PLACEMENT CRITERIA FOR MATHEMATICS ACCELERATION

THE ACCURACY OF THE ADVANCED MATHEMATICS PLACEMENT
CRITERIA IN IDENTIFYING STUDENTS FOR MATHEMATICS COURSE
ACCELERATION

A Doctoral Capstone Project
Submitted to the School of Graduate Studies and Research
Department of Education

In Partial Fulfillment of the
Requirements for the Degree of
Doctor of Education

Kristin M. Deichler
California University of Pennsylvania
July 30, 2021

PLACEMENT CRITERIA FOR MATHEMATICS ACCELERATION
Table of Contents
List of Figures
Abstract

viii
xi

CHAPTER I. Introduction

1

Background

1

Purpose of Study

2

Potential Outcomes

3

Financial Implications

3

Research Questions

4

Presumptive Outcomes

5

CHAPTER II. Literature Review

6

Historical Progression of Mathematics in Public Schools

6

Course Sequencing of Mathematics Curricula

11

Figure 1

14

Figure 2

16

Figure 3

17

Course Sequencing at a Local District

17

PLACEMENT CRITERIA FOR MATHEMATICS ACCELERATION
Figure 4
Acceleration, Ability Grouping, and Tracking
Figure 5
Approaches to Mathematics Acceleration

iv
18
19
22
23

Figure 6

27

Figure 7

29

Outcomes of Mathematics Acceleration
Figure 8

30
35

Mathematics Teachers’ Perceptions of Acceleration

36

Conclusion

39

CHAPTER III. Methodology

42

Purpose

43

Setting

47

Community Demographics

48

School District History and Demographics

49

Participants

54

Researcher

55

PLACEMENT CRITERIA FOR MATHEMATICS ACCELERATION

v

Research Plan

56

Fiscal Implications

60

Research Design

61

Data Collection

63

Ethical Concerns and the Institutional Review Board

65

Validity of Research Plan

66

Summary

67

CHAPTER IV. Data Analysis and Results

68

Data Analysis

69

Results

72
Figure 9

73

Figure 10

74

Figure 11

75

Figure 12

77

Figure 13

78

Figure 14

80

Figure 15

82

PLACEMENT CRITERIA FOR MATHEMATICS ACCELERATION

vi

Figure 16

83

Figure 17

85

Figure 18

86

Figure 19

88

Figure 20

89

Figure 21

91

Figure 22

93

Figure 23

95

Figure 24

96

Figure 25

97

Figure 26

98

Discussion

100

Summary

104

CHAPTER V. Conclusions and Recommendations

106

Conclusions

106

Figure 27
Limitations

113
113

PLACEMENT CRITERIA FOR MATHEMATICS ACCELERATION

vii

Recommendations for Future Research

115

Summary

119

References

121

APPENDIX A. Teacher Perception Survey

139

APPENDIX B. Student Data Spreadsheet Template

144

APPENDIX C. South Fayette Township School District Approval Letter

145

APPENDIX D. Institutional Review Board Approval Application

146

APPENDIX E. First Response from the Institutional Review Board

159

APPENDIX F. Researcher’s Response to Institutional Board Review Request

162

APPENDIX G. Approval from Institutional Review Board

164

APPENDIX H. Current Screening Process Rubric

166

APPENDIX I. Initial Screening Process Rubric

167

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viii

List of Figures
Figure 1. Percentage of High School Graduates with Earned Credits in Mathematics
Courses

14

Figure 2. Percentage of High School Graduates Progressing Through Mathematics
Courses

16

Figure 3. Percentages of High School Graduates and Their Different High School
Mathematics Course Sequences

17

Figure 4. Secondary Mathematics Course Sequencing, South Fayette Township School
District

18

Figure 5. Percentage of Schools with Tracked Courses by Content Area in Grade Eight
22
Figure 6. Percentage of White Students and the Prevalence of Tracking

27

Figure 7. Percentage of Low-Income Students Receiving Free or Reduced Lunch and
the Prevalence of Tracking

29

Figure 8. Distribution of Grade 12 Mathematics Course Taking Based on Grade Seven
Mathematics Grade Averages

35

Figure 9. Correlational Analysis- Total Points from Screening Compared to Pre-Algebra
Cumulative Grades

73

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ix

Figure 10. Correlational Analysis- Total Points from Screening Compared to Algebra I
Cumulative Grades

74

Figure 11. Correlational Analysis- Total Points from Screening Compared to Honors
Geometry Cumulative Grades

