored the differences in
test scores produced by students of mathematics NBCTs and non-NBCTs on the
AP Calculus AB Exam in 2014. The study also explored if there were any interaction
effects between NBCT status and gender, NBCT status and student grade level, and
NBCT status and student FRL status among those students taking the AP Calculus AB
Exam. . Moreover, through this study, it was the researcher’s interest to add knowledge to the relationship of student success on the AP Calculus AB Exam and NBCTs to the findings of previous studies that investigated the relationship between student achievement and mathematics NBCTs and non-NBCTs. Additionally, the researcher performed a quantitative analysis of mathematics teachers’ beliefs on classroom practices in relationship to the NCTM standards. More precisely, the researcher used Zollman and
Mason’s (1992) Standards’ Beliefs Instrument (SBI) to explore and measure the level of alignment of teachers’ beliefs with the NCTM.
Finally, through a qualitative interview process, this study revealed how NBCTs
deliver their classroom instruction and how they incorporate best mathematics teaching
practices that may lead students to succeed on the AP Calculus AB Exam.
This study was guided by the following research questions:
96 1. Do NBC mathematics teachers produce better results than non-NBC mathematics teachers on the AP Calculus AB Exam?
2. In respect to test score performance, is there an interaction between NBCT status and student gender, NBCT status and student grade level, or NBCT status and student FRL status?
3. What is the difference in perception of best mathematics teaching practices between NBCTs and non-NBCTs?
4. How do NBCTs incorporate best mathematics teaching practices in their daily instruction?
Sampling Plan, Instrumentation, and Data Collection
Prior to the beginning of the data collection process, the approval of two institutional review boards were required. Following the receipt of permission to perform research, the researcher contacted the Research Office of the pertinent county in South
Florida to obtain archival data of AP Calculus AB Exam scores, as well as information regarding students’ and teachers’ characteristics. Each NBCT and principal of the pertinent schools was contacted to determine his or her willingness to participate in this research study. Purposeful sampling was utilized for this study. The researcher contacted
NBCTs in the target county from an online directory of NBCTs offered by the NBPTS website. The first five NBCTs who agreed to participate were selected for this study. It is important to consider that all five of these teachers at their respective schools deal with students from different socio-economic backgrounds in the selected county in South
Florida, although students in the AP Calculus classes may not be representative of their schools’ overall demographics. This fact provides richer information due to the diversity
97 of blends of high school students and teachers. Informed consent forms were signed by
each NBCT before being interviewed. After the five interviews were finalized, the
researcher proceeded to create five case studies based on the data obtained from each
interview.
To answer Research Question 1 (Do NBC mathematics teachers produce better
results than non-NBC mathematics teachers on the AP Calculus AB Exam?), the
researcher contacted the Research Office of the school district and requested the test scores of the AP Calculus AB Exam from 2014. In addition to the students’ test scores, this report contained information about all NBCTs and non-NBCTs who taught AP
Calculus AB during the year 2014. Additionally, this report provided the score of each student who took the AP Calculus AB Exam along with other variables such as the student’s school site, grade level, AP score, gender, race, and FRL status. The report allowed the researcher to generate two independent non-random samples of the teachers who taught and prepared students for the AP Calculus AB Exam. The two samples were broken down as follows: (1) a sample of high school mathematics NBCTs with their pertinent students’ AP scores; and (2) a sample of high school mathematics non-NBCTs with their pertinent students’ AP scores. For this study, only teachers from the selected county were considered.
Research Question 2 (In respect to test score performance, is there an interaction
between NBCT status and student gender, NBCT status and student grade level, or
NBCT status and student FRL status?) was also addressed through the report from the
Research Office; it provided information on these particular variables for each student
who took the AP Calculus AB Exam during the year 2014. To answer both Research
98 Question 1 and Research Question 2, the researcher worked with archival data provided by the Research Office, which means that the researcher did not interact with teachers or students at any time.
In order to answer Research Question 3 (What is the difference in perception of best mathematics teaching practices between NBCTs and non-NBCTs?), the researcher asked NBCTs and non-NBCTs to voluntarily complete the Standards’ Beliefs Instrument
(SBI) (see Appendix F). The SBI is a 16-item instrument that can be utilized to explore and assess teachers’ beliefs with respect to mathematics practices in reference to the
NCTM standards. Responses from 31 NBCTs and 31 non-NBCTs grades 6-12 were collected. The data collected from the different SBI surveys served the purpose of providing information regarding the level of agreement between the two groups of teachers regarding how they believe mathematics should be taught according to the standards of the NCTM.
Research Question 4 (How do NBCTs incorporate best mathematics teaching
practices in their daily instruction?) was answered by collecting qualitative data from
face-to-face interviews (See Appendix D). The researcher contacted principals and
NBCTs from various schools in one county in in South Florida to ask for their consent to
participate in a voluntary interview that focused on how teachers incorporate the use of
best mathematics teaching practices on their daily instruction to effectively instruct their
students to learn calculus and eventually prepare them to succeed on the AP exam. The researcher contacted the participants by e-mail, and they were incorporated in the sample as they provided their respective consent to participate in this study. The data collected from the interviews served the purpose of allowing the researcher to create five different
99 case studies. The researcher met five NBCTs who taught AP Calculus AB during the year
2014. The interviews were performed outside the school environment. The interviews
focused on how NBCTs incorporate the use of best mathematics teaching practices in
their AP Calculus AB instruction. The researcher previously informed each teacher that case studies would be created based on the information that they provided on their interviews. Additionally, the researcher informed participants that he would assign a pseudonym to each of them in order to protect their identity when the case studies were written.
The researcher kept the collected data secure in order to ensure monitoring as well
as privacy of the subjects. The researcher is sensitive about loss of data and breach of
confidentiality. The data has been electronically kept at the university in the principal
investigator’s office under lock and key. The researcher also used a backup USB drive to
safeguard the data; this portable USB drive is also kept in a secure cabinet at the researcher’s domicile.
Subjects
The entire population (N = 1,162) of students taking a yearlong AP calculus AB
course in the selected county was used for this study. An additional section of AP
Calculus BC is available for either accelerated students or students who previously took
AP Calculus AB. Students for this additional section were not included in this study. The
population consisted of 1,162 students, of whom 591 students (51%) were male, and 571
students (49%) were female. Regarding student grade level, 501 students (43%) were seniors, 623 students (53.5%) were juniors, and 38 students (3.5%) were sophomores.
Socio-economic status is defined according to student eligibility for the FRL program.
100 Among the reported students, fewer than one third (31.5%) were FRL eligible, and the
remaining students (68.5%) were not eligible. Enrollment specifics are displayed on
Table 6
Characteristics of Student Subjects
Student Subjects N % % District wide Gender Male 591 51 1.9 (N = 30,714) Female 571 49 1.9 (N = 29,100)
Grade Senior 501 43.0 2.4 (N = 20,149) Junior 623 53.5 3.1 (N = 19,741) Sophomore 38 3.5 <1 (N = 19,924)
Economic Status FRL eligible 366 31.5 1.0 (N = 34,006) FRL ineligible 796 68.5 3.1 (N = 25,808)
Research Design
In order to answer Research Question 1 (Do NBC mathematics teachers produce better results than non-NBC mathematics teachers on the AP Calculus AB Exam?), the
researcher collected test scores from the 2014 AP Calculus AB exam from the different
teachers who prepared the school district’s students for this particular AP exam. These teachers were categorized into two different groups: NBCTs and non-NBCTs. To
compare results, the researcher utilized an independent samples t-test to test the
hypotheses about the difference in two population means.
Hypothesis Test # 1. The hypotheses for the comparison of two independent
groups are:
Ho: u1 = u2 (means of the two groups are equal)
Ha: u1 ≠ u2 (means of the two group are not equal)
The alpha level adopted for this study was α = 0.05.
101 To answer Research Question 2 (In respect to test score performance, is there an
interaction between NBCT status and student gender, NBCT status and student grade
level, or NBCT status and student FRL status?), the researcher performed three factorial
analysis of variance F-tests to address the three different interactions. The following
hypotheses were tested:
Hypothesis Test # 2.
H 0 :There is no interaction between NBCT status and student gender.
H1 :There is an interaction between the two factors.
Hypothesis Test # 3.
H 0 :There is no interaction between NBCT status and student grade level.
H1 :There is an interaction between the two factors.
Hypothesis Test # 4.
H 0 :There is no interaction between NBCT status and student FRL status.
H1 :There is an interaction between the two factors.
The Standards’ Beliefs Instrument (SBI)
In order to answer Research Question 3 (What is the difference in perception of
best mathematics teaching practices between NBCTs and non-NBCTs?), the researcher utilized the SBI instrument to assess teachers’ general beliefs in regards to mathematics instruction aligned with the NCTM standards. Zollman and Mason (1992) stated that in response to a crisis in mathematics education, a national reform in mathematics education was initiated. The NCTM created the Curriculum and Evaluation Standards for School
Mathematics . Zollman and Mason also stated that:
102 This document presents NCTM’s vision of how mathematics should be learned,
taught and evaluated by making a shift putting more emphasis in conceptual
development, mathematical thinking, reasoning, and incorporating more problem
solving while having students take a more active role in the classroom. (p. 359)
The SBI is a document that consists of 16 four choice questions that utilizes a four-point Likert-type scale and provides a view on how teachers perceive mathematics education. As noted by Futch and Stephens (1997), inappropriate teaching practices may be related to inadequate teacher beliefs about mathematics. Teachers’ adherence to a particular set of beliefs may result in limits on mathematical learning by students. In this study, the SBI provided important information as to how NBCTs and non-NBCTs perceived mathematics education in today’s classrooms. The scores on the SBI were derived from the summation of the 16 items.
The researcher collected data by assigning the SBI to NBCTs (N = 31) and non-
NBCTs (N = 31). After their responses were collected, the researcher performed a t-test.
By using a t-test, the researcher explored the level of agreement in alignment with the
NCTM standards between the two different groups of teachers. Furthermore, a Cronbach reliability analysis was incorporated to complement t-test results, investigate the average correlation of the 16 items in the SBI, and measure the internal consistency. The following hypotheses were tested:
Hypothesis Test # 5.
H 0 : NBCTs and non-NBCTs produce equal scores on the SBI.
H1 : NBCTs and non-NBCTs do not produce equal scores on the SBI.
103 Interviews
The researcher performed teacher face-to-face interviews in order to answer Research
Question 4 (How do NBCTs incorporate best mathematics teaching practices in their daily instruction?). As cited in Opdenakker (2006), Kvale (1983) defined the qualitative
research interview as an interview whose purpose is to gather descriptions of the life-
world of the interviewee with respect to interpretation of the meaning of the described
phenomena. Interviewing is necessary when we cannot observe behavior, feelings, or
how people interpret the world around them. It is also necessary to interview when we are
interested in past events that are impossible to replicate (Merriam, 2009, p. 88).
Interviewing NBCTs provided an in-depth snapshot on how best mathematics practices are being implemented on a regular basis. The researcher believes that the interviewing processes allowed NBCTs to express their ideas more clearly, especially when they needed to talk about more complex or abstract topics and how they deliver their teaching.
Interview Protocol/Procedures. The researcher sought the principals’ approval
from each school to interview each NBCT. Additionally, the researcher asked each NBCT
for consent to be interviewed. The method of voluntary participation in this study
required participants to be willing to be interviewed in regards to how they teach their
mathematics lessons and prepare their AP Calculus AB students in their respective
schools. After obtaining IRB approval from the school district, the researcher contacted
participants via e-mail asking for their participation. Then, the researcher gave the
participants a consent form (see Appendix C), extended an invitation containing pre-
interview questions (see Appendix A), and also sent an initial self-report/introductory
letter (see Appendix B) to those who agreed to participate in the study. No data were
104 collected until the consent form was signed and returned. The reason for the pre-
interview questions was to get the participants thinking about best mathematics practices
in order to get them better prepared to provide more elaborate and detailed answers
during the face-to-face interviews. Each interview lasted between 50 and 60 minutes.
Participants were informed that they could be asked to be interviewed on more than one occasion in additional information was needed. During the interview process, the researcher asked each teacher for consent to record each interview. After each interview was performed, the researcher transcribed the interview in order to analyze the emerging
themes.
Interview Questions. The interview protocol in this study focused on open-ended
questions. The researcher selected open-ended questions in order to give interviewees the opportunity to provide more elaborate and in-depth answers. Additionally, using open- ended questions allowed NBCTs to express themselves more openly and to share their ideas and concerns more honestly. Each question in the interview protocol was extracted from a mathematics classroom observation tool called the Inside the Classroom
Observation and Analytic Protocol (K-12) (see Appendix E), one of the three best instruments to evaluate math teaching instruction, according to Educational Research
Newsletter and Webinars (2014), along with the COEMET (pre-k) and the RTOP (k- university). The researcher designed all the questions in the interview protocol based on the different domains of observation contained in the Inside the Classroom Observation and Analytic Protocol (Horizon Research, 2000, available at http://www.horizon- research.com/instruments/clas/cop.pdf). Moreover, each of the questions obtained from this document is related to a best mathematics teaching practice presented in the literature
105 review for this study. This instrument has goals that are specifically aligned with the
NTCM standards and examines the lesson, teaching, and classroom characteristics
important in a measure of quality (Educational Research Newsletter & Webinars, 2014).
The interview questions served to evaluate teacher quality in AP Calculus AB instruction
and to provide understanding of the extent to which NBCTs use different mathematics
best teaching practices.
Quantitative Data Analysis
In order to test hypothesis 1, the researcher performed a t-test of two independent means. The t-test allowed comparison of students’ AP Calculus AB Test scores of
NBCTs and non-NBCTs. The alpha level adopted for this study was α =.05. After performing the t-test, the researcher conducted a Cohen’s d procedure to complement the results after rejecting the null hypothesis. This test allowed the researcher to calculate the
effect size of the differences between the two means on the t-test and explore if the
differences in the means from the t-test had a practical significance. The calculation of this effect size revealed how much more effective the difference was in the means of students’ test scores from both groups of teachers.
In order to test hypotheses 2, 3, and 4, the researcher performed three different factorial analysis of variance F-tests in order to explore if there was a significant interaction effect between NBCT status and student gender, NBCT status and student grade level, and similarly to explore if there was a significant interaction effect between
NBCT status and student FRL status. The researcher investigated if the results in
students’ test scores on the AP Calculus AB Exam depend on the interaction of the two
106 different NBC levels and the student gender, student grade level, and also student socio- economic status associated with FRL eligibility.
In order to answer Research Question 3, the researcher used the responses from
31 NBCTs and 31 non-NBCTs who had completed the SBI instrument. The data collected were utilized to perform a comparison about the beliefs regarding the NCTM mathematics standards between the two different groups of teachers. The researcher utilized descriptive statistics. In order to test hypothesis 5, the researcher performed a t-test and later on a Cronbach alpha reliability analysis in order to compare the teachers’ agreement with the NCTM standards. The reliability analysis was based on the idea that individual teachers should produce consistent results across the different questions inside the SBI instrument.
Qualitative Data Analysis
In order to answer Research Question 4, the researcher performed interviews with five NBCTs who taught AP Calculus AB during the year 2014. Each of these teachers prepared students who took the AP Calculus AB Exam in that year. After the interviews were completed and transcribed, the researcher created five different case studies.
According to Walton (1972), “the primary attribute of case studies are that they include sensitive descriptive material about human beings and they are illustrative in a way that leads to credibility” (p. 77). Merriam (2001) stated that “unlike experimental design, survey, or historical research, case study does not claim any method for data collection or data analysis. Any and all methods of gathering data, from testing to interviewing, can be used in a case study” (p. 42). Bromley (1986), as cited in Merriam (2001), wrote that case studies by definition get as close to the subject of interest as they possibly can, partly by
107 means of direct observation in natural settings, partly by their access to subjective factors
(thoughts, feelings, and desires) (p. 46). These case studies served the purpose of
providing rich and in-depth information about how NBCTs in the selected school district in South Florida utilize best mathematics teaching practices on a daily basis that may lead students to effectively learn AP Calculus. The data from the case studies also provided
more depth in understanding how NBCTs perceive that AP Calculus AB should be taught
and learned in order for their students to succeed on the AP exam.
Summary
The methods, participants, sampling plan, data collection, and data analysis were presented in this chapter. The purpose of this study was to compare student performance of NBCTs and non-NBCTs on the AP Calculus AB Exam and also to explore how best
mathematics teaching practices are incorporated in NBCTs’ classrooms.
For the quantitative analyses, the results of the statistical analyses of students’ test
scores were presented in two dimensions: (1) difference of test scores between students
of NBCTs and non-NBCTs; and (2) the analyses of interaction effects between NBCT
status and student gender, student grade level, and also student FRL status. Three
factorial analyses of variance F-tests were conducted to address such interaction effects.
Additionally, the researcher investigated if NBCTs and non-NBCTs depict a similar level
of alignment on mathematics teaching approaches with the NCTM standards by
performing a second t-test. Finally, a reliability alpha Cronbach analysis was
incorporated to explore how reliable the SBI instrument was in regard to the different
groups of teachers.
108 For the qualitative component of this study, five teacher case studies were utilized in order to explore how NBCTs incorporate best mathematics teaching practices in their daily instruction; common patterns and differences in teaching approaches were
identified among the five interviewed NBCTs. The results of this study are discussed in
Chapter 4.
109 Chapter 4. Analysis of Results
Introduction
This study was designed to examine the impact of NBCT status on the students’ test scores on the AP Calculus AB exam in one county in South Florida. The researcher measured the impact of the NBC based on students’ test scores on the 2014 AP Calculus
AB Exam. In this section the results of statistical analyses performed on quantitative hypotheses are discussed. The qualitative research provided an in-depth depiction of
NBCTs’ instructional approaches and teaching philosophies through the different case studies. The research in this study has addressed the following questions:
1. Do NBC mathematics teachers produce better results than non-NBC mathematics teachers on the AP Calculus AB Exam?
2. In respect to test score performance, is there an interaction between NBCT
status and student gender, NBCT status and student grade level, or NBCT status and
student FRL status?
3. What is the difference in perception of best mathematics teaching practices
between NBCTs and non-NBCTs?
4. How do NBCTs incorporate best mathematics teaching practices in their daily
instruction?
110 Descriptive Statistics
Table 7 presents the means and standard deviations of the participating county
students’ test scores on the AP Calculus AB Exam of 2014.
Table 7
Means and Standard Deviations for Student Test Scores of National Board Certified and Non-
National Board Certified Teachers
Students of NBCTs Students of Non-NBCTs (N = 260) (N = 902) Variable N M SD N M SD Gender Male 140 3.74 1.417 451 2.90 1.579 Female 120 3.67 1.434 451 2.57 1.525
Grade Level Grade 10 16 2.63 1.544 22 3.82 1.296 Grade 11 141 4.01 1.251 482 3.06 1.534 Grade 12 103 3.46 1.507 398 2.28 1.560
FRL Status FRL Eligible 59 3.02 1.514 307 2.53 1.517 Non-FRL Eligible 201 3.91 1.333 595 2.84 1.560
The data were desegregated according to the researcher’s three variables of
interest (gender, grade level, and FRL status). It can be observed that in almost every single category of each variable, the corresponding means of the students taught by
NBCTs are higher compared to the means of those taught by non-NBCTs except on the
10th grade level. The inferential statistics in this chapter provide an analysis of statistical
significance of the difference in means of students of NBCTs and non-NBCTs.
111 AP Calculus AB Scores 2014 350
300
250
200
150 Non National Board Certified Frequency National Board Certified 100
50
0 0 1 2 3 4 5 Scores
Figure 11. Distribution of AP Calculus AB scores for NBCTs and non-NBCTs.