75

Figure 12. Correlational Analysis- Total Points from Screening Compared to Honors
Algebra II Cumulative Grades

77

Figure 13. Correlational Analysis- Total Points from Screening Compared to Honors
Pre-Calculus Cumulative Grades

78

Figure 14. Correlational Analysis- Total Points from Screening Compared to Advanced
Placement Calculus AB Cumulative Grades

80

Figure 15. Retention Rates in Accelerated Courses

82

Figure 16. Teacher Responses to Accurate Placement in Advanced Courses

83

Figure 17. Teacher Responses to Accurate Placement of Non-Accelerated Students

85

Figure 18. Teacher Responses to Characteristics of Advanced Students Who May Not
Have Been Appropriately Placed

86

Figure 19. Teacher Responses to Characteristics of Non-Advanced Students Who May
Not Have Been Appropriately Placed in General Mathematics Courses

88

Figure 20. Each Criterion’s Exiting Student Report Based on Scores

89

PLACEMENT CRITERIA FOR MATHEMATICS ACCELERATION

x

Figure 21. Amount for Each Type of Exited Student Based on Rubric Scores for Each
Criterion

91

Figure 22. Comparison of Students Who Exited and Did Not Exit Accelerated Pathway
Based on Rubric Scores for Each Criterion

93

Figure 23. T.O.M.A. Raw Scores of Exited Students

95

Figure 24. Curriculum-Based Assessment Results of Exited Students

96

Figure 25. Cumulative Fifth Grade Averages of Exited Students

97

Figure 26. Most Accurate Criterion from Teachers’ Perspectives

98

Figure 27. Distribution of Grade 11 Mathematics Course Taking Based on Grade Seven
Mathematics Grade Averages (Cohorts 1 and 2 only)

113

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xi

Abstract
The purpose of this mixed methods study is to provide data points that address the
accuracy of the screening process for students to be accelerated in mathematics courses
prior to entering middle school in the South Fayette Township School District. This
screening process is a decision that happens prior to sixth grade but has long-lasting
implications related to the courses students can take in high school and their trajectory for
college and career goals. Prior to this study, the district had never conducted a review of
the process that was internally developed. The three research questions that drove this
study led to an evaluation of student data related to the five most recent cohorts of
students that had been accelerated (150 students), as well as a review of participant
feedback from a mathematics teacher perspective survey in which 17 teachers completed.
The outcome of this study indicated that the screening process has shown to be successful
for over 90% of the students that had been identified through the screening process.
However, there was a recommendation that resulted from both the student data and
teacher feedback for the district to consider changing one of the criteria in order to
enhance the screening process.

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1

CHAPTER I
Introduction
Background
South Fayette Township School District has had an accelerated mathematics
pathway that begins in sixth grade in place for over twenty years. However, eight years
ago, coupled with the implementation of the new Pennsylvania Core Standards for
Mathematics, the district established a new set of criteria for determining whether or not
students would qualify for placement in this accelerated pathway. The district chose to
maintain utilization of the criteria during the timeframe when students are exiting fifth
grade and prior to them entering sixth grade. The criteria include three components:
cumulative grade average for fifth grade mathematics, the raw cumulative score from two
subtests from the Third Edition of the Test of Mathematical Ability (T.O.M.A. 3), and the
percentage correct on a comprehensive summative assessment based on the general sixth
grade mathematics course’s standards. Since these three criteria are from different
sources and use different scales and metrics, the performance on each is converted to
points ranging from 0 to 5, and students earn a point total out of 15 points (Appendix H).
For the first two years of the new screening process, the Second Edition of the Test of
Mathematical Ability (T.O.M.A. 2) was used, and the raw point totals were distributed
slightly differently on the scoring rubric (Appendix I).
As the coordinator of the screening process, principal of South Fayette Middle
School, and former middle school mathematics teacher in the district, I have been

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involved in all aspects of the criteria process, from the development of the criteria to the
implementation of their use. Since these criteria have been in place, approximately 10%
of a grade level’s student population each year has qualified for the accelerated
mathematics course in sixth grade, ranging from 26 to 32 students. The accelerated
mathematics course in sixth grade is Pre-Algebra. If students remain on this pathway,
they have the ability to take Advanced Placement (AP) Calculus BC in their senior year.
This is not the case for students who do not qualify; they can only reach AP Calculus BC
by forgoing an elective in high school in order to double-up on taking mathematics
courses.
Purpose of Study
The mathematics course trajectory that is set into motion by the outcomes of the
students’ performances according to the criteria can allow or prevent students from
enrolling in the highest level of mathematics coursework by graduation. The intention of
this criteria is to appropriately identify students for acceleration, yet there has been no
research conducted thus far to support the accuracy of this criteria.
Since the qualification process is an exclusionary academic decision that occurs
once as students enter sixth grade, this research will be informative as to whether or not
the criteria are accurately identifying the mathematics placement of students.
Additionally, as a former mathematics educator, I am interested in analyzing the data
from multiple cohorts of students who have already been accelerated in mathematics to
identify if the use of the current criteria is accurate in their placement determination and
to determine if a certain criterion is a stronger indication of successful placement than the

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3

other two. The mathematics teachers’ perspectives, voiced through survey responses,
will also be valuable as they will provide a set of observations related to the
appropriateness of student placement in their accelerated mathematics courses.
Potential Outcomes
One potential outcome of the research would be the confirmation that the
established criteria have a strong alignment with identifying the correct students and that
those students are achieving success in the accelerated mathematics sequence. If the data
and research support this, then there would be no impact to the existing system. However,
if the data and research indicate that one or more of the criteria does not correlate to
accurate placement of students and subsequent success, then the qualification process
may need to be altered by the district.
Financial Implications
Due to the importance of identifying the most accurate criteria for determining the
proper students for acceleration, the selection of and investment in the right tools must be
achieved. The cost to conduct the current screening process for students to qualify for the
accelerated mathematics sequence is very minimal. Of the existing three criteria, two are
data pieces that are generated at no or minimal cost. One criterion, the students’
cumulative grade averages in fifth grade math, comes from a query in the district’s online
grading system. The second criterion, the comprehensive, summative assessment tool of
the general sixth grade mathematics course, has already been designed by district
mathematics educators. This assessment tool only needs to be copied each year for the

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4

students to take. An expense that the district has to incur related to the criteria is the
purchasing of the Third Edition of the Test of Mathematical Ability (T.O.M.A. 3). Copies
of these exams are budgeted for annually. Additionally, six teachers are paid an hourly
rate each year to score the two assessments. Three fifth grade teachers score the
T.O.M.A. 3 tests, and three sixth grade teachers score the comprehensive sixth grade
assessments. These teachers are paid an hourly rate and are not permitted to exceed 5
hours of grading. The hourly rate as of the 2020-2021 school year is $40.75.
Should the data and research from this Capstone Project indicate that the criteria
are not accurately identifying students for the accelerated mathematics sequence, there
could be a financial impact on the district of selecting an alternate (or multiple alternate)
screening tool(s). The funds allotted to pay the six teachers may also be impacted if new
tools would require less or more time for scoring.
Research Questions
To guide the research, the following questions have been identified:
1) Is the screening process for advanced mathematics coursework accurately
identifying students for acceleration based on the criteria?
2) Do teachers perceive that students are accurately placed in advanced
mathematics courses based on the qualification process that occurs prior to the
start of sixth grade?

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5

3) Of the three criteria used in the screening process, does a pattern exist as to a
certain criterion indicating a greater likelihood of success in mathematics
advancement?
Presumptive Outcomes
Based on these research questions, my professional involvement in the
qualification process, and the analysis of the quantitative and qualitative data, I anticipate
that the accuracy of the placement criteria for accelerated mathematics will align with
this current set of criteria. I expect that nearly all students who have been accelerated in
sixth grade will continue on this advanced pathway in subsequent years. If I were to
anticipate one of the criteria that would be most indicative of success in the accelerated
mathematics pathway, I would identify that as the T.O.M.A. 3. However, due to a
potential lack of awareness of this assessment tool by most mathematics teachers, I
believe that the teachers’ observations and feedback will emphasize the use of a different
criterion, particularly the comprehensive sixth grade summative assessment.

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CHAPTER II
Literature Review
Historical Progression of Mathematics in Public Schools
Research has found that when students take advanced level mathematics courses
in high school the result correlates to positive outcomes. Students taking such courses
yield higher assessment scores, a higher likelihood of enrolling in college and completing
a bachelor’s degree, greater returns in the labor market, and increased career satisfaction
(Altonji et al., 2012; Bozick & Lauff, 2007; Chen, 2009; Nord et al., 2011; Pellegrino &
Hilton, 2012). Completing Algebra I in eighth grade sets students on a trajectory in high
school to surpass the completion of Geometry and Algebra II, leading to enrollment in
courses that are deemed advanced level. Due to this trajectory, there has been a recent
nationwide effort to increase student engagement in advanced level courses and, most
notably, in having students take Algebra I by eighth grade (Bernhardt, 2014; Reed, 2008;
Domina, 2014, Simzar & Domina, 2014). Furthermore, Finkelstein et al. (2012) noted in
their research that “success in high-level mathematics in high school is predictive of postsecondary success and careers in STEM fields” along with finding that a close
relationship exists between a student’s level of success in middle school mathematics and
his/her level of performance in high school courses (p. 1). However, historically,
mathematics courses such as Algebra I and Geometry were not always courses deemed
important for all students.