Figure 11 provides a snapshot of the frequencies of AP scores corresponding to
students taught by the two different groups of teachers. In the group of students taught by
NBCTs, the highest frequency of scores is a 5. In regard to the students taught by non-
NBCTs, the highest frequency of scores is a 1.
Inferential Statistics
In order to answer Research Question 1 (Do NBC mathematics teachers produce better results than non-NBC mathematics teachers on the AP Calculus AB Exam?), the
researcher examined the difference in test scores between students taught by NBCTs and
non-NBCTs by conducting an independent sample t-test. Given a violation of Levene’s
test for homogeneity of variances F (1, 1160) = 17.33, p < .001, a t-test not assuming
homogeneous variances was calculated. The results indicated that there was a significant
difference in test scores between the two groups, t (454.065) = 9.462, p < .001. These
112 results suggest that students taught by NBCTs (M = 3.70, SD = 1.423, N = 260) tended to score higher than students taught by non-NBCTs (M = 2.74, SD = 1.560, N = 902). The
Cohen’s d was calculated with a pooled standard deviation. The size of this effect was d = .6429. As per Cohen’s (1988) convention, the coefficient d was found to be a moderately large effect. This implies there was a practical significance between the means of the test scores produced by the two groups of students.
In order to answer Research Question 2 (In respect to test score performance, is there an interaction between NBCT status and student gender, NBCT status and student grade level, or NBCT status and student FRL status?), the researcher performed three different factorial analysis of variance F-tests. The first ANOVA test relates to the following hypotheses:
H 0 : There is no significant interaction effect between NBCT status and gender.
H1 : There is a significant interaction between NBCT status and student gender.
Table 8 depicts the results for the interaction effect between certification status and student gender.
A two way factorial analysis of variance, with a power for the interaction effect of only .23 reveals that there was not enough evidence to reject the null hypothesis. There was no significant interaction found between NBCT status and student gender. It can be concluded that the effect of certification was consistent between genders.
The second ANOVA test relates to the following hypotheses:
H 0 : There is no significant interaction effect between NBCT status and student grade level.
H1 : There is a significant interaction between NBCT status and student grade level.
113 Table 8
Analysis of Variance of Interaction between Certification Status and Gender
Type III of Sum of Mean Partial Eta Observed Source Squares df Square F p Squared Power Certification Status 187.529 1 187.529 80.659 .000 .065 1.000 Gender 8.192 1 8.192 3.523 .065 .003 .466 Cert. Status*Gender 3.547 1 3.547 1.526 .217 .001 .235 Error 2692.283 1158 2.325 Total 2891.551 1162
Table 9
Analysis of Variance of Interaction between Certification Status and Grade Level
Type III of Partial Sum of Mean Eta Observed Source Squares df Square F p Squared Power Certification Status 6.661 1 6.661 3.052 .549 .000 .415 Grade level 82.104 2 41.052 18.810 .776 .000 1.000 Cert. Status*Grade 46.579 2 23.290 10.672 .000 .019 .990 Error 2522.860 1156 2.182 Total 2658.204 1161
The analysis as shown on Table 9 indicated with a power for the interaction effect of .99, that there was enough evidence to reject the null hypothesis. Although there was a significant interaction between NBCT status and student grade level, the effect of
certification is not consistent across the different student grade levels. It can be observed
that the means of students taught by NBCTs are higher for grade 11 and grade 12 but not
for grade 10. On the other hand, the 10th graders taught by non-NBCTs scored higher
than those taught by NBCTs (see Table 7).
114 The third ANOVA test relates to the following hypotheses:
H 0 : There is no significant interaction effect between NBCT status and student FRL
status.
H1 : There is a significant interaction between NBCT status and student FRL status.
Table 10
Analysis of Variance of Interaction between Certification and FRL Status
Type III of Partial Sum of Mean Eta Observed Source Squares df Square F p Squared Power Certification Status 89.203 1 89.203 38.813 .000 .033 1.000 FRL 53.692 1 53.692 23.362 .000 .020 .998 Cert. Status*FRL 12.351 1 12.351 5.374 .020 .005 .644 Error 2659.082 1157 2.298 Total 2814.328 1161
Based on the results, with a power of .644 for the interaction effect, there is
enough evidence to reject the null hypothesis. It can be observed that there was a
significant interaction between NBCT status and student FRL status. The effect of
certification is not consistent across the two levels of student FRL status. Even though
students of NBCTs scored higher regardless of their FRL status, the difference between
the means of NBCTs and non-NBCTs student scores is greater for higher student
economic (not eligible for FRL) status (see Table 7).
In order to answer Research Question 3 (What is the difference in perception of
best mathematics teaching practices between NBCTs and non-NBCTs?), the researcher examined the difference in SBI scores between NBCTs and non-NBCTs by conducting a two independent sample t-test, and explored if such difference in belief scores regarding
115 the level of agreement with the NCTM’s standards is statistically significant. Given a
violation of Levene’s test for homogeneity of variances F (1, 60) = 4.41, p < .01, a t-test not assuming homogeneous variances was calculated. The results indicated that there was a significant difference in scores between the two groups, t (53.102) = 3.184, p < .001.
These results indicate that NBCTs (M = 33.48, SD = 4.373, N = 31) tended to have a
stronger level of agreement with the NCTM standards compared to non-NBCTs
(M = 36.52, SD = 2.999, N = 31).
After performing the t-test, the researcher decided to explore the reliability of the
SBI instrument by calculating a Cronbach alpha reliability coefficient. As explained in
Santos (1999), an alpha value of 0.70 is acceptable. The researcher calculated three reliability Cronbach alpha values. The results showed that the Cronbach’s alpha value for the two groups of teachers combined was .422, which represents a low alpha value indicating that there is low internal consistency; 42.2% of the true score is estimable for the sample. Nearly 58% of the variability is due to error. The Cronbach alpha for the non-
NBCTs was .707, which indicates that the reliability is acceptable. Finally, the alpha value for the NBCTs was .370, indicating that the reliability was very low. The reduction in the coefficient alpha for the subjects on the NBCT group was possibly due to their restriction in range as there was less variance in this group. This result indicates the
NBCTs’ choices answering the 16 items of the SBI were consistently in higher agreement with the NCTM standards producing less variance in the selection of their answers. Less variance may lead to a smaller alpha value. It has been observed that the reliabilities for the SBI instrument in this study range from acceptable to low. Even though the reliability
116 was low, in the end the t-test was significant and reliable enough to detect the difference
of the two groups.
In order to answer Research Question 4 (How do NBCTs incorporate best mathematics teaching practices in their daily instruction?), the researcher conducted interviews with five NBCTs who volunteered to participate in this study. Later on, five different case studies were created by utilizing the data obtained from the interviews.
In order to ensure that interview questions were effective in targeting data, the researcher pre-tested the interview protocol on two mathematics teachers from a high school in Fort Lauderdale, Florida. These math faculty members were not part of the actual study. The researcher did not utilize any of the two interviewee’s responses for the findings in this study. The goal of these trial interviews was to evaluate the strength of the interview questions and to ensure that the actual interviews would not be too long
(not more than 60 minutes). The interview process consisted of two parts. The first part consisted of answering pre-interview questions that were representative of the forthcoming face-to-face interview. The purpose of these pre-interview questions was to get the interviewees to start thinking of the type of questions that they would be asked during the face-to-face interview. More specifically, the pre-interview questions allowed
interviewees to be better prepared for the face-to-face interview, which resulted in
obtaining richer and more in depth responses from the interviewees. Before each
interview, the researcher tried to build a rapport and a welcoming environment with the
interviewees instead of going straight to the interview questions. All interviews were
scheduled at the teachers’ own convenience, and the participants were allowed to select
their out-of-school interview site.
117 Sample of Interviewed NBCTs
Purposeful sampling was utilized for this study, and the participants consisted of
five NBCTs in one school district in Florida. Each of the participants was purposefully
chosen from an online directory of NBCTs offered by the NBPTS website. Two
participants had been teaching AP Calculus AB for 10 years, one for 3, one for 7, and one
for 4. Their years of teaching experience for this particular calculus class ranged from 4
to 10 years; however, all of these NBCTs had previously taught other mathematics
classes for several years. None of these participants started their teaching careers teaching
AP Calculus. It is interesting to note that all of the participants hold masters degrees.
Table 11 describes the NBCTs who participated in this study; pseudonyms were
used for the participants.
Case Studies of Insights from National Board Certified Mathematics Teachers Who
Teach AP Calculus AB
These five case studies illustrate the interrelationship between the NBC teachers’
beliefs and the incorporation of mathematics best teaching practices in their daily
instruction. Each teacher was interviewed for an average time of sixty minutes, and every interview was audiotaped. Their answers were utilized to assess commonalities and differences in the teachers’ instructional approaches and teaching styles regarding how they prepare students to pass the AP Calculus AB Exam and how they believe calculus can best be taught and learned. Moreover, it was the researcher’s intent to reveal NBCTs’
pedagogical repertoire as they attempt to promote mathematical learning and provide
opportunities to help students think critically.
118 Table 11
Characteristics of Interviewed National Board Certified Teachers
# of Years Total # of Teaching at Time Years NBC Teacher Year of National Board Degree Teaching AP Grade Name Certified Certification Level Calculus AB Level Ms. Johnson 2006 6 Masters 10 K-12 Ms. Doval 2001 4 Masters 10 K-12 Ms. Furgeson 2008 23 Masters 3 K-12 Ms. Morrison 2006 13 Masters 7 K-12
Mr. Glenc 2005 26 Masters 4 K-12
In order to protect the participants’ confidentiality, a pseudonym was assigned to
every interviewee. The researcher conducted each interview based on the participants’
voluntary consent to participate in this study. Written consent was required as the
researcher contacted each participant to participate in this study. The researcher was
required to obtain approval from the FAU Institutional Review Board (IRB) and also
from the Research Office IRB of the school district from which the teachers were
selected.
The NBCTs in this study bring an exceptional amount of experience. They are
veteran teachers with more than 10 years of teaching experience. They come from
different backgrounds such as engineering, psychology, the business industry, and
teacher education training.
In these case studies, the researcher discussed five domains regarding best
mathematics teaching practices: (1) design; (2) implementation; (3) mathematics context;
(4) classroom culture; and (5) assessment. The researcher designed all the questions in
the interview protocol based on the above domains of observation contained in the Inside
119 the Classroom Observation and Analysis Protocol which can be retrieved from http://www.horizon-research.com/instruments/clas/cop.pdf. Horizon Research
(2000) noted that “this instrument was designed to measure the quality of observed k-12 science or mathematics classroom lessons and to allow researchers to rate the lesson content and quality of pedagogy” (para. 1). Each of the questions derived from this protocol is related to a best mathematics teaching practice presented in the literature review for this study. The researcher utilized the interview questions to evaluate teacher quality in AP Calculus instruction and to provide understanding of the extent to which
NBCTs use different mathematics best teaching practices on a regular basis.
Case Study of Ms. Johnson. Ms. Johnson has been teaching mathematics for 14 years in grades 9 through12. She has taught all types of levels of mathematics high school courses ranging from Algebra 1A to currently teaching AP Calculus. She is currently an adjunct college professor at a community college teaching trigonometry, college algebra, and statistics. The first impression that one gets from Ms. Johnson is that she seems to be very enthusiastic and motivated about teaching. Behind her teaching philosophy she stated that “anybody can be taught; whether they want to be taught or not is something different, but they can be taught.” This perfectly aligns with the NBPTS philosophy that every child can learn.
Ms. Johnson does not come from a teaching preparation program. She has a degree in engineering, and because of her inability to get a job as an engineer, she applied for a job as a teacher. She experienced frustration due to the fact that she had obtained a degree in engineering, but because she did not have any contacts to get a job in the engineering profession, she decided to apply for a teaching position as she needed to do
120 something with her professional career. “I came into the teaching profession not knowing
what to expect,” she said. Even though she started teaching less motivated and lower
mathematics students, she expressed that she loved teaching right from the beginning.
She said, “I was pretty good at it but they gave me the worst of the worst students.” A
striking statement that Ms. Johnson revealed is “they gave me the worst of the worst. I
came home crying every day but I still loved math and teaching.” It is evident that despite
her frustration teaching these kinds of students, she very much felt that the teaching
profession was something that she really wanted to take on as her profession after all. In fact, she did not know that she liked teaching so much. She stated that being a good teacher and loving the teaching profession involve innate qualities. As she stated,
“teachers are partly born and partly made, but really partly born I think.”
In regards to AP Calculus, she believes that a teacher needs to know the subject well as the students in this class are very sharp and will challenge the teacher. This is also another standard from the NBPTS which states exactly the way Ms. Johnson feels about how a teacher should be prepared to lead and teach a class like AP Calculus. She is an advocate of professional development as she is a strong believer that learning should be an ongoing process in every teacher’s career.
Ms. Johnson is a proactive teacher. Before meeting her students on the first day of school, Ms. Johnson expects them to be prepared with certain algebraic skills. She assigns a summer package that contains mathematics problems that students should be able to solve in order to be prepared to start AP Calculus. She likes to check on prior knowledge before starting a new lesson by reviewing concepts from previously learned topics. She likes to design her classes in a way that everyone can participate and have a
121 voice in the classroom. Students look at each other’s work and discuss what is wrong as
they learn from one another. She strongly believes in cooperative learning and through
her various mathematics activities, she likes to do a great deal of pair sharing and
especially peer tutoring when some students are struggling with concepts while others
understand such concepts. She said this is an excellent way to help others and to help the
teacher as well; “That’s what I like to do because I can’t do it all. I can’t do everything.”
Ms. Johnson is indeed an advocate of using technology in the classroom. She sees
technology as a very important tool to accomplish what she wants in her lesson plans.
She incorporates a variety of digital resources such as graphing calculators, navigators,
the Promethean Board, and websites that lead students to have a better understanding of the subject matter. She believes that technology integration is highly necessary in upper level mathematics classrooms. Ms. Johnson stated that in the calculus class, she does not
incorporate a lot of project-based learning, but she does concentrate on the application of
solids of revolution. She tries to adapt this project according to each individual’s interest;
for example, if she has some students who want to become architects, then they can adapt
this project to whatever they think interests them in their fields of study. She stated, “AP
Calculus is very theoretical,” meaning that this class does not allow much room for
creating projects as part of the learning process.
When she asks questions during her lessons, she always expects students to
defend their answers verbally, and if they are unable to answer, her strategy is to utilize
probing questions that lead to higher order thinking questions. When planning her
lessons, she understands that sometimes she does not get the ‘elite’ students and she
needs to plan accordingly. She explained that at her school they filter students coming
122 into AP Calculus, but regardless if they are capable or not of facing the AP Calculus challenge, going through the experience of an AP mathematics class is always very positive for the student. She develops her lessons by knowing the type of students she has, but there is no recipe to anticipate how long the lesson will take. She stated that
“sometimes the lesson takes longer, sometimes it takes half of the time, and sometimes it takes as long as I expected.”
One interesting characteristic about Ms. Johnson’s teaching style is that she tries to add humor in order to make her lessons more interesting and capture students’ attention. She said, “When you are going through a certain particular theorem and it seems so very dry, so dry, I sometimes joke and say this is boring, isn’t it? All right, this is where we are going to use this.” Then she tries to bring a real-world application to make a connection between theory and practice in order to make learning more interesting. Ms. Johnson emphasizes that in calculus there are many symbols and Greek letters. This scares students right from the beginning. “Math is like a language and you have to instruct math in a way like it is a foreign language” she said. “No one is going to forgive you.” She sees theory and practice at the same level of importance. She tries to make her students understand that there is some concrete application after the theory.
While teaching calculus, she explained, “You need concrete because you need to keep the students focused instead of just being all abstract in order for them to know that there’s something at the end and there’s concrete.”
Ms. Johnson encourages and embraces students who ask questions, but she always creates an atmosphere of respect by making students confident that they will always be respected when asking any question. She likes to validate students’ questions
123 by saying something like “that’s a really good question” or “maybe in this group of 20 people, they don’t ask it, but you asked it and people do ask that question all the time.”
When having her students work in groups, she believes that a teacher needs to be very focused as students may deviate from doing the work. It is important to monitor that students are always on task. Cooperative learning is an important aspect of her teaching style. She likes students to cooperate with each other during her five minute daily quizzes. She allows students to create their own groups and she wants her students to compare their work, especially the symbolic calculus part and how they write or display their work. However, she believes that individual work is as important as working in groups because “nobody will help you to solve the problems during the AP Calculus test.”
Ms. Johnson sees informal assessment as a highly beneficial strategy to pass the
AP exam. These assessments come in different forms such as daily quizzes, multiple choice, free response, or just oral discussions, and such daily informal assessments serve the purpose of tracking students continuously. As she said, “I’m just gonna say that the daily quizzes help me make sure they are not falling behind as we go along.” However, everything that students have to solve has to come in the AP exam format. She always plans to finish the entire curriculum at least three months before the real AP exam. This allows students the opportunity to work on their project and review for the exam. Ms.
Johnson kindly provides a ‘calculus study hall’ for students. Once a week, she makes herself available after school to help out students who need additional help. She does it voluntarily, and she sees the study hall as a remediation strategy to reach out to those students who are not performing at the desired level. She also incorporates differentiated
124 instruction in her teaching, so she provides opportunities for those students who did not get a good grade, such as allowing them to explain the errors they made on the test.
Ms. Johnson sets high expectations for all her students, and it is evident that she very much cares for her students and wants to see them succeed. Her passion for her students clearly comes out as she speaks about her classroom and daily instruction. Table
12 depicts a summary of Ms. Johnson’s philosophy across the five different teaching domains under consideration for this case study.
In summarizing Ms. Johnson, she is a mathematics teacher who emulates the very best in math education instruction. She is very devoted to her students and confident that any student who works hard can pass the AP exam, which is part of her teaching philosophy.
She believes that teachers should constantly learn to become better professionals; and students benefit from such growth. She strongly supports cooperative learning and sees this strategy as an effective tool for students to learn from one another. For Ms. Johnson,
AP Calculus is a very theoretical class; she thinks that incorporating technology in her instruction helps students to visualize and understand AP Calculus concepts more efficiently. Additionally, her engineering background may be an indicator that she likes to see abstract concepts materialized and for that reason, she enjoys having her students make connections between AP Calculus and other areas where calculus concepts can be immediately applied.
125 Table 12
Summary of Ms. Johnson’s Teaching Approaches
Design * Students receive a summer package so they can get tested on their algebraic abilities. * Connections between new topics and old topics are made, often revisiting topics of algebra and pre-calculus. * Daily quizzes to check on students' progress. * Cooperative learning and pair share must be present while teaching. * Students learn from each other's mistakes.
Implementation * Technology: graphing calculators, navigators (students communicate via calculator), text messages, and calcchat (www.calcchat.com). * Projects: solids of revolution (doughnut + banana) to demonstrate volume. * Students defend their answers verbally. "I'll ask them a question and then another, probing, and get them through the steps. * Lesson plans: "there is no cookie cutter to state how long a lesson will take." * A teacher needs to modify lessons and pacing according to how students react to the new content.
Mathematics * Teachers need to be precise and specific about the content they are Content going to teach and not confuse students with unnecessary formal notations. * "Calculus is very theoretical. I use humor to draw students' attention. * Students are not used to higher level mathematics formal language. * Both theory and practice should never be separated, but students need to adapt to a new formal and symbolic way of thinking.
Classroom * It is important to praise students when they ask questions. "No Culture question is a stupid question." * Cooperative learning is essential but needs to be monitored to ensure students are working on task. * Students need to share their work and learn from each other's mistakes.