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Klein (2001/2003) claims that educational leader William Heard Kilpatrick had
the greatest influence on elementary and secondary mathematics instruction in the early
1900s. Kilpatrick believed that mathematics instruction should only be relevant to
practical value and that the traditional high school mathematics curriculum should only
be available as an indulgence to a select few (Klein, 2001/2003; Loveless, 1998).
Kilpatrick’s approach to mathematics instruction reigned in the public-school system,
although not unchallenged, through to the 1950s. This practical approach to mathematics
instruction was strongly reinforced by the Life Adjustment Movement in education in the
1940s.
It became apparent during World War II that there was a lack of basic skills
needed for bookkeeping and gunnery by army recruits (Klein, 2001/2003). As a result,
the Life Adjustment Movement had mathematics programs in schools focus on real-life
skills such as “consumer buying, insurance, taxation, and home budgeting, but not on
algebra, geometry, or trigonometry” (Klein, 2001/2003, pp. 178-179). In a separate
account of the progression of education and the Life Adjustment Movement, Ravitch
(1983) found that at the height of this approach to education, the curriculum across the
country lacked so much rigor and intellect that a high school principal described the
approximate 30% of his students enrolled in academic courses as wasting their time.
These practical approaches to mathematics instruction during the early 20th century
shifted later in the late 1950s to what is described as the New Math period (Klein,
2001/2003).

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Klein (2001/2003) posits that, in response to national embarrassment of the
U.S.S.R.’s launching of the first satellite into space, the United States wanted to improve
the quality of mathematics instruction in public schools. The New Math movement
yielded new curricula for elementary, junior, and senior high schools, as well as the
introduction of calculus as a high school course (Klein, 2001/2003; Loveless, 1998). The
New Math period waned by the early 1970s when the nation shifted to return to basic
skills instruction in mathematics. This period of mathematics education is labeled as the
Open Education Movement and was a reappearance of the fundamentals presented by
Kilpatrick in the beginning of the 20th century (Klein, 2001/2003). Lasting less than a
decade, the Open Education Movement received great criticism due to national
recognition of the poor quality of mathematics education it delivered in public schools.
In 1983 a commission under the leadership of the U.S. Secretary of Education
produced a report on the status of public education. The report, A Nation at Risk: The
Imperative for Educational Reform, highlighted numerous issues in education,
specifically stating the inadequacies in mathematics education (Klein, 2001/2003). A
Nation at Risk reported that only 31% of the nation’s graduates were completing
intermediate algebra by the time of graduation (United States, 1983; Klein, 2001/2003).
Notably, in 1989 the National Council of Teachers of Mathematics (NCTM), with
support from the National Science Foundation (NSF), produced the Curriculum and
Evaluation Standards for School Mathematics, which was “comprised of sections
devoted to general standards for the bands of grades: K-4, 5-8, and 9-12” (Klein,
2001/2003, p. 185). The NCTM Standards placed an emphasis on the use of technology

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and manipulatives in mathematics instruction, particularly calculators, as well as the
concept of constructivism, which promoted student-centered learning and mathematical
principles being taught through real world problems and discovery (Klein, 2001/2003).
Although the NCTM Standards were widely utilized and implemented in the 1990s, the
development of mathematics curricula informed by these standards did not uniformly
occur nationwide, nor were all states and districts following the guidance constructed by
NCTM.
By the turn of the 21st century, states and local school districts were still the
authorities governing mathematics standards and curriculum. However, when the No
Child Left Behind (NCLB) Act was passed in 2001, all public schools became
accountable for their students incrementally reaching certain achievement levels in the
areas of reading and mathematics through standardized testing in grades 3 through 8, as
well as once in high school (Klein, 2015). The goal of this act was for all students to
reach the proficiency level on their state assessments by the year 2013-2014. According
to Klein (2015), many critics feel that the scope of the mathematics curriculum during
this period became too narrow and heavily focused on preparing students for the
standardized test due to the NCLB Act.
The current state of mathematics instruction comes as a result of the development
of the Common Core State Standards for Mathematics (CCSSM) in 2010. Again, these
standards were not a national curriculum, but another attempt to provide standards
nationwide that, according to Akkus (2016), “were shaped to guarantee that all students
graduate from school with the necessary skills and knowledge to achieve in school,