Assessment * AP Style daily informal assessments, especially quizzes. * Continuous assessment ensures that students are on the right track. * Students should attend Calculus Study Hall if they are struggling, failing, or falling behind. * Differentiated instruction is necessary. * Opportunities are given to improve low tests grades if the student is able to explain why mistakes were made and how to get to the solution of a problem. * Mock AP exams are practiced close to the exam date.
126 Case Study of Ms. Doval. Ms. Doval is a very outgoing person. Because of the
way she presents herself, one gets the impression that she enjoys talking to people. She
showed a smile on her face as she talked about her past and background. She seems to
care a great deal for the teaching profession. She even stated that she liked videotaping herself in order to see how she could become a better teacher. She truly believes that she became a better educator by recording and watching herself teach. Surprisingly, that is something she would face at a later time in her teaching career. Going through the NBC process required her to videotape herself to show how she could deliver instruction effectively while following the standards of the NBPTS.
Ms. Doval is a person who comes from the world outside of teaching. She originally studied at her university to become an architectural engineer. She did not graduate because after two and a half years of studying engineering, she realized that she had no passion for that. She also was highly concerned that she would not be able to get a job in that field even if she had the engineering degree. She stated, “The job market in the
mid ‘80s was not good for engineers. So, at that time, I had many friends coming back to
get their teaching certification in mathematics to teach ‘cause they could not find a job.”
Moreover, Ms. Doval was convinced that she did not want to work in an office by
herself; she then decided to enter a state university in Kansas that offered a program in mathematics education. She went through the program and became a mathematics school
teacher. Ms. Doval had a very interesting beginning in her career. She began teaching in a
rural high school in Kansas, and she was the only mathematics teacher who taught every
single mathematics subject in that town. She appreciates the fact that she was exposed to
such an experience while being able to teach all types of mathematics classes; “the nice
127 thing is that I got my feet into every subject.” In the 1990s she moved to Florida, and she
taught mathematics and computer literacy at a completely different type of school. She
was hired to work for a high performing upper socioeconomic high school in Davie,
Florida. After settling down in Florida, she got her masters in mathematics education.
Ms. Doval believes that teacher mentoring was highly beneficial for her to be able to teach classes such as pre-calculus and calculus; “I got started teaching pre-calculus
under an amazing AP Calculus teacher and I just learned the ropes forwards and
backwards about what you needed to be prepared for that class.” Ms. Doval has now been
working as a teacher for 24 consecutive years.
Ms. Doval’s teaching philosophy is not that schools should prepare students for
mandatory state exams. She said that her teaching philosophy “does not necessarily agree
with the End of Course exam (EOC). “You want students to be successful in society.”
Additionally, and as part of her teaching philosophy, she believes that a teacher should be
constantly reflecting on his or her best teaching practices. This perfectly fits with the philosophy of the NBPTS. Ms. Doval is continuously looking backwards to see how she can become a better teacher. Besides teaching AP Calculus to help her students pass the
AP exam, she wants them to be well prepared for college. She likes to ask questions before teaching a new concept in order to check for prior knowledge. Asking lots of questions and probing is part of her teaching style. She emphasizes that “calculus is very cyclical. You are always spiraling backwards as you’re teaching a new concept to make it stick in their heads; but that’s also how you prepare for an AP test.” She believes that incorporating the Socratic Method to make connections with a new lesson is paramount
128 because one can create discussions where students verbally display if they are ready or
not or if remediation is needed before starting a new lesson or math topic.
She is a strong believer in research. She constantly wants her students to visit
mathematics websites. She gives students problems and tells them that the answers are
out there, and that they need to do some research in order to find the answers. She
appreciates the fact that the College Board offers students a test that is research based and
allows the opportunity for students to express their answers in multiple formats, which
can accommodate the different Calculus learning styles of each student. For sense
making, she also likes to assign problems and tasks to get students thinking before the
next lesson. This strategy makes students think, elaborate, and reflect on what they had
previously learned; Ms. Doval said, “If I give them something the night before the beginning of the next class and they know it’s due, they will come in and just start arguing with each other. I just sit back and let it go.” Ms. Doval realizes that Calculus students “are not the conventional learners; some are very visual, some can’t turn in
homework, some are very quiet” so it is important to adapt to the different learning styles
by having an open door policy listening to the students’ demands, questions, and making
herself available to tutor them one on one for extra help.
Ms. Doval confessed that she is not “as tech oriented as some of them are” but
technology plays an important role in her daily instruction. She embraces the use of
graphing calculators, a document camera, and the Internet mainly for research purposes.
Surprisingly, she mentions that PowerPoints are not important. She does not like using live Internet when delivering instruction but she makes students aware of where they can find a video or a PowerPoint from other AP Calculus teachers that they can see on their
129 own. Interestingly, Ms. Doval said, “I don’t fight the cell phone thing. I let them use it as a research tool as the textbooks are online and the textbook itself is huge.” Ms. Doval’s teaching does not seem to concentrate on project based learning a great deal, but as a
project and end of the year assignment (after they take the AP exam), she likes to have
her students write ‘letters of advice’ for future AP Calculus students. In this manner,
students at the end of the year get to express their difficulties; their good moments, what
they enjoyed the most, and they provide suggestions on how to deal with the class. As an
alternate choice, students are allowed to write a calculus song or poem using calculus terminology. Ms. Doval stated that checking on prior knowledge and using lower order questions is an essential technique to bridge old and new content. She is very organized in regards to her calculus curriculum; “you really have to chart your time and you have to plan for the year.” She always plans accordingly before teaching a lesson, but in order to determine if the pace of a lesson is appropriate for the needs of the students, she confessed that is a skill that she has learned over time from other more experienced teachers by going to math workshops and competitions. She believes that horizontal teaming is highly necessary to grow as an effective teacher, and she regularly keeps in touch with other AP Calculus teachers to ask them “where are you now?” and to see if she is developing good pacing as she teaches mathematics.
For Ms. Doval, a mathematics lesson becomes significant and worthwhile as long as the standards are followed and real-world applications are incorporated in the lesson;
“you have to throw in the real life application so they see the need in the world for that.”
When the real world comes in, she believes students tend to understand that the teacher is not just teaching to the test, but is getting them ready for other math courses when they
130 get to college. Because she took a wide range of engineering classes, Ms. Doval said that teaching calculus while making connections with physics happens in a natural way;
“That’s all integrated in my discussion; it is not a big deal.” Additionally, she has students who take both AP Calculus and AP Physics and “physics becomes so much easier when you understand calculus.” In Ms. Doval’s daily instruction, the use of manipulatives is sometimes present. She uses some three dimensional manipulatives in order to exemplify volume such as party balls or pipelines, but she thinks theory is very important and said, “you try to give them a little bit of theory and you try to show them the beauty of this.” She is aware that calculus involves a lot of theory and cares to make her lessons as interesting as possible.
Ms. Doval has created a culture of having students ask questions in her AP
Calculus class. She likes praising students’ questions by saying something like “that’s a great question” or “I’m glad you asked that. Last night I was thinking about that too.” By praising questions, she ensures that students feel respected and that they feel that they bring some contribution into the class. She said asking questions is a great strategy to initiate mathematics discussion, and this provides opportunities to have students work in groups where they can have discussions and argue with one another. She loves the fact that on multiple occasions she has observed her calculus students leave the classroom discussing and arguing things that were brought up in the classrooms thanks to the questions that were previously asked. Once a month, Ms. Doval holds what she calls the
‘Continental Calculus Competition’ which allows students to work in groups and attempt to solve problems. She gives each group 15 minutes at a time to discuss how certain calculus problems can be solved and the winner group gets extra points. When group
131 work is assigned, Ms. Doval likes to walk around the room to make sure that students are
on task. She added, “you have to pay attention; you can’t ignore what they are saying.” If
she hears something really interesting, then it becomes part of the classroom
conversation. She likes to pair up top students with lower performing students as she
believes this is a strategy that accommodates different learning styles.
In order to assess her students, Ms. Doval explained, “you have to assess them in
every direction.” She likes to give quizzes, and at the beginning of the school year she
likes to give students ‘the big ones’ as she calls it, in which students tend to run out of
time. She wants them to experience the ‘run out of time’ situation so that they can get
better training with their timing, given that the AP Calculus AB Exam presents time constraints. Many students tend to run out of time, and she does not want them to experience that when they take the actual exam. She directs her students to take the AP
Calculus AB Exams from previous years that can be found on the College Board website.
Her midterm exam is like a “mini AP Calculus exam.” When it comes to informally assessing her students, Ms. Doval likes to have verbal interaction. She considers informal assessments as anything that is not written on paper. Assigning students AP exam type problems is a way of making sure that students are not falling behind. That is how she monitors if students are prepared or if they need some additional help. If she sees that a student shows deficiencies, then she tries to get them to stay after school to tutor them one on one. Additionally, she holds review sessions for any students who want to attend.
Table 13 depicts a summary of Ms. Doval’s philosophy across the five different teaching domains addressed within the case study interviews.
132 Table 13
Summary of Ms. Doval’s Teaching Approaches
Design * Students should be taught to pass the exam but also to be successful in their future studies. * Mathematics should be taught always making connection with real world situations. *It is extremely important to check on prior knowledge before moving on to a new concept. * Incorporation of Socratic method to create discussion is a great method to check on students’ preparedness. * Student must do research and become more independent learners.
Implementation * Technology: graphing calculators, document cameras, websites, cell phones (to do research). * Students write letters of advice for upcoming AP Calculus students. * Teacher should check on prior knowledge to make a connection between old and new topics. * Get information from peers on how to implement curriculum pacing.
Mathematics * Follow AP College Board standards when teaching a lesson Content * “you have to throw in the real life application” in order to make the lesson meaningful. * AP Calculus is highly connected with physics. Having an understanding of calculus makes physics learning come easier. * Theory is highly necessary in AP Calculus. In her curriculum, there is not much room to incorporate manipulatives.
Classroom * Praise students’ questions. No question is a bad question. Culture * Questions are great to create group discussion. * Group competitions once a month to engage students in group work. * Walk around the room when students work cooperatively. It is important to pay attention to what students say in case they bring up something important and meaningful for the lesson. * Teacher provides freedom to create the groups.
Assessment * Quizzes * Midterm exam is a mini AP. * Practice AP Calculus tests given by the College Board in previous years. * Assess students informally through verbal interaction. * Assign AP exam type of problems continuously to make sure that students are not falling behind. * She makes herself available after school to help students who need reinforcement.
133 In summarizing Ms. Doval, she is a mathematics teacher who came into the
teaching profession for job security reasons, not knowing that she would find great
passion in teaching children. Her love for teaching mathematics is reflected in how she
cares for the future of her students. She loves learning from other peers and she translates
that philosophy of peer learning with her own students. She is a teacher who constantly
reflects on her practices to better herself and believes that teacher mentoring plays an
important role in the success of a teacher. For her, teaching to the test is not the key to
make successful citizens; teachers should teach to prepare students for the future. She
believes that creating discussion and learning from one another makes a difference in
how students will be prepared for the AP exam.
Case Study of Ms. Furgeson. Ms. Furgeson is a mathematics teacher who earned a college degree in pure mathematics. Before she became a teacher, she was in the business industry for several years. She did not feel a passion for her job and for that reason, at the age of 45, she decided to make a turn in her career and become a teacher.
Her teaching career began in a middle school in Florida, but unfortunately she was not lucky enough to have a great start. She said, “I got hired in the middle of the year. I did not like the discipline; this wasn’t for me. There was graffiti in the classroom and gang issues. My own children urged me to teach high school.” Eventually, things took a better turn for Ms. Furgeson, and after she decided to make the switch into the high school level, she was fortunate to be hired by a high school in Fort Lauderdale; she has been
teaching there for 17 years now.
An interesting fact about Ms. Furgeson is that she wanted to apply for the NBC as
soon as she started to teach. She stated, “I could not do it the first year because you had to
134 teach – round it up to three years.” Nevertheless, Ms. Furgeson admitted that the purpose of going through the NBC process was mainly because of the bonus money that would increase her paycheck if she got such certification. Additionally, she explained that the application for the NBC was subsidized and “it seemed to me the only way I could make more money as a teacher.” Most important is that she realized that it was highly beneficial to experience the NBC process. She said, “It happened to be a great program as opposed to many masters’ courses I took that I considered meaningless.” Later on,
Ms. Furgeson was asked to teach AP Calculus in her second year of teaching high school.
She said she felt nervous because she had taken calculus many years ago, and she did not remember it all very well. However, after her first year of teaching calculus she decided that teaching such a subject changed the way she thought about students. She found working with motivated and hardworking students was very rewarding, and she repeated that she had not been very successful teaching low level students.
Ms. Furgeson believes that when students are hardworking and they put effort into the class, that becomes a gateway to success. She said that learning by doing or learning through practice is a key component to understanding calculus; “You have to learn by doing. Like somebody telling you how to throw a javelin or throw a football is not gonna help. You have to get out there and do it, and realize what can go wrong.” As part of her teaching philosophy, she stated that a person needs to be reflective; “I’m definitely into reflecting and I am also into changing my mind.” In order to check for prior knowledge, she likes asking questions by picking students. She recommends not asking the brightest or the worst students on a regular basis because the brightest may already know the answer and the worst might not. Picking a ‘C’ student is always a good
135 idea. Ms. Furgeson tests her students’ knowledge a great deal, advising, “Do not assume
too much about prior knowledge under any conditions.” She believes that frequent
testing is a way to evaluate her students and her own teaching. Interestingly, she noted, “I
have reservations about too much accommodation of learning styles. When they go to
college, they’d better open their ears and take notes; they need to learn how to
mainstream.” Additionally, she feels a teacher should be a ‘guide’ and not just do
everything for the students at this level of mathematics. She wants students to partially
teach themselves, as she noted, “I don’t really teach them that much because they really
have to do it themselves; I am not gonna cut their meat for them now that they’re in high
school.” She explained that this teaching strategy is useful because classes last for only
50 minutes, which means that when a concept is semi-taught, then the bell is almost about to ring, and students need to discover how to get the end result of such concepts.
Ms. Furgeson leans toward a more traditional teaching style. She is not a big fan of technology. Most of her use of technology concentrates in using the graphing calculator as it is an essential tool for the final examination. She does not like to incorporate PowerPoints or instructional videos. She believes they do not contribute significantly to learning. When it comes to project-based learning, Ms. Furgeson considers she is an old-school type of teacher. Once a year she implements a graphing project that consists of graphing 12 different graphs on paper and corroborating results with the graphing calculator. She said, “It is not innovative. It’s very old fashioned. I’m sure people were doing it in calculus classes in the 1940s.” She had tried a couple of other projects but they did not work; “they took too much time and students did not learn that much.” Furthermore, she believes that “calculus is very theoretical, and the old-
136 fashioned way of teaching calculus and all mathematics classes is the most effective way to teach.” An interesting comment that Ms. Furgeson made is that “you have to have good materials and understand your subject well” in order to make a transition from lower to higher order thinking questions. She believes that knowing the subject matter well helps the teacher to know how to effectively interact with students and ask different types of questions. For example, she likes improving the thinking process when solving a problem. She said, “You can’t do too much prepacking. They have to see how you think about a problem.” She does not rehearse any problems that she presents to the class, and if she happens to make a mistake, she stated that it is beneficial for students to see that mistake and correct it. The questions are generated by experiencing situations of this kind so there is no formula to start with lower order thinking questions and end up with higher order thinking ones. She considers prepacking is not important when it comes to teaching; she explained, “I don’t do a pacing guide too much even though at the beginning of the year, I know where I’m gonna be at the end of the year. I don’t know exactly where I’m gonna be in January.” It is important to note that many AP Calculus teachers follow a pacing guide suggested by the College Board, but Ms. Furgeson goes with the flow of her own class; she explained, “you can’t allow them to just swallow and not get a concept because then they just get immobilized.”
Ms. Furgeson thinks that teachers should identify textbooks that are good resources to help students and, if teachers see that students can be well taught with those books, then they should keep them as a reference and a tool for their teaching. She explained that the county has adopted textbooks that are awful, saying that “the development of the material is ridiculous.” She explained that new books sometimes
137 bring new topics, and perhaps these are topics that come and go. Those topics that never
change, such as the Fundamental Theorem of Calculus, are the ones that are significant
and worthwhile learning, and calculus is a subject that is quite immune to other changes.
When it comes to dealing with real world problems, Ms. Furgeson’s philosophy to
prepare students to face this type of mathematics work is to teach them how to read. She
stated, “you can give a kid a math problem that is entirely a math problem and they’ll do
it. As soon as you veil it in words, many of them will panic.” She thinks the most
important strategy is to teach them how to read and understand what is happening before
they proceed to calculations. She believes that she does not have as much time as she
would like to have to work on real-world applications and for that reason, she tries to concentrate on physics problems where students can see the rate of change for the derivative. She reiterates that learning how to read is paramount to work on problems of this kind. For Ms. Furgeson, theory is always more important than concrete elements, and
as she explained, “theory takes you anywhere.” She added, “Students need to understand that calculus is theoretical – let’s go back to Algebra 1A or 1B; too much use of manipulatives is just turning them into low level students.” Ms. Furgeson has never seriously used any kind of manipulatives to teach AP Calculus. She used pecans and potatoes to show the use of solids of revolution but only for 5 minutes and then just quickly turned to the theoretical part of the lesson. She is aware that other mathematics teachers do incorporate manipulatives to teach, but she realizes that her kids do not need them to understand math concepts; she said, “my kids always understand by thinking
about it.”
138 Ms. Furgeson believes that the use of humor is a positive strategy to create a
climate of respect among students. She said, “I make fun of my students all the time,” but
she selects the smartest students so that everyone can feel comfortable. She does not want
students to feel the pressure of classroom rules. Her students do not need to ask for
permission to go to the bathroom; they can just go regardless of the type of class she
teaches, whether that is a class of upper or lower level students. Ms. Furgeson even
allows them to eat in class. The only rule she has is that students are not allowed to listen
to music or pull out their cell phones. She does not believe in imposing rules in the
classroom; as she said, “it’s a way of gaining mutual trust,” and she knows it may take a
couple of months to create this kind of atmosphere as well as creating an atmosphere of
total respect for students’ ideas and opinions. In contrast, she stated that she yells at
students when she sees them off track. She said, “It’s very natural for them to start talking about other stuff; I don’t allow them to go off task.” She admitted that she is known to be a ‘yeller,’ but she yells in a way that students are not afraid of her, and they understand this is an approach to go back to doing their work. On Mondays, she likes to pair up students who struggle with students who understand. She likes creating this idea of collegiality. Additionally, the idea of working in groups is very important for
Ms. Furgeson, and she explained that “it’s so reassuring to have somebody next to you if
there’s so much work in calculus that you don’t understand when you start, and you don’t
even know how to start.” She does not like big groups, so everyone has a say. She wants
to see that every single member in the group participates and contributes to the group.
Ms. Furgeson explained that she assesses her students on a regular basis; “I’m like a natural assessor. I didn’t learn it so as to be a teacher, I was just born – and I come from
139 a family of teachers.” She said that she likes having ‘calculus conversations’ with her students individually in order to see how much progress they are making. She considers this as a way to informally assess her students.
Training her students to learn how to read and interpret multiple choice questions is necessary right from the beginning of the school year. She keeps a huge test bank for students to practice and get acquainted with the AP style questions, and as stated before,
on Mondays, she pairs up students to work on multiple choice questions as well as pairing up students who are good with the calculator and those who are not. Later on, she makes the transition of training her students on how to work different ways and evaluate various routes on how to solve free response problems. She considers multiple choice and
free response AP style tests to be excellent assessment instruments. Nevertheless, she
believes free response questions are paramount because, as she said, “I give them the free
response and not only do I see what their answers are, but I’m seeing how they do their
work. That’s when I can tell what they don’t know.” She can even evaluate if students
are having problems with basic concepts from Algebra II, and from that point she decides
if a basic or previous concept needs to be retaught. She cannot see such deficiencies on
multiple choice questions. When students are falling behind, she does provide students
with an opportunity to retake quizzes and tests as long as they are paying attention.
Additionally, she makes herself available every single day of the week at lunch time and
twice a week after school to provide additional support to students who need help. She
stated, “I haven’t had a lunch all year long.”
140 Table 14
Summary of Ms. Furgeson’s Teaching Approaches
Design * Teacher and students work in a collegial manner. * Students “learn by doing.” (learning through practice and hard work). * Teachers should reflect on their own practices. * Pick average students to ask questions when starting a lesson. Never assume students possess too much prior knowledge. * Frequent testing is good to evaluate students and your own teaching. * At the calculus level, it is important to help students with deficiencies to mainstream in order to be prepared for college. * Teacher does not teach it all. Students need to discover and teach themselves as well.
Implementation * Technology is not important to make students understand calculus. “Videos are too slow. I can do the same on the board as what a PowerPoint does.” * Graphing calculators are essential in AP Calculus. * Old fashioned project on paper is a good way to learn. “People learned a lot in the past from that project.” Projects sometimes are not beneficial to calculus learning as they take time and students do not get much out of them. * Teaching calculus the old-fashioned way is the most effective strategy. * It is beneficial to improvise when solving calculus problems. If the teacher makes a mistake, the class should solve it and benefit from it. * Pacing guides are good as references but every class is different so it is more important to see how a class reacts to the AP Calculus content rather than follow a pacing guide.
Mathematics Content * Teachers need to identify which textbooks are in fact good resources to work with students and stick with them (as supplementary resources) when the county adopts new books. “Sometimes new adopted books are a nightmare.” * Teach students how to read and interpret real life mathematics word problems. * Theory is always more important than concrete elements. Too much use of manipulatives deteriorates mathematics thinking.
Classroom Culture * Using humor helps to create a good classroom atmosphere with respect. * It is not necessary to implement classroom rules. Teacher and students need to gain mutual trust. * Yelling is an effective approach to keep students on task. * Pair up students to create a collegial and productive working atmosphere. Small groups are beneficial for learning as long as every member participates.
Assessment * Informal and ongoing informal assessments having calculus conversations individually. * Start working at first on how to read multiple choice questions; then make a transition in working different ways to solve free response questions. * Free response questions are an excellent tool to see if students have calculus or previous mathematics content deficiencies. * Provide retake opportunities for those students who are falling behind. 141 It is quite remarkable to see how much Ms. Furgeson cares for her students. She is
also aware of the high demands of her high school and of her students taking too many
AP classes at the same time or having jobs after school, so she is very careful about not
stressing her kids out by giving them too much enrichment. Reteaching and retaking is a
formula that has worked for her. Finally, as part of her teaching philosophy, she believes
that “anybody who does their work can pass the AP exam” and she reiterated,
“everybody can pass.”
In summarizing Ms. Furgeson, she is a teacher who believes that students will
succeed through their hard work. She thinks a teacher gets better while reflecting on his
or her own practices. She wants students to be prepared for college, and she wants to
make them aware that when they get to college or graduate school, they will face an
academic world where they won’t receive accommodations for their learning. She
strongly believes that calculus is very theoretical and that project-based learning or manipulatives do not contribute a great deal to the learning of calculus. Teaching calculus the old-fashioned way is the most effective way but it is highly important that the teacher has a great command of the subject content. She believes that working in groups is a way of making students feel supported by one another to work toward success and she emphasizes the importance of continuously assessing students. If a student is falling behind or failing, she is a strong believer that providing new opportunities for students to succeed is a key to success on the AP Calculus AB Exam.
142 Case Study of Ms. Morrison. Ms. Morrison is a person who does not come from a teaching background. At first, she decided to study psychology at Florida International
University (FIU), but she was not fully convinced if that was what she wanted for her future professional career. Later, she decided to change careers and switched to engineering but she was unsure whether that would have a better fit in her career. She truly did not know what she wanted to do with her life. One thing she knew was that she has “always been good at math without trying” as she said, and that opened her mind to take another turn in her career. Then, she decided to go for a degree in mathematics education at FIU. After a few years of teaching mathematics, Ms. Morrison became interested in either going for a master’s degree or trying to accomplish the NBC process.
She opted to go for the NBC and admitted that originally she went for it because of financial benefits. To her surprise, she never expected to learn as much as she did by going through the process of becoming an NBCT.
Ms. Morrison’s teaching philosophy is based on “making the students see the whole picture in mathematics” as she said. She wants students to take what they learn in the classroom to other classes, subject areas, or situations where they can see the application of mathematics. She believes it is very important to talk to students individually and as a whole group while making sure that no question is unanswered.
Modeling and interaction should complement each other in order to deliver proper instruction. Ms. Morrison stated, “I do reflect. I think one of the things the National
Board did teach me to do just naturally on a daily basis is to reflect.” She always looks for ways to improve her lessons after reflecting on what she has done, and she is always trying to see what can work better the following year.
143 Ms. Morrison stated that before she starts teaching a new concept, she likes to
revisit old topics or concepts that may be necessary to understand the new lesson. She has
a separate board (sideboard as she calls it) on a side wall of her classroom that she utilizes to review old content. She said, “I go to the sideboard of my classroom when it’s something that’s prior knowledge.” She likes to assume that students are always forgetting something. She does not take for granted that when she starts a new lesson, they remember all previous concepts to perfection. She also likes to take a look at previous assessments to see how they performed, and she uses them as indicators to see where her students stand. Additionally, she likes assigning questions and walking around the classroom to see who is getting the right answers and who is not. She takes reviewing very seriously. She adds, “…sometimes I have to make two sweeps around the room to verify that they’re with me.” Ms. Morrison admits that she does not know each individual’s learning style, but she can always predict who is not going to get a passing grade on the AP exam. After targeting those students in danger of not passing, she pulls them aside and tries to convince them to visit her during lunch time; however, they must make an appointment to see her in advance. She said that she does not have many students under these circumstances as she works in a high economic high performing school, but still the one-on-one tutoring strategy has worked very well to have her
students succeed.
In regards to her teaching style, she considers herself a ‘traditional teacher.’ Her
approach is to use whole lecture and complement her lessons with group work activities.
She said, “I write my notes on the Promethean Board and then I upload all those notes
onto a website.” She has access to a website called “Weebly”, which is a free website for
144 teachers, and she has the capability to upload documents and files with all the notes that
she writes on the Promethean Board daily. Whenever Ms. Morrison teaches a lesson, she
likes to break the lessons into different components. She teaches part of a section, and
then she gives students a quick and short assessment or a group activity. She stated that it
is important that a teacher knows his or her objectives well so that the chunking of the
lesson can be properly delivered. She said, “…if for example I’m teaching section 4.1, I
don’t plan on teaching all of it in one day. I break it up into three days of parts that they
would get okay. They have to get each part before they go to the next level – it’s a
building process.” She added that having taught AP Calculus for several years helps her
to predict when students are going to have difficulties facing a new topic, or when she
needs to do a quick review.
When it comes to incorporating technology in the classroom, Ms. Morrison is a
teacher who sets the example for many educators. She possesses a wide repertoire of
technologic tools that she utilizes on a daily basis in order to improve her instruction. “I
use a lot of applets and a collection of websites” she said, and she likes the ‘object
moving interaction’ to teach various calculus concepts. Additionally, whenever she
discovers something new, she likes to share it immediately with other AP Calculus
teachers. She very much enjoys taking workshops with other teachers and maintaining
regular communication with them to exchange ideas and discuss ways to make lessons
fun and more interesting. This clearly depicts a reflection of NBPTS standards represented by her teaching approaches. PowerPoints are also part of her technology repertoire; however, she does not incorporate any instructional videos like Khan
Academy or others. She has a different approach to the use of graphing calculators. She
145 considers they are ‘a must’ to incorporate in AP Calculus, but she changed the way she
incorporates using the calculators. She used to introduce graphing calculators at early
stages, but now she waits until the month of January. She prefers to have her students
develop more mental mathematical thinking during the first four chapters, and then in
January she introduces the use of graphing calculators. She said, “they pick it up very
quickly, there aren’t many commands that they need to know.” Ms. Morrison admits that
she is not highly creative with projects. She would rather utilize other teachers’ projects
that she knows will have a positive impact on students’ learning. She waits until near the
end of the school year to incorporate a ‘volume’ project, which is a group project that
consists of having students calculate volumes with cross sections as they attempt to find
the function. She explained, “They have to figure out the width of each of those pieces
and they have to come up with the area by integration versus the exact area.” She likes
the idea of having a 3D model for students “to touch and see” and she has taken on this
project now for the past few years based on the recommendations of other more
experienced AP Calculus teachers who think this project is beneficial as some students
may struggle with understanding and visualizing concepts of cross sections.
Ms. Morrison’s questioning strategies are based on a gradual progression checking on her
students’ prior knowledge. Before asking questions, she likes students to write things
down and identify “what we are looking for” as she models the lesson. As the lesson progresses, she aims to evaluate how much her students know; then she decides if she will ask a lower or a higher order level question. When she finally sees that her students have understood the concepts, then she starts delivering more complex questions that lead to critical thinking. For Ms. Morrison, any lesson that encompasses any kind of physics
146 applications is extremely important for the developmental needs of the students. She said,
“…a lot of them are taking physics, so a lot of them can connect the physics concepts
fairly quickly, for example, when we talk about acceleration.” These are the types of
lessons that have a higher degree of importance for her.
Ms. Morrison also believes that calculus is a theory based subject and sometimes kids have problems seeing the connection between the theory and the real world applications. She said, “They rarely ask, when are we going to use this? I think our students are overwhelmed.” She explained that nowadays her students do not ask the question of ‘when are we going to use this’ as they used to; they just want an “A” and they want to obtain the college credit. Regardless of whether the students ask or not, whenever she plans her lessons, she believes that any content that can be applied in other areas is a meaningful lesson. Even though she is an advocate of real-world applications, she still believes that theory is more important than concrete practice at this level of mathematics. She is a supporter of incorporating manipulatives or interacting with geometric objects at earlier stages of the mathematics learning process but at the calculus level, your kids have to open their minds to start thinking abstractly. She stated, “Math becomes very abstract as you go up in the levels, I don’t know what manipulatives to use.”
Ms. Morrison’s approach to creating a classroom atmosphere where everyone feels respected is by making students feel that they are respected by the teacher as individuals first, and then they tend to give her respect back. She said, “Respect is mutual.” She does not feel comfortable reprimanding students, and when a student says something wrong, as she said, “I give them a look or go talk with them.” When it comes
147 to have her students voice their opinions, she always ensures that the students’ answers
and opinions are always welcome. If a student gives a wrong answer, she does not say
“you are wrong” but instead she would say, “you are on the right track, but let’s try this instead.” This is how to establish an environment where everybody feels safe to express themselves. Furthermore, Ms. Morrison has students who may not participate on a regular basis, and for that reason, she wants to provide positive feedback to those who rarely participate to encourage them to keep on participating and bringing ideas to the class. She is aware that sometimes she makes mistakes but she feels rewarded in seeing that her students are able to correct those mistakes and learn from them.
Walking the classroom is a strategy that has beneficially impacted Ms. Morrison’s classroom atmosphere. She believes that all teachers should walk the classroom to make sure that everything is under control. Moreover, monitoring students contributes to a more positive impact in the way that students behave. Ms. Morrison likes having a plan in place when arranging students to work in groups. She assigns and distributes work to everyone in the group with the purpose of having every single student busy on a specific task; and she continues to walk the classroom to check that students are doing what they are supposed to. She stated, “You know, it is AP Calculus and you get the crème of the crop” meaning that students will naturally work, stay on task, and behave properly; however, the classroom needs to be monitored often times. As students work together,
Ms. Morrison believes that it is important to create different groups at all times, sometimes homogenous and sometimes heterogeneous groups. Students need to adapt to work with different people on a regular basis. Moreover, this group making variety creates more fairness by not constantly having a high performing group and a low
148 performing group that work together at all times. Ms. Morrison said, “It’s good to create variety.” One example of cooperative learning in Ms. Morrison’s class is when students attempt to work and solve free response calculus questions. She said, “Through the free response problems, especially at the beginning, they work in groups as they have never read those problems before. They’re very intimidating.” Her strategy is to create support with these types of questions that they may be unfamiliar with. Cooperative learning is a very important practice for Ms. Morrison, but as much as she enjoys seeing her students working in groups, she confessed that she may incorporate this kind of practice only about once or twice a month. “Time is a constraint and there’s a lot of curriculum that I have to present.”
When asked about the types of assessments that she incorporates in her curriculum, Ms. Morrison readdressed her habit of walking the classroom; this practice provides her with the opportunity to see what students are doing individually and to observe if they understand the concepts or not, especially those shy students who would never participate or raise their hands to ask questions. She sees this strategy as a way to informally assess her students. Her formal assessments consist of quizzes and tests. She stated, “the formal is at least once or twice a week while they’re getting some type of formal feedback whether it’s a quiz or a test.” She added, “On quizzes, they have a reflection piece that they have to turn back in. They have to reflect on their learning.
They not only have to correct their mistakes but also to analyze and reflect on what they could have done better.” Ms. Morrison utilizes various instruments to make sure that students are on track. She uses the College Board website and has her students solve the latest online recommended practice problems. Additionally, she has loads of free
149 response exercises that she has collected all through the years. This combination of
exercises, in addition to quizzes and tests, are indicators of how students are performing.
For their midterm and final exam, she uses portions of the real AP Calculus exams given
in previous years, and she believes that these exams serve the purpose of depicting if
students are ready for the exam or not. Whenever she sees that students are under
performing, Ms. Morrison assigns extra practice and monitors students more closely. If
students are willing to work for remediation, then Ms. Morrison suggests that students
attempt to solve double the number of problems assigned for homework. Then she tries to
work with them individually as long as they come up to her to ask for help. She also said,
“if there’s some type of assessment that everybody did poorly on, then I reassess.”
In summarizing Ms. Morrison, she is a very passionate teacher who believes that
teachers should continuously cooperate with each other to become better rounded in their
instruction. At the same time, she believes this interaction is key to leading students to
success on the AP Calculus AB Exam. She believes that calculus becomes more
meaningful and interesting to students when they see that the concepts can be applied to
physics or real world situations. She believes that reflecting on her own practice is
beneficial to improve her teaching.
Ms. Morrison is incredibly versatile with the use of technology. She incorporates multiple technological tools in order to accomplish her overall goal of ‘making students see the whole picture in calculus,’ which is the basis of her teaching philosophy. She is
very open minded to listen to other teachers’ suggestions and always willing to improve
herself as a teacher. She strongly believes that attending workshops is an effective way to
grow.
150 Table 15
Summary of Ms. Morrison’s Teaching Approaches
Design * Math should be taken and applied in other subject areas and in different real- world contexts. * Communicating with students individually and in groups is highly beneficial to effectively learn. * Teachers need to reflect on a regular basis * Teachers need to produce better lessons from year to year looking back at what they did from previous years. * A sideboard to review old concepts is a good tool starting a new lesson. *Always check on prior knowledge and never assume students are 100% ready for the new lesson. * Students in danger of not passing the AP exam must make an appointment to have one-on-one tutoring during lunch time. * Teacher has her own website and publishes her classroom notes. She uploads files and documents continuously. When teaching a lesson, it is important to divide it in different chunks and not teach the whole concept in just one day.
Implementation * Use of Promethean Board. * Use of various applets and multiple websites to improve her lessons. * Share tech resources with other AP Calculus teachers. * PowerPoint presentations are useful. * Graphing calculators are a must. * Put emphasis in teaching in physics applications when teaching AP Calculus.
Mathematics * Lessons are meaningful when their content can be applied in other Content areas/subjects. * Theory is more important than concrete elements at this level of mathematics. * Students should adhere to abstract thinking.
Classroom * Respect starts with the teacher. When students feel respected, they give Culture respect back. * Students’ ideas and opinions are welcome. Try to provide positive feedback even if the answer if wrong, especially with those who rarely participate. * Walk the classroom. Monitor your students for assistance and also to make sure that appropriate behavior occurs at all times. It should be a natural trait in each teacher. * Assign specific work to each student when group work takes places. Walk the classroom to ensure that students are on task. * Create different groups to accomplish work. Have students adapt to different groups over the course of the year.
Assessment * Walking the classroom is a way to check on students’ performance individually and can be considered an informal way to assess. * Formal assessments consist of quizzes and tests. Students must reflect on their mistakes and analyze what could have been done better. * Use AP Calculus exams for midterm and final exams. * Exams, quizzes, and College Board exercises are a great tool to examine if students are at the desired level and ready to pass the real examination. Assign extra practice to those who are underperforming. * Provide reassessing opportunities when most of the students do poorly on a test or quiz. * Provide enrichment activities for those who are succeeding.
151 Walking the classroom is a strategy that has multiple approaches in Ms.
Morrison’s classroom. She uses it as a way of monitoring students to have them
accomplish work, to behave properly, and to assess them informally. Under her
philosophy, all teachers should monitor the classroom on a daily basis. She is a strong
believer that students should learn from their mistakes and grow as they reflect on those
mistakes. Ms. Morrison is also willing to help and provide opportunities to those who
need additional support as well as encourage them to work harder in order to accomplish
success on the AP Calculus AB Exam. She really advocates student success.
Case Study of Mr. Glenc. It is quite a rare event to encounter someone with so
much enthusiasm about the teaching profession as Mr. Glenc. Mr. Glenc is from Indiana.
He graduated from Ball State University with a master’s degree in secondary education
with a minor in mathematics before starting his teaching career. His face shines with joy,
and perhaps with some pride, when he talks about his background and how he made himself the type of educator he is today. Mr. Glenc is currently teaching AP Calculus, but he had a very different career start compared to what his current daily teaching job looks like. He started teaching low level classes. He said, “I taught for six years in Indiana with low-level classes because that’s what you start with, it is at the low level in the totem pole.”
Later Mr. Glenc moved to Florida with his family, and he was hired by a high school to teach geometry. He considers moving to Florida as sort of an upgrade in his teaching career; he stated, “I was teaching geometry, which doesn’t sound that high level but at that time for me it was like, wow this is so much better!” Moreover, in this new school where he was teaching, he saw the opportunity to teach high level classes if he
152 displayed a good performance. Gradually, his objective materialized, and he moved up to
teaching higher level classes such as Algebra II and Algebra II Honors.
After six years of teaching at this high school in Florida, he got hired by a high
socio-economic and high achieving high school where he has been teaching for 24 years.
Mr. Glenc currently teaches AP Calculus and Pre-calculus at this school, and he has
accomplished a great deal of success in having most of his students pass the AP exam
year after year. He adds, “I have probably the best schedule of any math teacher in the
country – I have two AP Calculus and four pre-calculus classes.” He indeed feels very
lucky to be in the position of teaching the best students in his school.
Mr. Glenc is another educator who has gone through the NBC journey because of financial benefits to increase his paycheck. He admitted, “I did it strictly for the money.”