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10

profession, and life, regardless of where they live” (p. 1). Pennsylvania was one of the
states to adopt the CCSSM; however, the standards were not officially adopted until
2014, after undergoing modifications to fit the needs and desires of the state’s education
department. These standards were also renamed as the Pennsylvania Core (PA Core)
Standards (Pennsylvania School Boards Association, 2015). The modifications were
made in order to withhold control of educational standards at the state level, as well as to
include certain standards that the state’s department of education believed to be crucial
(Pennsylvania Department of Education, 2013). Districts, including South Fayette
Township School District, had to make a quick transition to adopt and phase in the new
PA Core Standards in one year, causing rapid and simultaneous curricular and
instructional changes within Pennsylvania’s public school. Pennsylvania joined over 40
other states who currently use the CCSSM standards (or a modified version), as well as
the accompanying mathematical practices to guide their schools’ K-12 mathematical
curricular framework (Akkus, 2016; Polikoff, 2017). In a collective statement by leading
mathematical educational organizations, the National Council of Teachers of
Mathematics (NCTM), the National Council of Supervisors of Mathematics (NCSM), the
Association of State Supervisors of Mathematics (ASSM), and the Association of
Mathematics Teacher Educators (AMTE) describe the Core’s curricular framework as
“the foundation for the development of more focused and coherent instructional materials
and assessments that measure students’ understanding of mathematical concepts and
acquisition of fundamental reasoning habits, in addition to their fluency with skills”
(National Council of Teachers of Mathematics et al., 2010).

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Over the course of history, mathematics instruction has vacillated between the
need to prepare students for their practical futures and the desire to develop students who
are globally competitive. After the tribulations of trying to achieve one or the other by
way of different instructional eras, many practitioners and scholars believe that the
CCSSM accomplishes both the need and the desire- the practical and the competitive.
Course Sequencing of Mathematics Curricula
Because the United States still does not have a national curriculum, even with the
CCSSM in place in over 80% of the states, the sequencing of mathematics courses at the
secondary level can vary by district and state. The Common Core Curriculum does
recommend a sequencing of mathematics courses after fifth grade, and this sequence is
absent of tracking or ability group placement (Summer, 2011). The sequencing takes all
students, heterogeneously grouped, through one course to the next, by grade level not
ability, and integrates mathematics coursework (e.g. Algebra I, Geometry, Pre-Calculus,
etc.) at the appropriate levels. The sequencing outlined by the Common Core Curriculum
after fifth grade is Math 6, Math 7, Math 8, Secondary Math I, Secondary Math II,
Secondary Math III, and Secondary Math IV. California is an example of a state that
provided this sequential coursework, as well as course nomenclature, for its schools to
adopt. California does not mandate this sequencing in high school, and in 2015 a review
by EdSource of the state’s 30 largest school districts found that only about half of those
large districts moved away from a traditional sequence to the integrated design of the
Common Core (Harlow, 2015). California is not alone in not having schools fully adopt
the Common Core’s integrated approach. For example, in Pennsylvania, some schools

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did realign their courses to the Core’s recommended sequence, at least through eighth
grade, but nearly all schools maintained their traditional approach of single-topic courses
in high school such as Geometry, Algebra II, and Pre-Calculus. This pattern of traditional
sequencing tends to be common practice across the country, especially due to the
familiar, tiered nature of mathematical concepts.
At the middle school level, an analysis of school tracking programs conducted by
Schmidt (2009), reveals that 27% of U.S. eighth grade students attend a non-tracked
school in which only one mathematics course is available to all students. The remaining
73% of eighth grade students attend a tracked school that offers two or more different
mathematics courses or tracks. At the high school level, the National Center for
Education Statistics, under the United States Department of Education, conducted the
High School Transcript Study (HSTS) in 2009, gathering data about course pathways in
mathematics from a sample of 37,700 high school graduates from approximately three
million public and private schools from across the nation (National Assessment of
Educational Progress, 2009/2018). From this study, it was determined that the most
common pattern of course sequencing in mathematics for high school students was
Algebra I in ninth grade, Geometry in 10th grade, Algebra II in 11th grade, and higherlevel courses (Trigonometry, Precalculus, or Calculus) in twelfth grade (National
Assessment of Educational Progress, 2009/2018). However, course selection and
achievement levels in middle school, along with available mathematics courses in high
school, local and state requirements for graduation credits, and students’ interests and
levels of motivation also impacted the course sequencing for students at the high school

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level (Lee et al., 1998; Newton, 2010). As a result of these factors, the common course
sequencing pattern previously described may not be prescriptive for all high school
students. Based on the data collection from the HSTS, Figure 1 displays the distribution
of percentages of students enrolled in various mathematics courses in each of the four
high school years.