It is interesting to note that Mr. Glenc does not consider that the NBC made him a better teacher. He explained, “… when I applied, I had been teaching for over 20 years. I tried everything, and I had been tuning and refining what I had done; you know, after so many
years I’ve tried it all. So I don’t think the National Board did anything except get me a lot
of money.” Mr. Glenc also thinks that he had already mastered the reflective part of the
teaching career even before going through the certification process. He is a person who heavily relies on the experience factor to make a teacher a better educator. He described his teaching philosophy as “most kids can learn up to a certain level.” Under his
philosophy, there are students who can do basic mathematics and they can perform
mechanically without understanding. He stated, “Everybody can do Algebra I or
Algebra II and go through them, but as far as actually understanding what they’re doing
or having the knowledge they need to have, I don’t think all kids are capable of doing
153 that.” He believes that facing AP Calculus is not for everyone. He believes that even
students who have been ‘A’ students during their entire school career may not be capable
of handling AP Calculus concepts. He said that four to five weeks before the real AP
examination, he knows who is going to succeed or fail. For those who are not performing
at the desired level, Mr. Glenc offers additional support in the early morning before
school starts and monitors those students in need more closely during regular
instructional time.
Mr. Glenc thinks it is important to review concepts of Algebra II or Pre-calculus
in the event that students are to face a lesson that requires the knowledge of topics that
they have not utilized in quite a while. Reviewing helps to check on their prior
knowledge and to ensure they are ready for the new lesson. He would rather do lesson
review and have teacher-student conversations than give an assessment to check on
students’ readiness. He said, “I know they’ve already forgotten, and there are times that I
get frustrated, but I try my best not to let it show.” In order to accommodate different
learning styles, Mr. Glenc incorporates supplemental calculus books such as ‘Calculus
for Dummies’ and other materials that can be good resources available to those students
who encounter challenge with calculus concepts. In that way, these students can work on
their own time and have some additional support to figure things out. Furthermore, he
assigns classwork as a strategy to assist students. He explained, “I give them work and let them start in class so that if the kids can take off, you can take off. I’m there from kid to kid to kid just walking around helping them. Part of his teaching philosophy as he described it is, “‘I never say no. I don’t give an assignment, sit down and do whatever like checking my e-mail. That doesn’t happen.” Mr. Glenc stated that some lessons are
154 very long and may take the entire 50 minute class period. In order to provide students
time to make sense of the taught material, he allows time for students to work together
and solve problems. If there is no time on a specific day, then he will assign problems to
take home and start discussion the next day. He knows there are kids who will not ask
him any questions; however, they will work with their ‘smart buddies’ as he describes. So
he starts the day assigning fifteen minutes to work in pairs, and then he allows time for
questions and discussion. Mr. Glenc also added, “At the level I teach, parents can’t help
98% of the time, and for that reason I want them to go home and see if they can do the work I assign.”
Mr. Glenc stated that the use of graphing calculators is highly important. He believes, “the sad assumption by most of us is that kids know how to use their calculators. They don’t. It is really odd to me how trusting they are and how willing they are to play on that cell phone, but when you give them a calculator, they don’t even know how to graph a line.” For that reason, he thoroughly teaches how to use graphing calculators. Calculators are the only tech tool that Mr. Glenc incorporates in his AP
Calculus classes. Mr. Glenc is not highly engaged with the use of technology. He does not believe in other tech resources such as PowerPoint presentations, videos, or websites.
He said, “I’m the old dog that doesn’t believe you can learn new tricks. I don’t want to learn how to make a PowerPoint. I want to be interactive with the notes, writing notes on the board, pausing, and listening to what I am saying.” He reiterated that he is not a big
fan of technology. No project based learning is accounted for in Mr. Glenc’s classes. His
preference is to create a portfolio with the goal of having his students performing
extensive review. This portfolio consists of a wide repertoire of practice tests, and it takes
155 place right before the AP exam. Mr. Glenc wanted to emphasize that “I’ve never run into
another calculus teacher who said ‘oh, I tried this project and the kids learned so
much’…” However, he said that if he ever hears from any teacher in any state who has created a project that could have a huge positive impact in AP Calculus students’
learning, then he would be very open to try it out. The way Mr. Glenc builds on students’
knowledge to transition from lower to higher order question is to “Start with things they
already know, like vocabulary. Start with something simple and basic.” Then he
suggested making connections by using questions as ‘how do you use this?’ or ‘how does
this affect…?’ He structures his transition questioning method as a ‘repeat, state,
describe’ model parallel to the Marzano model. Even though Mr. Glenc is a highly
experienced teacher, he confessed that he would like to attend other schools where the
AP Calculus teachers have a passing rate close to a 100%, talk to the respective teachers,
and observe their classes to see what they are doing. He said, “I’d love to do that but we
just don’t have time.” He admitted that he does not know if his questioning strategies are the most appropriate but he said, “whatever I’m doing, it must be working” as in the end, most of his students are able to pass the AP exam and then when they get to college, they come back to visit and thank him because they are successful in more advanced college mathematics classes. When asked about the pace of his teaching and the developmental levels and needs of the students, he explained that when he began teaching AP Calculus for the very first time at this high performing school, there was a culture created by a previous head of the mathematics department. This person wanted every AP Calculus teacher to adopt at the same pace, teach the same material every day, give the same quizzes, tests, and related assessments. He followed the teaching pattern that was
156 essentially imposed on him for several years. He stated that through trial and error he was
able to identify the things that did not work for him and how the kids would struggle the
most. He said it took him about two years to comfortably catch up and identify how to plan for the developmental needs of the students for this particular class.
Mr. Glenc believes that the content of a lesson is significant and worthwhile as
long as students can get any true interest or excitement about the lesson as they learn it.
He explained that there are lessons where students see real world applications and can relate that learning with other classes they took or are currently taking such as physics for example. He added, “…some of the things you have to learn, that’s a rule; but some other things you can show real world examples and build their interest.” Mr. Glenc likes to talk about volumes, rotations, parabolas, video games, policeman speed radars; he likes to build three dimensional things and incorporate instructional elements that are bound to increase students’ attention. It is evident that Mr. Glenc likes to bridge his lessons making appropriate connections with real world situations. Moreover, he thinks that vertical teaming is important to prepare students appropriately for the next level of math. He would like to see calculus teachers talking to pre-calculus teachers and discussing what students need at the next level. In the same way, he would like to see pre-calculus
teachers talk to Algebra II honors teachers to let them know what is expected in pre-
calculus. This vertical teaming philosophy is a strategy that, according to Mr. Glenc,
could place students correctly in AP Calculus at early stages. He believes that sometimes
there are a number of students who should never have taken AP Calculus due to
deficiencies which had originated in previous classes.
157 Mr. Glenc has an interesting way of seeing theory and practice. He believes that theory becomes more important than practice based upon the future mathematics paths that the students will follow. As he said, “It’s all individual kid-based.” He explained that if a student is going to follow a career that requires taking business calculus, then the student needs to see the application more than the theory. On the other hand, if that student will do something research based, then the theory becomes very important.
In order to ensure that students’ ideas and questions are welcomed and respected, even if a question is too basic and students are supposed to know the answer, he would say,
“that’s a good question.” He would never say, “you should already know that” as he wants to create a classroom atmosphere where all students feel welcomed and respected.
Mr. Glenc added, “I try to be personal with them, concerned about them. Sometimes there are kids that who sit there and you never even know their voice; so when you’re engaging with them, you create an atmosphere not of friendship but of ‘I’m concerned about you.’”
Even though Mr. Glenc teaches for a school with high performing students, he is faced with the reality that sometimes he finds himself in need of sending a student out of the classroom. When working in groups, sometimes he may observe some student being off task and disrupting the collaborative work experience. Then he sends the student out of the class and directs him or her to stay in a lower level class. He said, “They don’t like that and all it takes normally is just one time of that; the good kids don’t want to do that.”
This is a strategy he uses to set the example and keep students behaving properly and working on task with their respective groups. He stated that mostly this situation can be caused by seniors who came into AP Calculus as a last choice. Mr. Glenc added,
158 “because there’s some of them that they’ve already gotten into their school. They don’t
care. They’re guaranteed one. They’ll do nothing if you let them.” He also removes their
cell phones if they happen to be using them while they should be working.
An activity that Mr. Glenc likes to integrate when working with groups is to have
students teaching each other how to use the graphing calculator. He added, “…there are kids who have new types of calculators like the TI-Nspire; I’ve got no clue how to use them.” He also indicates that it is a natural reaction for his students to get together in groups without being told to do so when they are reviewing for a test. As students work together, Mr. Glenc goes from group to group and asks questions of each group member, sometimes even individually in order to assess their knowledge. According to Mr. Glenc, cooperative learning is an important teaching strategy to incorporate but requires the teacher to be alert to what is happening as the students do the assigned work. He stated
that it may bring the disadvantage of having students getting distracted and going off
task. He said, “…the disadvantage is that when you talk to a group, you need to turn your
back to make sure that the other groups are working; occasionally you have somebody in
a group teaching other students how to do something which is wrong.”
Mr. Glenc implements an assessing plan that is set up in two different stages
during the school year according to the frequency of how students are exposed to tests and quizzes: (1) assessing students until the middle of April, and (2) after mid-April
during the weeks leading up to the AP exam. Regarding the first assessment stage, on
average Mr. Glenc’s students receive a quiz or a test every six to seven days. The second
stage comes when the time gets closer to the real AP exam and there is a change of
strategy that consists of exposing students to daily assessments. Nevertheless, he plans
159 for approximately three and a half weeks of continuous daily review as he thinks this type
of extensive review can be mentally exhausting. He stated, “...after two and a half weeks
of reviewing, students are already burnt.” To supplement these daily assessments,
Mr. Glenc provides a review package with AP style problems. He also likes to work on
previously given AP Calculus exams which are published on the College Board website.
As far as informally assessing his students, Mr. Glenc uses not just the results of quizzes and tests as evaluation instruments, but he also examines the results in his daily discussions, thereby covering the different types of problems to be solved from the review package. In a similar fashion, he evaluates how well his students performed on the previous College Board tests during instructional time in addition to the type of doubts and questions that may arise from the students. He likes to continue assessing his students by making himself available in the early morning before school starts or during lunch time. Then he can get an even better understanding of the deficiencies that some students in particular may have.
Mr. Glenc’s main remedial strategy to help those who are not performing at the desired level is to meet with every one of those students and pick the tests and quizzes where they made the most mistakes. Then he has his students try to identify what the problem was; his goal is to see if the student can identify what the mistakes were. It is evident that this strategy requires a lot of personalized attention, but as stated before,
Mr. Glenc is a teacher who makes himself available before school, sometimes during his planning time, during his lunch time, and whenever possible after school. Moreover, he considers this is a great strategy to prepare his students for the major tests. When this
160 type of review occurs during instructional time, Mr. Glenc likes to have students
reviewing their mistakes in groups.
One of his goals as a teacher is to make students see that he is helpful for whenever they need him. He wants his students to feel that they are always going to be supported and accommodated and that their requests will never be ignored. As he said, “I
would hate my students to say that I am not working enough for them.”
In summarizing Mr. Glenc, he is a highly motivated, self-confident, and enthusiastic teacher. He possesses immense experience in the teaching profession, and he believes that experience is what makes a teacher become a master teacher. He does not believe that the NBC makes a teacher grow professionally or pedagogically. Mr. Glenc came into the teaching profession with a master’s degree in mathematics education and was able to accomplish his goal of transitioning from teaching low level students to currently teaching highly advanced motivated ones. Even though he utilizes older and traditional teaching methods of instruction, he always tries to bring the best in his students. He has been successful in having most of his students pass the AP Calculus AB Exam.
Under Mr. Glenc’s teaching philosophy, not every student is up to the AP Calculus level in high school. Those who really understand how to make appropriate real world connections with mathematics are the ones who truly understand the concepts and can grow and succeed academically at higher levels. Mr. Glenc does not concentrate in incorporating the use of technology for AP Calculus instruction except for using graphing calculators. He considers that technology is not highly beneficial to help students pass the
AP exam. He does not believe in project based learning either.
161 Table 16.
Summary of Mr. Glenc’s Teaching Approaches
Design * Help students in need and tutor them in the morning before school starts. Monitor such students more closely. * Revisit concepts of less advanced classes to ensure that students are well prepared to learn more new advanced topics. * Provide supplemental material/textbook, especially for those who struggle with calculus concepts. * Monitor students close and individually during instructional time. *Assign problems to take home to make sense of the lesson and start discussion on the following day.
Implementation * Incorporate the use of graphing calculators. Never assume that the student knows how to use it regardless of other technologies he or she may know. * Graphing calculators are the only technology component to be incorporated in teaching calculus. Old traditional teaching methods with no technology are effective to help students succeed. * No projects. Have students take multiple practices tests and create a portfolio of such assessments. Keep an open mind in case you see any projects from another teacher that may have impact in students’ learning. * Start lessons with basic and simple concepts. Have students make connections with new concepts and explain/justify how connections are made. * Follow the Marzano model. * Ideally, it would be great to observe other highly successful AP Calculus teachers at other schools.
Mathematics * Lessons that catch the students’ attention and interest are significant and Content worthwhile. It is important to make connections between calculus concepts and real world applications. * Vertical teaming is beneficial to better prepare students to be ready for the next level of mathematics. * Theory becomes more important than practice based upon the future career of the student will follow.
Classroom * Every question is always a good question. Respect students’ questions and ideas. Culture * Engage in conversations with students and show you care about them. *Remove a student if he or she is not at task, or disrupting cooperative learning activities. This strategy sets the example to make students aware of the teacher’s expectations when working in groups. * Group students to assist each other on how to use the graphing calculator. * When working in groups, walk the classroom and talk to each group. It is fine to ask questions to group members individually to assess their knowledge. * It is highly important that the teacher understand how to monitor how students work in groups to have the entire classroom on task and making sure the students are teaching one another without incorrect information.
Assessment * Assess students with quizzes and tests during the entire year. A few weeks before the real examination, provide daily assessments. * Offer review packages with AP style problems. Solve previously given AP Calculus exams. * Assess the problems students have in solving previously given College Board tests. Analyze the type of questions that students come up with. * Meet students in the morning before school or during lunch time in order to assess their deficiencies. * Review quizzes and have students identify their mistakes. In this way, they can be better prepared for major tests.
162 Mr. Glenc is caring and is always trying to show his students that he is interested in their
personal lives. He wants to let them know that he cares for them; however, Mr. Glenc is a teacher who likes to continuously see order and respect in the classroom. He is very strict when it comes to following classroom rules, especially while students work in groups. He entirely disapproves of students being off-task or disrupting other students when he uses cooperative learning. His passion for seeing his students succeed becomes evident in the way he makes himself available. He devotes a great amount of his school time in order to assist students who are underperforming or need extra help. His supplemental assistance is a remedial strategy that he likes to put in practice. In addition, Mr. Glenc likes assigning extra homework or supplemental books that can help students who struggle to work on their own to aid them. He believes in continuously assessing his students in order to ensure that they all get enough practice and remediation to be prepared for the
AP Calculus AB Exam.
Summary of the Findings from the Case Studies
The five NBCTs presented in these case studies came from either educational or non-educational fields into the teaching profession. All of these teachers have been teaching AP Calculus for several years, and throughout the different interviews, it became evident that they all found a passion for teaching. The case studies clearly revealed that these NBCTs have been making a difference in their students’ lives by helping them to succeed in school, on the AP exam, in their preparation for anticipated college careers, and to be competent citizens in the future. All of these teachers made their way to teaching AP Calculus after several years of teaching other mathematics classes. All of them started teaching low level mathematics classes, which evidently was
163 beneficial for them to gain teaching experience and become more effective at higher levels of math. Another commonality found among the NBCTs is that they presented themselves as very meticulous about their teaching methods and highly committed to their profession. An additional common characteristic that the researcher found is how focused and intensely each of the NBCTs presented themselves as they spoke about their daily teaching mathematics practices. They all seem to take the teaching profession very seriously and responsibly. Each of them also showed a great deal of concern as they spoke about their students’ success, but at the same time some of them displayed a great deal of pride as they revealed how much success they have produced by having their students pass the AP Calculus AB Exam year after year.
The decision to pursue the NBC is voluntary. Candidate teachers need to go through a very rigorous process. The process requires teachers to work on a regular basis beyond their teaching hours. It represents an opportunity to develop and grow as a well- rounded teacher. Most of the interviewed teachers expressed that they experienced growth by going through the certification process; except for one who mostly relies on experience in regards to growing as a successful teacher. Four of them found the NBC process to be a great program that helped them to grow professionally and become better and more productive teachers. Teachers are feeling a lot of pressure from the constant changing educational system, and it is important to note that all of these teachers have adapted successfully to the continuously changing teaching profession regardless of whether they came from a teacher training background or from other professional areas.
Additionally, these teachers had to pass a rigorous high level mathematics content test.
Passing this test required them to possess extensive knowledge of advanced mathematics,
164 concepts of physics, and statistics. Having a high level of content knowledge in these areas is a common factor among all of these NBCTs, and knowing their subject well is something that they have officially proved. The researcher also found that being humble about their knowledge of their content area was a trait that they all had in common. It is interesting to note that all of the interviewed NBCTs possess an educational level holding a master’s degree. Regardless if they pursued the NBC for financial enhancement or to better themselves as professionals, most of them found reward in going through the process, and they admitted that they became more aware of how their teaching performance could be improved. They also went through a very rigorous commitment of working beyond schools hours, evenings, weekends, or holidays in order to accomplish the required demands that the NBC application sets upon its candidates. Moreover, they have demonstrated that they have adapted their teaching styles to the NBPTS standards as they teach on a daily basis. These are common patterns that were observed among the five NBCTs as they shared about their instructional methods.
The information acquired from the interviews depicts, specifically calculus instruction related, that all of the NBCTs do not spend too much time on projects. This commonality was found based on the teachers’ rationale that AP Calculus is a very theoretical course. They expressed that AP Calculus is not a class that allows much time for project based learning due to the massive volume of abstract concepts that are part of the College Board curriculum for this particular class. It also became evident that in general these NBCTs do not incorporate manipulatives or concrete objects in their daily instruction with the exception of one or two occasions during the entire year when they have their students build a model and work on it as a one time project for a specific
165 activity. However, in AP Calculus there are many physics applications that make an
immediate connection with real-world situations; these teachers described how having their students see calculus connected to real world applications makes those lessons worthwhile and significant. Moreover, the researcher noticed that most of the NBCTs possess a very ‘caring’ trait when it comes to helping their AP Calculus students. These are teachers who, on a regular basis, make themselves available in multiple ways in order to provide additional support and help their students succeed on the AP exam. It became evident that they utilize this support strategy, which often involves working during their non-instructional time, as a strategy for remediation. They all clearly proved that teaching does not begin or end during school teaching hours; they go above and beyond to aid their students and continuously provide additional support. Furthermore, these NBCTs incorporate multiple types of ongoing formal and informal assessments that help to ensure that students are continuously monitored on their performance and mastery. This also helps to keep their students on track. It has been observed that for those students who are falling behind, all of the NBCTs implement differentiated strategies in order to support and reinforce students with subject content.
There are variations in their teaching approaches regarding the way the interviewed NBCTs see the impact, benefit, and effectiveness of incorporating technology to teach AP Calculus. For three out of the five NBCTs, technology is an effective and interactive way to help students visualize abstract concepts and materialize what they learn. For the other two, traditional teaching techniques without technology are seen as more beneficial to get students to understand more effectively. The researcher noticed that the younger NBCTs were more likely to be in favor of incorporating
166 technology into their AP Calculus instruction. The older NBCTs prefer and implement
instruction using traditional teaching methods. Regardless of their teaching approach,
most of their students do meet success in passing the AP exam. Their teaching approach
is inevitably subject to focusing on the use of graphing calculators, which is a
requirement for every student. Students are expected to use a graphing calculator when
they take the AP exam; it is recommended for them to have graphing calculators
throughout the entire school year in order to develop and practice the proficiency skills needed. In regards to cooperative learning, all the NBCTs incorporate group work in their
AP Calculus instruction, and they all see the benefit of utilizing this mathematics instructional practice. As students work in groups, the researcher noted that the NBCTs opt to set a classroom atmosphere in which students feel that they are being monitored, supported, and provided with the opportunity to be assisted by their own peers. Overall,
NBCTs set up rules in order to gain mutual trust and to create an effective classroom atmosphere where everyone is respected.