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Figure 1
Percentage of High School Graduates with Earned Credits in Mathematics Courses
70
60

Percentage of Students

50
40
30
20
10
0

¥
Calculus

Precalculus

Other
Algebra II
advanced
mathematics

Grade 9

Grade 10

Geometry

Algebra I

Grade 11

Below
Algebra I

No
mathematics

Grade 12

Note. ¥ Reporting standards not met. Adapted from “Paths Through Mathematics and
Science: Patterns and Relationships in High School Coursetaking” by J. Brown, B.
Dalton, J. Laird, and N. Ifill, 2018, National Center for Education Statistics, p. 10.
Copyright 2009 by the National Center for Education Statistics.
Figures 2 and 3, also from the HSTS, display the variety of actual course
sequences that represent the students in the study. In Figure 2, each semi-circle, by its
size, represents the percentage of students in a given course at a particular grade level.
Then, following the arrows from each semi-circle, the percentage of students taking
various subsequent mathematics courses is revealed. Figure 3 is a chart displaying

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15

common high school mathematics course sequences and the percentage of students from
the HSTS that followed each of these pathways.

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Figure 2
Percentage of High School Graduates Progressing Through Mathematics Courses

Note. Reprinted from “Paths Through Mathematics and Science: Patterns and
Relationships in High School Coursetaking” by J. Brown, B. Dalton, J. Laird, and N. Ifill,
2018, National Center for Education Statistics, p. 10. Copyright 2009 by the National
Center for Education Statistics.

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Figure 3
Percentages of High School Graduates and Their Different High School Mathematics
Course Sequences
Most frequently taken mathematics pathways

Percentage

Algebra I – Geometry – Algebra II – No math

10.1

Algebra I – Geometry – Algebra II – Precalculus

9.7

Algebra I – Geometry – Algebra II – Other advanced mathematics

7.4

Algebra I – Geometry – Algebra II – Algebra II

3.2

Geometry – Algebra II – Precalculus – Calculus

7.8

All other mathematics pathways (1,015 Total)

61.8

Note. Course names may include other labels. For example, Algebra II includes courses
such as Linear Algebra and Secondary Math 3. Adapted from “Paths Through
Mathematics and Science: Patterns and Relationships in High School Coursetaking” by J.
Brown, B. Dalton, J. Laird, and N. Ifill, 2018, National Center for Education Statistics, p.
10. Copyright 2009 by the National Center for Education Statistics.
Course Sequencing at a Local District
South Fayette Township School District is a suburban school district in
southwestern Pennsylvania with over 3400 students. The district has consistently had
high achievement scores on the state standardized assessment since the PA Core
Standards were adopted in 2014. Based on the South Fayette Middle School’s Program of

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Studies, two tracks of mathematics course sequences are offered, similar to the majority
of middle schools found in Schmidt’s (2009) study. South Fayette High School’s
sequencing is similar in progression to that described in the HSTS; however, there is a
more extensive set of course offerings. The mathematics curriculum and sequencing,
beginning in sixth grade, is shown in Figure 4. The middle school pathway consists of
two fixed sequences, including the accelerated pathway and the on-level pathway. As
shown in Figure 4, the options for students expand in the high school, resulting in
multiple, flexible pathways.
Figure 4
Secondary Mathematics Course Sequencing, South Fayette Township School District
On-level Pathway
6th Grade
7th Grade
8th Grade
Transition to High
School Pathways
9th Grade
10th Grade
11th Grade
12th Grade

Mathematical
Functions
(General Course)
Pre-Algebra
Algebra I
On-level Pathway

Geometry
Algebra II
Algebra
III/Trigonometry
Precalculus or
Honors
Precalculus

Accelerated
Pathway
Pre-Algebra

Algebra I
Geometry
Accelerated
Pathway (On-level
during Middle
School)
Honors Geometry
Honors Algebra II
Honors Precalculus

Advanced Placement
Pathway (continuation
of Accelerated Pathway
from Middle School)
Honors Algebra II
Honors Precalculus
AP Calculus I