In conclusion, even though these teachers have various teaching approaches in common, they all have proved to be effective educators who continuously ensure that students get the most out of their AP Calculus AB instruction. It is true that some of these
NBCTs went through the certification process in order to achieve an increase in their salaries; however, their passing rates depict their effectiveness in preparing their students
to succeed on the AP Calculus AB Exam. Even though these teachers utilize slightly
different teaching approaches, particularly with the use of technology, regardless of their
teaching philosophies and strategies, they are highly successful teaching AP Calculus
AB, and they annually accomplish their goal of having most of their students pass the AP
167 Calculus AB Exam. Due to the teachers’ strong content knowledge, teaching approaches,
and their NBC status, these may be considered good indicators related to student success.
Figure 12 summarizes the similarities found in specific components among the interviewed NBCTs in regards to their behaviors, commitment to the teaching profession, and pedagogical approaches.
The NCTM (2000) noted that their principles and standards outline the essential
components of high-quality school mathematics and emphasize the need for well-
prepared teachers. All of the NBCTs proved to be well qualified for teaching AP
Calculus and also to have accomplished a graduate level of education by holding master’s
degrees. According to Potterfield and Majerus (2008), both the National Science
Education Standards and the NCTM standards recommend appropriately incorporating
the use of mathematics in other disciplines. The NBCTs clearly emphasized the
importance of applying calculus concepts to physics applications and to other areas. One
of the standards of the NCTM relates to the learning environment. Slovin (1996) stated:
The learning environment refers to the classroom context and conditions in which
mathematics teaching and learning takes place. It encompasses all the conditions
that structure the social aspects of classroom life as well as the view of
mathematics and learning that is promoted. The environment is shaped by the
tolerance for diverse thinking and ideas, the values about mathematical thinking
conveyed, pacing and timing of activities, and interpersonal relations. (p.10)
168 They are passionate Possess caring about their teaching. traits, which Highly committed to clearly reflect in making their students their actions succeed They all have master's differentiation degrees strategy
trait trait Provide continuous support, especially for level of education those in need They posses high level of adapting learning content styles knowledge math knowledge Commonalities
found among adapting teaching Have adapted their the NBCTs styles teaching styles to the national standards of the NBPTS They do not concrete objects applied AP Calc incorporate concepts manipulatives in their instruction group work mathematics They expect their content monitoring students to make connections between AP Calculus and real world applications Incorporate Not too much cooperative room for project They believe AP learning based learning Calculus AB is a very Assess students on strategies theoretical subject (due a regular basis to the volume of abstract concepts)
Figure 12. Commonalities among National Board Certified Teachers.
The NBCTs stated the importance of working with groups while creating a safe and organized atmosphere where students are focused, on task, creating discussion, solving problems, and helping one another. Through formative assessment, students develop a clear understanding of learning targets and receive feedback that helps them to improve (NCTM, 2015). The data clearly provide evidence on the importance of incorporating continuous AP Calculus AB style assessments, not only on paper but also by questions to students in class, creating discussion, and by providing them with
169 immediate feedback. In this manner, NBCTs receive evidence of students’
misconceptions, which they utilize to adjust their instruction, reflect on their practices,
adapt their teaching styles, and differentiate teaching. All the components depicted in
Figure 12 depict that NBCTs’ pedagogical practices at least partially align with the vision and standard of the NCTM regarding how mathematics should be taught.
Figure 13. Differences and discrepancies among National Board Certified Teachers. It is not a surprise to observe differences among educators. Teachers naturally
have different philosophies, and as a matter of daily practice, teachers in general
formulate different goals and set different priorities for their practices. However, one
170 must respect these NBCTs’ pedagogical ideas given the fact that they effectively produce
positive results by preparing their students for the AP Calculus AB Exam and for their
future careers. Rosenshine (2012) stated that “the most effective teachers ensure that
their students acquire, rehearse, and connect background knowledge by teaching new
material in manageable amounts, modeling, guiding practice, helping students when they
make errors, and providing for sufficient practice and review” (p. 12). The differences
that can be observed in Figure 13 among the interviewed NBCTs for this study are
irrelevant in the end as every single one of them has proved to be a successful educator.
Summary
Chapter 4 presented the data and analysis addressing the impact of teacher
certification on students’ test scores on the AP Calculus AB Exam. When the impact of
teacher certification was analyzed using a t-test, the impact of certification was significant. Moreover, the effect size calculated using Cohen’s d revealed that the result showed practical significance. The results also indicated that teacher certification status and gender did not produce a significant interaction. However, there was a significant interaction between teacher certification status and grade level as well as certification status and FRL status. Then the researcher investigated the level of agreement between
NBCTs and non-NBCTs in alignment with the NCTM standards and vision by
incorporating a second t-test. The researcher found the difference in SBI scores between
NBCTs and non-NBCTs to be statistically significant. A Cronbach alpha reliability
analysis was also incorporated. The researcher’s goal was to investigate if the SBI
instrument was reliable among NBCTs and non-NBCTs. The results showed that the SBI
is not a very reliable instrument with NBCTs due to a very low alpha value. However, the
171 Cronbach alpha depicted an acceptable alpha Cronbach value, which indicates that the
SBI is reliable with non-NBCTs, and there was acceptable internal consistency among
the item on the SBI. NBCTs tended to be in higher agreement with the NCTM standards
compared to non-NBCTS. For the two groups combined, the SBI was not found to be
reliable due to another low alpha level. Even though there was low reliability, the t-test
came out to be significant.
Lastly, the researcher reported several best mathematics practices discussed by
the interviewed NBCTs. Among some of the addressed best practices that NBCTs regularly incorporate are cooperative learning, problem solving, use of technology, classroom management, assessment, and differentiated instruction. NBCTs emphasized the minimal use of project based learning and exclusion of manipulatives. Similarities and discrepancies were found among the NBCTs’ teaching approaches, but the findings
revealed that NBCTs are successful in preparing their students to pass the AP Calculus
AB Exam.
172 Chapter 5. Discussion, Conclusions, and Recommendations
Introduction
This study used comparative mixed methods to identify best practices and student success associated with mathematics teachers who have earned the NBC. One focus was on the success of students of NBCTs and non-NBCTs on the AP Calculus AB Exam. The quantitative part of this study compared archival data of students’ test scores on the 2014
AP Calculus AB Exam with NBCT status in a large South Florida school. Given the potential presence of numerous confounding variables, no causal relationships can be assumed between NBCT status, the quality of mathematics instruction, or student success on the AP Calculus AB Exam. The research also explored the interaction effects between
NBCT status and student gender, student grade level, and student FRL status. An additional research interest was to explore and compare the level of agreement between the beliefs of NBCTs and non-NBCTs related to the standards and vision of the NCTM.
For the qualitative portion of the study, the researcher investigated how NBCTs incorporate best mathematics teaching practices in their daily instruction.
In this chapter, the researcher has summarized the data presented in chapter 4 addressing the four research questions. These summaries of the findings and conclusions based on the data analyses are discussed along with limitations, implications for practice, and recommendations for future research.
173 Summary of Findings
Research Question 1 queried whether NBCTs produce better results than non-
NBCTs on the AP Calculus AB Exam. The researcher found that students taught by
NBCTs produce statistically significant higher results in comparison to those students taught by non-NBCTs. The t-test indicated that the difference of means t (454.065) =
9.462, p < .001, was statistically significant. These results suggest that students taught by
NBCTs tended to score higher than students taught by their counterparts. The size of this
effect Cohen’s coefficient (d = .6429) was found to be a conventionally large effect. This means there was a practical significance between the mean differences in students’ test scores. The mean difference of the students’ test scores between the two groups was 0.96
(scale 1 to 5), a difference that was high enough to interpret results in favor of the
NBCTs. The greatest difference in test scores of students taught by the two groups of teachers occurred at the poorest performance level (see Figure 11). The lowest score frequency of students of NBCTs was 1; however, this was the highest frequency for the students of non-NBCTs.
Previous research revealed different findings about the impact of the NBC on student achievement. Most of the previous mathematics studies focused attention comparing the two different groups of teachers at non-advanced levels of mathematics.
This is the first known study exploring the effectiveness of the NBC at the AP Calculus
level. The results fall in line with previous studies because even though previous findings
have produced mixed results when comparing NBCTs and non-NBCTs; most of the
previous results played in favor of teachers having the NBC (e.g., Cavaluzzo, 2004;
Clotfelter et al., 2007; Goldhaber et al., 2004; Vandevoort et al., 2004, and Whitman,
174 2002). As Sawchuk (2015) stated, “The evidence continues to mount that teachers who
earn national-board certification are more effective than other teachers, both at the high
school and elementary levels” (p. 6).
Research Question 2 was related to exploring the interaction effect among the
different levels of NBCT status and (1) student gender; (2) student grade level; and
(3) student FRL status. Three two way factorial analysis F-tests were performed. With a
power of only .23 for the interaction, F (1, 1162) = 1.52, p > .05, it was found that the
effect of certification is consistent across the levels of gender. However, a significant
interaction was found between teacher certification status and grade level. With a power
of .99 for the interaction, F (1, 1161) = 10.67, p < .01, it can be said that the effect of
certification is not consistent across the levels of student grade level, and an interaction
was detected between non-certified teacher status and 10th grade students. Finally, an
additional significant interaction effect was found between NBCT status and student FRL status and with a power of .64 for such interaction, F (1, 1161) = 5.377, p < .05, it can be concluded that that the effect of certification is not consistent across different the levels
of FRL status.
Research Question 3 focused on exploring how NBCTs and non-NBCTs believe that they incorporate best mathematics teaching practices while aligning their teaching styles with the standards and vision of the NCTM. By using the SBI instrument and obtaining summation of the scores pertaining for each teacher in the two different groups, it was found that there was a significant difference in the SBI mean scores between
NBCTs and non-NBCTs, t (53.102) = 3.184, p < .001. NBCTs tended to have a stronger
level of agreement in alignment with the NCTM standards compared to non-NBCTs. The
175 consistency among the 16 items on the SBI revealed that this instrument is not reliable
for NBCTs with an alpha value of .370. The reduction in the coefficient alpha for the subjects on the NBCTs group is probably due to their restriction in range as there was
less variance in this group. However, the reliability of the SBI was acceptable for the
non-certified group with an alpha value of .70, which indicates that the instrument is
reliable. The overall α reliability value for the two groups was .42 which represented a
low alpha value, indicating that there was low internal consistency among the items in the
SBI instrument. Even though this reliability was low, the t-test was able to detect the
significant difference.
Research Question 4 focused on how NBCTs deliver instruction on a daily basis
to properly train and prepare their students to learn calculus and eventually succeed in
passing the AP Calculus AB Exam. All interviewed NBCTs felt they were limited to
incorporating a few best mathematics teaching practices. In the first place, they must
teach to the test and train their students on how to technically answer multiple choice and
free response questions. Because AP Calculus AB is a very theoretical subject, the
teachers were barely able to incorporate some best mathematics practices such as project-
based learning and the use of concrete elements of mathematics such as manipulatives.
As Ms. Johnson stated, “AP Calculus is very theoretical; you need concrete because you
need to keep the students focused instead of just being all.” The findings show that
NBCTs are teachers who provide opportunities for concept building, effective monitoring
strategies, problem solving, reasoning, and connections with the curriculum and the real
world in other disciplines. As Ms. Doval said, “…teaching Calculus while making
connections with physics happens in a natural way, that’s all integrated in my
176 discussion.” NBCTs showed they do not rely on heavy memorization and try to make
math meaningful. Mr. Glenc expressed, “…some of the things you have to learn, that’s a
rule; but some other things you can show real world examples and build their interest.”
This may be a way to have their students understand AP Calculus AB and equip most of
them to pass the AP exam.
It would not be possible to lead students to success without content and
pedagogical knowledge. All interviewed NBCTs have master’s degrees, and they all
showed themselves highly knowledgeable in regard to how well they know the concepts
of AP Calculus AB, which is a high level mathematics class. In different shapes and
forms, ongoing assessments came out to be a common denominator as a finding.
Ms. Morrison stated, “…the formal is at least once or twice a week while they’re getting
some type of formal feedback whether it’s a quiz or a test.” She added, “on quizzes, they have a reflection piece that they have to turn back in. They have to reflect on their learning. They have not only to correct their mistakes but also to analyze and reflect on what could have been done better.” The teachers find it paramount to expose their students to continuous ongoing assessment in order to prepare them to be able to solve
AP Calculus format questions and to develop their critical thinking in how they select their answers on the multiple choice portion; they are also required to be proficient in showing how they work step by step on the free response part of the examination.
A highly remarkable finding was the consistent support and feedback that these teachers provide by making sure that they are available for their students before, during, and after school. In this manner, they try to ensure that their students get to the AP
Calculus AB Exam with confidence and knowledge. Ms. Furgeson commented, “I make
177 myself available every single day of the week at lunch time and twice a week after school to provide additional support to students who need help. I haven’t had a lunch all year long.”
The findings depicted a few instructional discrepancies among the NBCTs. It is
important to mention that some teachers pursued the NBC to obtain an increase their salaries. Mr. Glenc admitted, “I did it strictly for the money.” Other NBCTs pursued the certification for personal improvement. Ms. Doval expressed, “I like video-taping myself
in order to see how I can become a better teacher. I truly believe that I became a better
educator by recording and watching myself teach.” The use of technology emerged as a
big discrepancy as there were split decisions about the benefits of incorporating different
technologies to help students effectively comprehend the calculus concepts.
Ms. Morrison said, “I use a lot of applets and a collection of websites,” and she likes the
‘object moving interaction’ to teach various calculus concepts. On the other hand,
Mr. Glenc stated, “I’m the old dog that doesn’t believe you can learn new tricks. I don’t want to learn how to make a PowerPoint. I want to be interactive with the notes, writing notes on the board, pausing, and listening to what I am saying.” Nevertheless, all NBCTs are subject to incorporating the use of graphing calculators because it is mandatory for the examination. Some of the NBCTs incorporate one or two yearly projects, but others do not. Table 17 displays the quantitative data, and Table 18 summarizes the qualitative interview findings related to the study’s research questions. These provide the basis for the discussion that follows.
178 Table 17
Summary of Findings (Quantitative Portion)
Research Questions Findings
R. Q. 1: Students taught by NBCTs produced statistically Do NBC mathematics teachers produce significant better results in comparison to those better results than non-NBC mathematics students taught by teachers who do not hold the teachers on the AP Calculus AB Exam? certification.
There was a practical significance between the differences in means of students’ test scores.
The results agree with some of the research cited but may be impacted by compounded factors not addressed in this study.
R. Q. 2: In respect to test score performance, is The effect of NBCT certification is consistent there an interaction between NBCT status across the levels of gender. and student gender, NBCT status and student grade level, or NBCT status and A significant interaction was found between NBCT student FRL status? status and grade level. The effect of certification is not consistent across the levels of student grade level. An interaction was detected between non- NBCT status and 10th grade students.
A significant interaction effect was also found between NBCT status and student FRL status. The effect of certification is not consistent across the two levels of student FRL status. NBCTs seem to have a better effect on higher SES students, but other factors may be involved. R. Q. 3: What is the difference in perception of There was a significant difference in the SBI mean best mathematics teaching practices scores between NBCTs and non-NBCTs. between NBCTs and non-NBCTs? NBCTs tended to have a stronger level of agreement in alignment with the NCTM standards compared to non-NBCTs. The SBI is not a reliable instrument for NBCTs.
The SBI was acceptable for the non-certified group, which indicates that the instrument is reliable.
The overall α reliability value for the two groups was low indicating that there was low internal consistency among the items in the SBI instrument.
The statistical t-test however was able to detect the significant difference in the mean scores between NBCTs and non-NBCTs.
179 Table 18
Summary of Findings (Qualitative Portion)
Research Question Findings
R. Q. 4: All interviewed NBCTs were slightly limited to How do NBCTs incorporate best incorporating a few best mathematics teaching mathematics teaching practices in their practices. daily instruction? They must teach to the test and train their students on how to technically answer multiple choice and free response questions.
AP Calculus AB is a very theoretical subject. This limits teachers to barely incorporate some best mathematics practices such as project-based learning and the use of concrete elements of mathematics, e.g., manipulatives.
The findings show that NBCTs are teachers who provide opportunities for concept building, effective monitoring strategies, problem solving, reasoning, and connections with the curriculum and the real world in other disciplines.
NBCTs showed they do not rely on heavy memorization and try to make math meaningful. All interviewed NBCTs have master’s degrees, and they all showed themselves highly knowledgeable in their knowledge of AP Calculus AB. Ongoing/continuous assessments are a common denominator among NBCTs. A great deal of support and caring traits were clearly evident. NBCTs provide continuous feedback.
Some NBCTs obtained the certification for financial reasons, others to better themselves as professionals.
Some rely on the use of technology; others prefer to incorporate old traditional methods that do not require the use of technology. All of them however must emphasize the use of graphing calculators as it is a requirement for this class.
Some of the NBCTs incorporate one or two yearly projects and others do not.
Regardless of their differences in teaching approaches, NBCTS proved to be effective teachers. It is important to consider that some of them came into the teaching profession with no pedagogical knowledge or teaching experience.
180 Not all teachers teach in the same manner, so it is a natural phenomenon to see slightly different approaches among all NBCTs. Outside the instructional scope, an observed difference among the interviewed NBCTs was the backgrounds they brought before they came into the teaching profession. Some came already trained pedagogically from teacher training institutions, and others came from other professional fields not related to the educational world, which means that they had to adapt and learn how to combine both content and pedagogical knowledge. Regardless of their backgrounds or instructional approaches, all of these NBCTs have proved that they are effective instructors, that they accomplish the goal of having most of their students pass the AP
Calculus AB Exam year after year, and most important, that they truly cherish and enjoy the teaching profession and care for their students.
Discussion
The quality of high school mathematics teachers and how they ensure that they properly prepare their students for the AP Calculus AB Exam is a concern to many students and parents. Passing the AP exam represents not only obtaining college credit while being in high school, but also represents an indication that the student went through rigorous and hard work at the advanced mathematics level and is ready for college. Often
times, colleges and universities have faith and accept students that unfortunately are
insufficiently prepared for the academic level that these entities require. Consequently,
students who pass AP exams are bound to be granted access to universities of their choice
based on the assumption that passing an AP exam is indeed a sign that the student has
gone through rigor and effort. Many universities and colleges have a mathematics
requirement; a high score, specifically on the AP Calculus AB Exam, will most of the
181 times satisfy such requirements for many universities. Moreover, pressure to raise test
scores has been a continuous burden for teachers and principals. When students pass a
standardized test, every passing score contributes to achieving a better grade for the
school. The importance of students’ success on the AP Calculus AB Exam contributes to
adding points toward the school grade and to the reputation of the school at the same
time. The implications of having students obtaining low or non-acceptable grades have a huge impact in the image of the school, and it is essential for a school to create that good image to attract new students in forthcoming years. Lawson (2005) stated,
Because of the current increased emphasis on accountability student achievement
has never been more important at the national, state or local levels than it is today,
the implication and consequence of lower than acceptable student achievement
may affect a school’s very existence. (p. 115)
This study was designed to analyze and discuss the impact of the NBC on student’s test scores on the AP Calculus AB Exam and also on teacher effectiveness in successfully preparing students to learn calculus sufficient to pass such an exam. The researcher addressed the following questions:
1. Do NBC mathematics teachers produce better results than non-NBC mathematics teachers on the AP Calculus AB Exam?
2. In respect to test score performance, is there an interaction between NBCT
status and student gender, NBCT status and student grade level, or NBCT status and
student FRL status?