Honors or AP
Calculus I

AP Calculus II or Honors
Linear Algebra

Regardless of the structure of the course offerings that vary from school to school,
sequencing of mathematics courses in middle and high schools rely on the use of

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prerequisite courses, like in the case that Algebra I is typically a prerequisite for Algebra
2 (Finkelstein et al., 2012). Finkelstein et al. (2012) recognize that not all students have
seamless course-taking patterns through their secondary careers. For example, a district
may require students to repeat a course, like Algebra I, if they do not reach a certain
grade expectation and, therefore, do not continue in the traditional pattern. Likewise,
students may accelerate beyond the typical course-taking pattern as early as middle
school, which has been shown to closely relate to continued achievement in accelerated
or advanced high school mathematics (Finkelstein et al., 2012; Stevenson et al., 1994;
Wang & Goldschmidt, 2003). Also, as Summers (2011) summarized, “students who are
placed in accelerated or advanced mathematics courses following elementary school are
better prepared for the postsecondary education of their choice” (p. 7).
Acceleration, Ability Grouping, and Tracking
In education there are several terms, at times used interchangeably, that describe
the learning experiences in which students in the same grade level are enrolled in
different levels of courses. Some of these terms include acceleration, ability grouping,
and tracking. Acceleration is defined by Pressey (1949) as “progress through an
educational program at rates faster or at ages younger than conventional” (p. 2). This
framing of acceleration has been reiterated by several researchers over time (Carafella,
2016; Colangelo et al., 2004; Ma, 2002; Smith, 1996). Expanding on Pressey’s definition,
researchers Southern et al. (1993) actually categorized 17 different types of acceleration.
Of those numerous types, single subject-matter acceleration is noted due to it being the
most commonly used practice for mathematics acceleration in schools. Specifically,

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single subject matter acceleration is when a student receives higher grade level
instruction by attending class in an advanced grade or by being in a class of similarlyaged peers in which the higher-grade-level content is instructed (Southern & Jones,
2015). In regards to ability grouping, Tieso framed the outcomes of Kulik’s research on
this practice by defining it as a method “that places students into classrooms or small
groups based on an initial assessment of their levels of readiness or ability (Tieso, 2005,
pp. 61-62). Tracking selectively places students into a different sequence of courses
based on ability (Chiu et al., 2008; Domina, 2014; Klapproth, 2015; Loveless, 1998;
Mulkey et al., 2005). As a result of Loveless’s (1998) research with tracking, he
identified three common tracks in the American public schooling system: a high-track,
with advanced level or honors courses that prepare students for colleges and universities,
a general track that serves the greatest population of students and provides them with
enough exposure for whatever their post-secondary plans may be, and a low track, with
low-level or vocational classes that prepare students for consumerism and basic adult
functioning. For the purposes of this study and further synthesis of research, these terms
may be interchangeably used.
The process of separating students into higher level courses through acceleration
has been in practice in the United States school system since the mid-1800s due to the
expanding school system as well as a result of industrialization, urbanization, and diverse
immigration populations (Chiu et al., 2008; Kozol, 1991; Loveless, 1998; Tyack &
Hansot, 1982). Notably, as early as 1862, the St. Louis public schools implemented a
flexible promotion program in which students advanced to higher grades or courses based

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21

on their individual level of readiness and achievement (Kulik, 2004). However, the
passing of Title IV, part of the Civil Rights Act in 1964, impacted accelerative practices
because the act prohibited these practices from being discriminatory in any way against a
student's race, color, or origin (Chiu et al., 2008; Loveless, 1998). For the last century and
through the evolution of tracking practices, these practices have existed with the intent to
accomplish three common goals: to increase the achievement level of students who
performed beyond the mixed, general ability level, to help students improve their selfesteem, and to assist teachers in being more effective at meeting different levels of
learners (George, 1988). Additionally, according to Oakes (1990), ability grouping occurs
more expansively in the United States than any other country as revealed by the Second
International Math Study (SIMS), which studied mathematical practices in schools across
the globe.
It is argued that mathematics is the subject area in which accelerative practices are
most frequently applied because proper grouping is most beneficial in these classes and
has the most significant implications on career attainment (Mulkey et al., 2005;
VanderHart, 2006). Additionally, according to Renzulli and Reis (2003), as well as
Passow (1996), accelerative practices are best applied to sequential content areas,
particularly mathematics. As shown in Figure 5, Loveless’s (2013) synthesis of NAEP 8th
grade data between 1990 and 2011, mathematics maintained the practice of tracking
longer and more consistently than any other content area. It can also be gleaned from
these statistics that tracking in the subject of mathematics has a significantly higher