3. What is the difference in perception of best mathematics teaching practices between NBCTs and non-NBCTs?
182 4. How do NBCTs incorporate best mathematics teaching practices in their daily
instruction?
Four statistical tests were utilized to provide a comprehensive report of the impact
that the National Board Certification has on student achievement on the AP Calculus AB
Exam. A t-test was employed first using the 2014 AP Calculus AB scores as the dependent variable. Cohen’s d was incorporated to accompany the t-test results to investigate the effect size of the mean difference.
When looking at the results of the first t-test, it is important to be aware of the
implications that may or may not fall behind the statistically significant results. The
researcher emphasizes that, given the presence of numerous confounding variables, it is
possible that there is no causal relationship between NBCTs and quality mathematics
instruction. For example, more motivated teachers are more likely to seek and be
successful with NBC, as well as more likely to pursue activities that make them better
teachers. Although one can postulate, it is not possible to know what variables may
confound the effect found in this study. It is important to understand that in finding an
association between the NBC status of a teacher and the scores of students of those
teachers, the element of causation is far from guaranteed although it is clear that NBCTs
in this study taught students who achieved higher scores on the 2014 AP Calculus AB
Exam.
Motivation levels of both teachers and students in AP Programs may be higher.
NBCTs are required to have advanced content knowledge and demonstrated pedagogical
skills that contribute to student outcomes. Additionally, going through the NBC process
is a process that requires a lot of commitment and motivation. It is possible that NBCTs
183 may be more motivated than their non-certified peers. As stated in Oko (2014) , “a
teacher who is motivated will put more efforts in the classroom to pass adequate
knowledge to his students in order to affect the behavior of the learner in the most
desirable and positive manner” (p. 9). Based on the student demographic information in
this study, one could question if there is some correlation between AP Calculus classes
and more privileged students. Two thirds of the students who took AP Calculus AB in
2014 (see Table 6) were not eligible to participate in the FRL program. Some of the
success regarding these types of students may be attributed to factors such as having the
means to access private tutoring, having parents with college degrees who can help and
motivate them, having access to computers with Internet, and also being students who are
self-motivated. These are among some of factors that may lead students to succeed in
passing the AP exam. Furthermore, there seems to be a tendency that NBCTs tend to
teach the upper level classes, which means that NBCTs tend to associate more with
highly motivated students who are more likely to possess higher mathematics abilities
and be more motivated.
Three factorial ANOVAs were used to answer Research Question 2 to test the
interaction effects between NBCT status and variables such as student gender, student
grade level, and student FRL status. The first interaction detected was between 10th
graders and non-NBCTs. The reason for this interaction may be due to the larger sample
size related to the non-NBCTs. These 10th graders are great achievers even at a young age, only likely to be in AP Calculus classes in the 10th grade because of superior math
ability or prior completion of high level math courses in middle school. Nevertheless,
these students generated higher passing rates.
184 The second interaction was detected between NBCTs and students who were not eligible for the FRL program. Based on the results, NBCTs seem to have a better effect on high economic status students compared to low economic status students. There is no literature in this study that explains this effect; however, it is the researcher’s interest to point such an interesting result revealed in this study.
Research Question 3 focused on whether there was some sort of level of agreement in regard to teachers’ beliefs of how they incorporate best mathematics teaching practices in alignment with the NCTM standards. To answer this question, the researcher assigned the Standards’ Beliefs Instrument (SBI) as a survey to NBCTs
(N = 31) and non-NBCTs (N = 31) to collect information and analyze the level of agreement between the two groups of teachers. A reliability analysis was performed to explore how reliable the SBI instrument was in terms of measuring its internal consistency. By summation of all the scores from the 16 items on the SBI, the 31 teachers in each group obtained a mean score, and a second t-test was utilized to explore if the difference between the two mean scores was statistically significant. Even though the Cronbach alpha coefficient was low for NBCTs, the t-test still detected the statistical significant difference in the perception of how mathematics teaching should be aligned following the standards of the NCTM.
Five cases studies were designed to answer Research Question 4, and the researcher decided to incorporate this qualitative component in order to get a more in-depth picture on how NBCTs deliver instruction and use best mathematics teaching practices on a daily basis. The researcher interviewed all five NBCTs to obtain as much detailed information as possible.
185 The results from the interviews revealed that NBCTs do not incorporate manipulatives, although they are highly recommended by most of the best mathematics studies presented in the literature review section of this study (e.g., Busatto, 2004; Furner et al., 2005; Steedly et al., 2008; Willig et al., 2009), and in Closing the Achievement
Gap: Best Practices in Teaching Mathematics (The Education Alliance, 2006). They are perceived to have a positive impact on student learning. However, perhaps because AP calculus consists of more abstract concepts, these NBCTS were less likely to use concrete objects or project-based learning. Interestingly, this commonality found among the interviewed NBCTs creates some conflict with their responses on the SBI survey, which found that the NBCTs tended to have stronger agreement with the NCTM standards.
However, not incorporating manipulatives or project based learning contradicts such agreement with the NCTM Principles and Standards which emphasize the importance of using manipulatives and visual representations. Moreover, all NBCTs expressed the importance of making connections between theory and practice. Not recommending the use or incorporation of manipulatives for AP Calculus AB highly contradicts the NCTM philosophy of teaching and the recommendation of the body of literature that recommends the incorporation of best mathematics teaching practices as well. This disagreement with the standards creates an interesting and conflicting result regarding what was observed in the SBI results which generally concluded that NBCTs are in more agreement with the NCTM standards compared to their non-certified peers.
The results observed from the case studies may suggest that if NBCTs are highly effective and possess traits that other non-certified teachers do not have, then these teachers could be used to make an impact in other teachers’ classrooms. They could
186 mentor new and veteran teachers and help them to improve their practices. The districts could use them to lead workshops where other teachers could learn how to incorporate effective mathematics teaching practices. One district leveraged the NBC by
“placing NBCTs in a variety of teacher leadership roles, including staff development and peer counseling, with the idea that those teachers will return to the classroom with an increased capacity to teach and collaborate” (National Board for Professional Teaching
Standards, 2010b, p. 2).
Furthermore, NBCTs could work together to make a contribution to the counties where they work by creating websites where they can provide supplemental materials for other teachers, teach them how to more effectively follow the standards of the NCTM, and make them more aware of the importance of sharpening their practices by following such standards. These are among some of the benefits that could help other teachers to grow, and at the same time NBCTs would be contributing to create more equity in many classrooms because it is evident that if teachers grow, then so do their students.
Limitations of the Study
There are a number of limitations to this study that should be acknowledged before making any final conclusions. Those include:
1. The researcher is an NBCT in mathematics; however, the research design allows objectivity throughout the entire study in order to avoid bias that may affect the results and/or findings.
2. The study is limited to only one school district in South Florida.
3. The study was performed at the AP Calculus AB level, and thus the results of this study are only applicable and unique to the teachers and students at this level.
187 4. The researcher evaluated test scores corresponding to the year 2014 only.
5. Most students placed in AP Calculus AB classrooms are not at-risk students.
6. The sole focus of the dependent variable was the AP Calculus AB Exam scores.
However, confounding variables may apply to teachers, students, and schools beyond the
scope of this study. For example, there are many outcomes associated with NBCTs other
than just achievement as measured by performance on an AP examination. Moreover,
there are indications of student learning which may be related to student achievement
such as motivation, past experiences, etc., that are important to measure but were not
included or captured in this study.
7. Among the study participants, two thirds of the student population is not
participating in the FRL program. There seem to be more advantaged students in AP
Calculus AB classrooms than in other high school classes.
8. It is important to note that more motivated teachers may enhance students’
academic achievement. Motivation could be an alternative explanation for the effect of
the first t-test.
9. The researcher interviewed only NBCTs so this study does not provide any
evidence of what strategies non-NBCTs utilize to lead their students to pass the AP
Calculus AB exam.
10. The interviewed NBCTs were White Americans. There was no diversity of race or nationality in this sample of teachers. Foreign NBCTs or NBCTs of color were not incorporated in this study.
188 Conclusions
The results in this study have focused on the practices and outcomes of mathematics NBCTs. Using archival data for high school NBCTs and non-NBCTs and their students from one particular school district in South Florida, the researcher found evidence that the NBC may be a signal of teacher effectiveness based on the students’
2014 AP Calculus AB scores. Due to what is involved in obtaining the NBC and passing the high level mathematics assessment to accomplish such certification, holding the NBC appears to be a sign that a teacher is well prepared with content and pedagogical knowledge, although some non-NBCTs may reflect similar qualities. When teachers know their subject well, students’ learning is enhanced, and they develop broader understanding of the subject. On the other hand, if teachers are unprepared, this is a disservice to their students, and unfortunately students too often are exposed to this reality from early stages. As Baker, Smith, and Abbott (2011) stated, “Research suggests that many teachers even in the elementary grades, where the real foundations of mathematical understanding are laid, do not know enough mathematics and in some cases are uncomfortable with math” (p. 6); this becomes a critical issue at all levels of mathematics education. Furthermore, these concerns are increasing with changes and reforms in educational practice. More productivity is being demanded of teachers as they are being exposed to higher demands and increased pressure. Although parents and students may presume benefits associated with being taught by NBCTs, there may be other factors that contribute to successful grades and high standardized test scores.
Regarding the significant interaction effect between teacher certification status and student grade level found in this study, it is important to observe that 11th and 12th
189 grade students of NBCTs scored higher; however, 10th graders of non-NBCTs had a
better performance. There is no previous research to compare this type of finding, but
there is the belief that 10th graders who are able to take advanced mathematics courses at
such early stages are considered among the most capable students. The sample of 10th
grade students of non-NBCTS (N = 22) was larger than the sample of 10th grade students
of NBCTs (N = 16), and perhaps the broader sample of this particular grade level of
students may have had a positive effect on the average test scores regardless of the
certification status. The significant interaction effect found between teacher certification
status and student FRL status depicted that teachers appear to have a more positive
impact on students who do not hold the poverty indicator, and such impact is more
evident with students of NBCTs. Ravitch (2012) remarked that research has convincingly demonstrated that NBCTs are effective, especially effective in low-performing schools, which supports state policies that provide stipends for NBCTs. The results in this study are in agreement with the results from previous research; however, in regards to student socio-economic status, the study could be an eye opener to those who might be interested in specifically researching the relationship between the NBC and low performing students. Beyond the scope of this study, there should be an initiative to support those schools whose population is represented by students with low resources or minority students. If NBCTs can make a difference at the AP Calculus level, perhaps they can make an even bigger difference with students in other mathematics classes.
When it comes to instructional practices, regardless of each individual NBCT’s characteristics, it can be concluded from the interviews that these teachers are well- rounded but face the challenge of teaching to the test while making learning AP Calculus
190 AB a meaningful and interesting subject to learn. Due to the high degree of abstract
concepts contained in this particular course, there are some limitations with incorporation
of some best mathematics teaching practices, which creates a contradiction with some of
the results observed regarding instruction alignment with the standards of the NCTM.
The interviewed NBCTs perceive that at lower levels of mathematics there is more
flexibility to incorporate a wider range of best practices. The question that arises is if
teachers teach higher level of mathematics courses, are they then getting more
disconnected from the NCTM standards? It is highly important for NBCTs to make a
seamless connection between abstract and concrete concepts related to other disciplines
or real world situations.
There is evidence that NBCTs have shown a great deal of devotion and love for
the teaching profession while talking about their AP Calculus AB teaching. However,
their remarks occasionally diminished the art of teaching lower performing students.
Some expressed it is difficult to work with low level students, a concept that is not
aligned with the philosophy of the NBPTS. Another NBCT expressed her concerns with
dealing with discipline problems, and another expressed that she was not successful
teaching lower level students. These data create conflict while trying to understand the
philosophy of these NBCTs and their perspectives toward what should be ethical and
professional.
NBCTs expect their students to succeed not only by passing the AP exam but also by learning calculus effectively so that they can be prepared for taking more advanced mathematics classes in the future. This study also compared how NBCTs’ and non-
NBCTs’ teaching philosophies align with the NCTM standards and vision. The level of
191 agreement of NBCTs with the NCTM standards was statistically higher, and it was revealed that NBCTs tended to agree in a more homogenous way creating less variance in the selection of the items on the SBI. To become an NBCT, one must go through the process adhering to the standards established by the NBPTS throughout the entire process. Adhering to the same standards and undergoing the rigor of the NBC creates a culture of work and mindset on how mathematics should be taught. This mindset might have been a reason why NBCTs were in more accordance with the NCTM standards due to the fact that all NBCTs have been trained in a similar fashion and adapted themselves to the NBPTS standards to accomplish the certification. This high level agreement was reflected in the reliability coefficient that measured how reliable the SBI was regarding the two different groups of teachers. It can be concluded that the SBI is not a good instrument to measure variances among NBCTs.
In this study the NBC has been associated with student success on the AP
Calculus AB Exam; however, many components need careful attention. Moreover,
NBCTs have demonstrated how they effectively prepare their students to learn calculus and eventually pass the AP Calculus AB Exam by incorporating best mathematics teaching practices. Regardless of the teaching techniques of each NBCT, it is important to note that the NBC focuses on core instructional practices. Student achievement on standard tests has not been one of the specific goals of going through the certification process. However, as reported by Sawchuk (2015), the President of the NBPTS, “There definitely seems to be a predictable trend that when you have teachers who have this understanding about their job, this depth of their knowledge, their skills set, they get these results and they’re replicable” (p. 26).
192 Implications
Focusing on the data that were gathered and analyzed in this study, the following implications are to be considered. Encouraging and supporting teachers to apply and attempt to achieve the NBC could represent a strategy to increase the quality of classroom teachers. Skilled teaching has a positive impact on student achievement not only on AP exams but all kinds of standardized tests and student preparedness for their future. “The quality of the teacher in the classroom is a primary catalyst behind the need for a complex, knowledge based system” (Darling-Hammond, 1996). School districts may re-evaluate the policies of providing financial support to those teachers who would like to go through the process but are unable or unwilling to pay the high cost that is currently involved in applying to become NBC candidates.
The most practical application of the findings in this study could be used to help
educators consider the importance and benefit of obtaining the NBC. The findings may
also serve as an eye-opener to principals and school administrators when they need to
consider hiring mathematics teachers to work for their schools. NBCTs seem to work
well with AP Calculus students; however, it is important to state that the philosophy of
the NBPTS is to reach out to all students in need, not just at the upper level of
mathematics. The researcher acknowledges that it is a sad but true reality, as stated in the
literature review in this study and as depicted from the results in this study, that low
performing students do not seem to be NBCTs’ first choice of students to teach. NBCTs
often times are given the choice of the classes they want to teach, and they end up
teaching the best students. Since there is a tendency for NBCTs not to teach low level
students in many cases, then there is an evident disconnection between what the NBC
193 represents and the equity that is supposed to be created under these circumstances. So the
question that arises is: how well can NBCTs prepare under-performing students and how
well can they incorporate the best mathematics teaching practices mentioned in this study
to make a difference with mainstream students?
Another important implication of this study deals with policy makers who should
be concerned about where the most effective teachers, including NBCTs, end up
working. Most financial incentives for these teachers are gone, and not giving appropriate
recognition or providing support to teachers who can make a difference may create
additional stress and inequities in our educational system.
The reported Cronbach alpha reliability coefficient was low for the NBCTs group.
In reference to this result, there may be reliability implications in this study. It is
important to mention that several commonalities were found among NBCTs that could
have had an impact in how the NBCTs’ beliefs were measured and how the teachers
approached the selection of their answers on the SBI. Having gone through a common
process of applying for the NBC, they may have had little variance in their responses. It
is also logical to think that a larger number of NBCTs may be more familiar with the standards of the NCTM, and some of them may be current or past members of the NCTM themselves. If these teachers know and closely follow the standards, then the reliability results would make complete sense.
Based on the results and findings of this study and previous studies that compared the effectiveness and impact of the NBC, school districts and educational administrators may become more aware of the benefits of having more NBCTs in the classrooms and use this as a strategy to produce a better impact on students and their schools. Schools
194 receive additional points for the school grade by having students meet success on standardized assessments, and specifically on the AP Calculus AB Exam. Students receive college credit and access to colleges and universities of their choice, so hiring
NBCTs to train these students may bring benefits to both the schools and their students.
To accomplish that, it is important to say that teachers with high content and pedagogical knowledge are needed to lead this type of mathematics course and present content to students in a meaningful way that fosters deep understanding of high level mathematics concepts.
Recommendations for Future Research
The results in this study revealed interesting findings about the impact of NBCT status with students’ test scores; the interaction effects between NBCT status, student gender, student grade level, and student FRL status; teacher’s beliefs about how mathematics should be taught in alignment with the NCTM vision; and how NBCTs incorporate best teaching practices in their daily instruction.
These results should be viewed as a result of a single study; however, it is important to look at single studies within the body of previous research and not in isolation as a single study. These results presented do not guarantee that being taught by an NBCT is the main indicator or a causative factor to passing the AP Calculus AB Exam or any other examination. Even though the selected county is one of the largest school districts in the entire nation, replicating this study with a bigger sample size of NBCTs could strengthen the validity of the results in this particular study.
Replicating this study would also bring very informative results. Because this study was limited to students enrolled in one particular school district, a future study
195 could be performed including a broader and more diverse student population. The results show that the effectiveness of having the NBC may have a higher impact on students of high economic status. The differential effectiveness of the NBC on students’ test scores for low economic and high economic status needs to be further researched. Replicating this study by exploring the effectiveness of the NBC with minority students, students with special needs, or low performing students may bring very insightful information to understand the effectiveness of the NBC on student achievement.
The case studies only captured a small homogenous group of NBCTs who volunteered to participate. An attempt to delve into a school district that has a greater number of NBCTs teaching AP Calculus AB may be beneficial to incorporate a bigger sample of NBCTs willing to be interviewed, and in that way create a broader compilation of case studies. Additionally, the researcher recommends incorporating a more diverse sample of NBCTs. For future research, it is also recommended to explore how NBCTs utilize best teaching practices at lower levels of mathematics. This may depict a more comprehensive picture of the incorporation of best mathematics practices, especially since the students in AP Calculus are mostly high-achieving and well-prepared students who have already gone through honors mathematics classes. The differentiation in instruction components gets limited by the homogeneity in student ability levels in AP classes.
Since this has been the very first study comparing NBCTs at the calculus level, it is evident that additional research needs to be conducted to determine the impact of the
NBC on student success on the AP Calculus AB Exam. It is the researcher’s intention that this study generates questions and interest for future research, and even methods that
196 could replicate the design of this study. Additional variables such as ethnicity, English of other language speakers, and GPA among others could be incorporated to add more value and understanding of student performance to this study. In short, the recommendations for future research in this study are as follow:
• Replicate this study utilizing a larger sample size of NBCTs.
• A future study could be performed including a broader and more diverse student
population by capturing more counties or counties in other states.
• Attempt to delve in a county that has a greater number of NBCTs teaching AP
Calculus AB in order to capture a bigger sample of NBCTs willing to be
interviewed. In this manner, a broader compilation of case studies could be
generated.
• Attempt to capture a more diverse sample of NBCTs.
• Replicate this study to investigate if the NBC has a higher impact on students of
higher economic status compared to students of lower economic status.
• Explore how NBCTs utilize best teaching practices at lower levels of
mathematics.
• Incorporate other variables such as ethnicity, English of other language speakers,
and GPA among others to add more value and understanding of student
performance to this study.
• The researcher recommends a future study to explore if the commonalities found
among the interviewed NBCTs can materialize while having these teachers
reaching out to disadvantaged students. Then this type of research design could
reveal if NBCTs could make a difference in the academic success of those
197 disadvantaged students and if more equity could be accomplished. Having a more
equal distribution of qualified teachers would create more fairness in the division
of classes of those students who possess different mathematics abilities or access
to financial resources.
Summary
Increasing student achievement is a major priority in today’s schools. The qualifications and characteristics of a teacher may be a potential indicator not only for the reputation of the teacher, but also for the processes necessary to increase student learning and achievement. Former studies that explored the effectiveness of the NBC suggested that students who are exposed to NBCTs demonstrate higher achievement (Cavaluzzo,
2004; Clotfelter et al., 2007; Goldhaber, Perry, & Anthony, 2004; and Vandevoort et al,
2004). It is important to note that none of the previous research focused on comparing the
NBC at the highest level of high school mathematics courses. This is the first study that was performed comparing NBCTs and non-NBCTs at the AP Calculus level.
The results in this study indicated that there were significant differences in student success among those students taught by NBCTs and non-NBCTs. The incorporation of best mathematics teaching practices and the processes implemented to guide students to succeed on the AP Calculus AB Exam served to identify the quality of instruction that most of the NBCTs who teach AP Calculus AB provide on a daily basis in the selected county. Results also indicate the NBCTs’ teaching philosophy tends to align more with the vision of the NCTM compared to their non-certified peers. Part of the NBCTs students’ success on the AP Calculus AB Exam may be related to how NBCTs align their teaching with the NCTM standards even though there were a few discrepancies on this
198 particular matter since there are teaching practices that disconnect their teaching with the
NCTM philosophy at the AP Calculus AB level. Additionally, NBCTs do align their
teaching practices based upon the College Board curricular guidelines (see Table 5).
They clearly demonstrated how they emphasize problem solving while connecting
mathematics instruction with other disciplines and avoiding having their students perform
unnecessary repetition with mathematical procedures. Iran-Nejad, McKeachie, and
Berliner (1990) stated, "The more meaningful, the more deeply or elaborately processed,
the more situated in context, and the more rooted in cultural, background, cognitive, and
personal knowledge an event is, the more readily it is understood, learned, and
remembered" (p. 511). Additionally, the ongoing alternative assessments are a paramount
factor to allow NBCTs to monitor students to lead them to success.
One of the most important aspects facing education is how to prepare students to
succeed in high school and be productive citizens in the future. Improving the quality of
the teachers that are placed in today’s classroom has been the main focal point of several
educational reforms. One way that teachers can demonstrate their skill level and that they
are indeed qualified to be successful in the classroom is by earning the NBC. In
conclusion, good teachers are among the most important factors in student achievement
regardless of the gender, grade level, or socio-economic status. Despite some contradictions that previous research may depict, the findings in this study suggests that it is well worth the effort for any teacher to attempt to accomplish the NBC as the teacher, the students, and the school will eventually benefit from such accomplishment by the teacher.
199 Appendices
200 Appendix A. Invitation and Pre-interview Questions Invitation
Dear Colleague,
Thank you for responding to my previous email/telephone call. Again, I am completing my doctorate in Curriculum and Instruction at Florida Atlantic University, and I want to investigate how best mathematics strategies and practices are currently being used by National Board teachers in their Advanced Placement (AP) Calculus classes. Specifically, I would like to explore what strategies, methods, and approaches work best for each individual teacher. It is my goal is to add base knowledge to the relationship of certification status and best mathematics teaching practices at the AP
Calculus level. Therefore, it is my pleasure to invite you to participate in an interview so that you can bring to light some of the great practices that you are incorporating in your teaching of mathematics.
This interview process will consist of two parts. The first part consists of answering pre-interview questions that you will find below. These questions are representative of our forthcoming face-to-face interview. The purpose of these pre- interview questions is to get you started to think of the type of questions that you will be asked during the face-to-face interview, which will be scheduled at your convenience.
The interview will last approximately 60 minutes. After I perform all my
interviews, I will compile and analyze all the information and present it on my
dissertation. In order to add more depth and precision to the findings from the
interviews, I may utilize some quotes your interview. Since my dissertation will be open to the public, I will protect all interviewees’ confidentiality by assigning a pseudonym during the interview. All the information that I will obtain will be safely under locked at
201 my university and in a password protected home computer. Thank you for personal time and for being such important part of my study.
Pre-interview Questions
1. What four words would students use to describe your mathematics teaching strategies?
2. What do you consider to be your strengths and how will you use them in your teaching to prepare students to succeed on the AP Calculus AB Exam?
3. Describe your knowledge and experience with the College Board content standards applicable to your content area.
4. Best practices are methods, techniques, or strategies that have consistently shown positive results such as increased student success and passing rates. Please describe a best practice that you have utilized in your AP Calculus math class that has increased student learning.
5. How can you be sure that this practice increased student learning?
6. Please describe any other best practices or strategies that you have utilized in your AP math class that has increased student learning.
7. Do you feel that your National Board Certification (NBC) as a mathematics teacher has contributed to you being a more effective teacher when you teach AP Calculus?
202 Appendix B. Initial Self-Report/Introductory Letter
Dear Colleague,
As many of you know, I am completing my doctorate in Curriculum and
Instruction at Florida Atlantic University. I have reached the dissertation phase of my doctoral degree, and one of the topics of my dissertation consists of exploring and describing how best mathematics teaching practices are currently being used by mathematics National Board certified teachers (NBCTs) who currently teach Advanced
Placement (AP) Calculus.
Basically, I want to explore how National Board certified mathematics teachers who teach AP Calculus identify and describe strategies, methods, and techniques that are working to enhance teaching and learning. In a few weeks, I will be contacting some of you to set up an interview, and you will be asked to share all the great stuff that you are doing in your classrooms on a daily basis to improve student learning and lead them to find success on the AP Calculus AB Exam. Additionally, you will be sent some “pre- interview questions” so that you can get acquainted with the type of questions that you will be asked. To help me get started, I would like to collect some information from you.
Name______Age____
School Phone Number______Preferred Email______
Address______Gender______Race______
Title at work location______
How many years have you been teaching overall?
Specifically, how many years have you been teaching AP Calculus at your school?
Year you obtained your NBTC______
203 Appendix C. Adult Consent Form
1) Title of Research Study: The Effectiveness of the National Board Certification on the Advanced Placement Calculus AB Exam
2) Investigator(s): Principal Investigator: Dr. Joseph Furner [FAU Professor, Diss. Chair] Co-investigator: Fernando Antunez [Doctoral Candidate in CCEI]
3) Purpose: (1) To gain knowledge about the relationship of student success on the advanced placement mathematics exam and National Board Certification (NBC) training; (2) To explore if some particular factors such as student gender and student grade level reveal a significant interaction with certification status on the Advanced Placement (AP) Calculus AB Exam, and (3) to reveal how best mathematics teaching practices are being currently utilized and incorporated in classrooms of NBC teachers in Broward County
4) Procedures: The method of voluntary participation in this study will require participants to be willing to be interviewed in regards as how they teach their mathematics lesson and prepare their AP Calculus students in their respective schools. After obtaining IRB approval, the researcher will contact participants via email and will give each of them a consent form and also extend an invitation containing pre-interview questions. No data will be collected by participants until the consent form is signed and received. The reason for the pre-interview questions is to get the participants thinking about best mathematics practices in order to get them better prepared to provide more elaborate and detailed answers during the face-to-face interviews. Participants will also receive the Standards’ Beliefs Form (SBI) in a word format via email. This will help the researcher gain insight of the participants’ mathematics teaching beliefs and will also help the researcher understand each participant’s mathematics teaching philosophy. Then, the researcher will proceed to arrange a date and time to conduct 60 minute long interviews with those participants who voluntarily agree to participate in this research study. Participants may be asked to be interviewed in more than one occasion in additional information is needed. The researcher will record the interviews.
5) Risks: Throughout the interview there will be discussion and reflection of teaching styles and practices that may develop critical consciousness of one’s performance in the classroom. I propose this risk to be channeled in a positive way with careful attention while providing positive feedback. There is a minimal risk of confidentiality when recording teachers’ interviews. Recordings will be kept separately in a password protected personal computer and will only be accessible to the researcher. Names and any other information collected from the interviews will be eliminated from the written transcript of the interview when the researcher performs the qualitative data analysis. Additionally,
204 pseudonyms will be assigned to each participant to protect confidentiality No physical risks or discomforts apply in this study.
Initials ______6) Benefits: The participation in this study does not provide participant benefit. Participants will be able to see if this study reveals whether or not students taught by National Board certified mathematics teachers makes higher significant scores on the AP calculus exam. Even though this is not a benefit, participants will receive feedback from their respective interviews, and the researcher will report to them what best mathematics teaching practices are found to be most utilized by NBCTs who teach AP Calculus in one particular county in South Florida.
7) Data Collection & Storage: Any information collected about you will be kept confidential and secure and only the people working with the study will see your data, unless required by law. The data will be kept for 1 year in a locked cabinet and a password-protected computer in the investigator’s office. After 1 year, paper copies will be destroyed by shredding and electronic data will be deleted. We may publish what we learn from this study. If we do, we will not let anyone know your name/identity unless you give us permission. The way in which I will collect the to answer my first two questions will be by contacting the Research Office of the selected participating county and requesting the test scores of the AP calculus year 2014. I will work with datasets provided by the county which means that I will not be interacting with subjects in real time. Additionally, I will contact the College Board and the Florida Education Data House for additional data reports on test scores and demographics.
I will also collect data from teacher interviews. I will meet seven county NBCTs outside the school environment to perform the interviews that will focus on the use of best mathematics teaching practices. I will inform each teacher that I will create case studies based on the information that they provide on their interviews, but I will let them know and make very clear that I will assign each pseudonym to every single one of them to protect their identity as I write the case studies.
I will privately keep the data collected in order to ensure monitoring as well as privacy of the subjects. I am sensitive about the loss of data and breach of confidentiality so if the data were to be lost, then the electronic data will be kept at the university in the principal investigator’s office under lock and key. I will use a backup pen drive to safeguard the data and this pen drive will also be kept in a cabinet at the researcher’s domicile.
8) Contact Information: • If you have any questions about the study about the study, you should call the principal investigators, Dr. Joseph Furner at (561) 460-5911; Mr. Fernando Antunez at (954) 560-4045.
205 • If you have questions or concerns about your rights as a research participant, contact the Florida Atlantic University Division of Research at (561) 297-0777 or send an email to fau.research@fau.
9) Consent Statement: I have read or had read to me the preceding information describing this study. All my questions have been answered to my satisfaction. I am 18 years of age or older and freely consent to participate. I understand that I am free to withdraw from the study at any time without penalty. I have received a copy of this consent form.
I agree ______I do not agree______to be audiotaped during this interview.
Signature of Participant:______Date: ______
Printed Name of Participant: First Name ______Last Name______
Signature of Investigator: ______Date:______
206 Appendix D. Face-to-face Interview Questions
Interview #______
Date______/_____/______
Interview Protocol
Name of Interviewer: Fernando Antunez
Name of Interviewee: Title:
School: Starting time: Ending time:
Thank you for participating in this interview today. You have been asked to participate in a case study whose purpose is to investigate how National Board certified teachers
currently utilize strategies and best mathematics teaching practices that lead to success on
the Advanced Placement (AP) Calculus exam. There are no wrong answers so feel free to
express your points of view. Keep in mind that positive and negative comments are
valuable and most welcome.
The information that you provide in this interview will be fully confidential. I would like
to ask for your permission to tape record today’s interview so that the information that
you provide can be accurately documented. Once again, the information obtained on this
interview will be confidential and specifically used for my personal research to complete
my doctorate program at Florida Atlantic University. The interview will last
approximately 60 minutes.
Interview Questions
Introduction
Tell me about yourself
207 Domain I: Design
1. What is your teaching philosophy for teaching mathematics?
2. Explain what kind of activities you incorporate in your lesson to check for prior
knowledge and evaluate students’ preparedness before teaching a new concept?
3. Describe your teaching style and how you accommodate the different learning styles
for the students in your calculus classes.
4. Describe how you provide adequate time and structure in your lesson for sense-
making.
Domain II: Implementation
5. How do you integrate technology into the math curriculum you teach in your AP
calculus classes?
6. Describe any innovative projects you have been involved in having your students
develop/demonstrate the understanding of AP Calculus.
7. Teacher’s questioning strategies are likely to enhance the development of student
conceptual understanding/problem solving. How do you transition from lower to higher
order questions as you teach your math lesson?
8. How do you determine if the pace of the lesson you plan to teach is appropriate for the developmental levels/needs of the students and the purposes of the lesson?
Domain III: Mathematics content
9. Explain how you can evaluate if the content of the lesson is significant and worthwhile.
10. When you teach calculus, how do you make sure that appropriate connections are made to real world contexts or other areas of mathematics?
208 11. Briefly provide a description of when elements of mathematics abstraction (e.g., symbolic representation, theory building) are more important than concrete elements of mathematics (e.g. use of manipulatives). When do you bring in math manipulatives and concrete into the instruction?
Domain IV: Classroom culture
12. Indicate how you ensure that there is a climate or respect for students’ ideas, questions, and contributions as well as promoting a positive classroom atmosphere.
13. In what ways do you keep students on task and well behaved during collaborative group activities?
14. Give an example of how you have used cooperative learning in your classroom. How important is this strategy to you and how often do you use it?
Domain V: Assessment
15. What sorts of assessment, both formal and informal, do you view as being important indicators of successful performance for students to meet success on the AP Calculus exam?
16. Describe the evaluation instruments you use to assess student learning and ensure that they are not falling behind?
17. Explain what remedial strategies you utilize when a student is not performing at the desired level. What differentiation do you provide for students in your classroom?
209 Appendix E. Inside the Classroom Observation and Analytic Protocol
Observation Date: ______Time Start: ______End: ______School: ______District: ______Teacher: ______PART ONE: THE LESSON Section A. Basic Descriptive Information
1. Teacher Gender: ___Male ___Female Teacher Ethnicity: ___American Indian or Alaskan Native ___Asian ___Hispanic or Latino ___Black or African-American ___Native Hawaiian or Other Pacific Islander ___White
2 Subject Observed: ____Mathematics ____Science
3. Grade Level(s): ______
4. Course Title (if applicable) ______Class Period (if applicable) ______
5. Students: ______Number of Males ______Number of Females
6. Did you collect copies of instructional materials to be sent to HRI? ______Yes ______No, explain:
Section B. Purpose of the Lesson: In this section, you are asked to indicate how lesson time was spent and to provide the teacher’s stated purpose for the lesson.
1. According to the teacher, the purpose of this lesson was:
2. Based on time spent, the focus of this lesson is best described as: (Check one) o Almost entirely working on the development of algorithms/facts/vocabulary o Mostly working on the development of algorithms/facts/vocabulary, but working on some mathematics/science concepts o About equally working on algorithms/facts/vocabulary and working on mathematics/science concepts. o Mostly working on mathematics/science concepts, but working on some algorithms/facts/vocabulary o Almost entirely working on mathematics/science concepts 210
Section C. Lesson Ratings In this part of the form, you are asked to rate each of a number of key indicators in four different categories, from 1 (not at all) to 5 (to a great extent). You may list any additional indicators you consider important in capturing the essence of this lesson and rate these as well. Use your “Ratings of Key Indicators” to inform your “Synthesis Ratings”. It is important to indicate in “Supporting Evidence for Synthesis Ratings” what factors were most influential in determining your synthesis ratings and to give specific examples and/or quotes to illustrate those factors.
Note that any one lesson is not likely to provide evidence for every single indicator; use 6, “Don’t know” when there is not enough evidence for you to make a judgment. Use 7, “N.A” (Not Applicable) when you consider the indicator inappropriate given the purpose and context of the lesson. This section also includes ratings of the likely impact instruction and a capsule rating of the quality of the lesson.
I. Design A. Ratings of Key Indicators 1. The instructional strategies and activities- 1 2 3 4 5 6 7 used in this lesson reflected attention to the students’ preparedness and prior knowledge. 2. Adequate time and structure were provided for 1 2 3 4 5 6 7 “sense making”
II. Implementation A. Rating of Key Indicators 1. Technology tools are used to enhance math 1 2 3 4 5 6 7 Education. 2. The teacher’s questioning enhances the Development of student understanding/problem 1 2 3 4 5 6 7 Solving; e.g. emphasized higher order Questions.
III. Mathematics/Science Content A. Ratings of Key Indicators 1. The teacher displayed an understanding of 1 2 3 4 5 6 7 Mathematic concepts. 2. Appropriate connections were made to other 1 2 3 4 5 6 7 Areas of mathematics, science, and to other Disciplines, and/or to real-world contexts.
211 IV. Classroom Culture A. Ratings of Key Indicators 1. There was a climate of respect for students’ 1 2 3 4 5 6 7 Ideas, questions and contributions. 2. Interactions reflected collegial working 1 2 3 4 5 6 7 Relationships among students (cooperative Learning).
212 Appendix F. The Standards’ Beliefs Instrument (SBI)
Directions: Bubble in the answers that best describe your feeling about the following statements. Use the following code: 1 = strongly agree, 2 = agree, 3 = disagree, and 4 = strongly disagree. Strongly Agree Disagree Strongly Agree Disagree 1. Problem solving should be a SEPARATE, DISTINCT 1 2 3 4 part of the mathematics curriculum. 2. Students should share their problem-solving thinking and 1 2 3 4 approaches WITH OTHER STUDENTS. 3. Mathematics can be thought of as a language that must be 1 2 3 4 MEANINGFUL if students are to communicate and apply mathematics productively. 4. A major goal of mathematics instruction is to help 1 2 3 4 children develop the beliefs that THEY HAVE THE POWER to control their own success in mathematics. 5. Children should be encouraged to justify their solutions, 1 2 3 4 thinking, and conjectures in a SINGLE way. 6. The study of mathematics should include opportunities of 1 2 3 4 using mathematics in OTHER CURRICULUM AREAS. 7. The mathematics curriculum consists of several discrete 1 2 3 4 strains such as computation, geometry, and measurement which can be best taught in ISOLATION. 8. In K-4 mathematics, INCREASED emphasis should be 1 2 3 4 given to reading and writing numbers SYMBOLICALLY. 9. In K-4 mathematics, INCREASED emphasis should be 1 2 3 4 given to use of CLUE WORDS (key words) to determine which operations to use in problem solving. 10. In K-4 mathematics, skill in computation should 1 2 3 4 PRECEDE word problems. 11. Learning mathematics is a process in which students 1 2 3 4 ABSORB INFORMATION, storing it easily retrievable fragments as a result of repeated practice and reinforcement. 12. Mathematics SHOULD be thought of as a 1 2 3 4 COLLECTION of concepts, skills and algorithms. 13. A demonstration of good reasoning should be regarded 1 2 3 4 EVEN MORE THAN students' ability to find correct answers. 14. Appropriate calculators should be available to ALL 1 2 3 4 STUDENTS at ALL TIMES. 15. Learning mathematics must be an ACTIVE PROCESS. 1 2 3 4 16. Children ENTER KINDERGARTEN with considerable 1 2 3 4 mathematical experience, a partial understanding of many mathematical concepts, and some important mathematical skills.
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