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22

prevalence than the other areas, maintaining that about three-quarters of schools use some
form of tracking.
Figure 5
Percentage of Schools with Tracked Courses by Content Area in Grade Eight
Year

Mathematics

English

Science

History

Language
Arts
2011

76

*

-

-

2009

77

*

-

-

2007

75

*

-

-

2005

73

*

-

-

2003

73

43

-

-

2000

73

*

26

-

1998

¥

32

¥

15

1996

71

35

21

¥

1994

72

37

19

17

1992

73

50

¥

¥

1990

75

60

29

29

Note. *Tracking question was not asked. ¥ No data available– Tracking was not reported.
The statistics shown are percentages of schools that were included in the dataset that
indicated the presence of ability grouping or tracking practices based on responses from

PLACEMENT CRITERIA FOR MATHEMATICS ACCELERATION

23

school principals. Adapted from “How Well Are American Students Learning” by T.
Loveless, 2003, Brown Center Report of Education, 3(2), p. 17. Copyright 2013 by The
Brookings Institution.
Approaches to Mathematics Acceleration
Although mathematics is the subject area in which acceleration happens most
frequently, there is not a uniform grade level or method for which acceleration occurs. At
the elementary level, since classes are mostly self-contained, acceleration typically
happens in the form of within-class ability grouping (Loveless, 1998, 2013; Mulkey et
al., 2005; Oakes, 1990; Slavin, 1987, 1988). Within-class ability grouping is when
students are placed into smaller groups for instruction and can be configured as
heterogeneous or homogeneous groups. The group’s composition is typically determined
by the teacher and based on the purpose for grouping, as well as the students’ abilities,
skills, and/or interests (Gentry & MacDougall, 2009). Unlike what typically occurs at the
secondary level, within-class ability grouping at the elementary level is intended to be
flexible where the teacher frequently assesses the students and reassigns them to different
groups based on the results of those assessments (Tieso, 2003). According to Loveless
(2013), over the last 20 years the occurrence of within-class ability grouping in
mathematics has increased from 40% to 61% at the upper elementary level. Loveless
(2013) believes that the increase in this grouping method is a pedagogical response by
teachers due to accountability being linked to standardized testing.
The stratification of acceleration for single-subject advancement, in which
students are placed into homogeneous groups and, for the duration of these school years,

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24

into inflexible sequences of courses, most commonly begins to occur at the middle school
or junior high level (Loveless, 1998, 2013; Lucas, 1999; Mulkey et al., 2005). Although
the stratification of ability grouping continues to occur at the high school level, the
rigidity of course sequencing and selection decreases and opportunities for a variety of
accelerative options increase for students at different achievement levels (Loveless,
1998).
Research indicates that acceleration beginning in middle school is largely
determined by achievement on standardized tests, teacher subjectivity, and parental
influence (Bitter & O’Day, 2010; Hallinan, 2003; Kelly, 2007; Loveless, 1998; Meehl,
1954; Oakes, 1985; Useem, 1992). For example, the Wake County Public School System,
which is the largest school system in North Carolina and the 15th largest in the nation,
utilizes the following criteria for acceleration in mathematics: nomination by a parent or
educator and a score of 80% or higher on a standards-based mathematics assessment
(Hemelt & Lenard, 2018). Similarly, in the state of California, data sources including
students’ scores on placement assessments, achievement in previous mathematics
courses, and the receipt of teacher recommendations determine students’ accelerative
placements (Huang et al., 2014; California Department of Education, 2015). In an urban
school district in Utah, the determination for acceleration after fifth grade occurs as a
result of students’ achievement levels on the Iowa Test of Basic Skills (ITBS) (Summers,
2011). Students’ results on the ITBS are compared to the achievement levels of students
across the United States, and, if a particular student’s results are high enough, he/she is

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25

accelerated into Pre-Algebra in sixth grade instead of the general course (Summers,
2011).
The South Fayette Township School District’s criteria for acceleration does not
fully align with all aspects of the aforementioned research. When students are considered
for acceleration upon entering sixth grade, the criteria include two placement
assessments, as well as the cumulative grade average from fifth grade mathematics.
Neither teacher nor parental recommendations play a role in the screening process for
acceleration. Additionally, the placement assessments that are utilized do not include
Pennsylvania’s state standardized assessment in mathematics. One of the assessments is a
curriculum-based assessment designed by the middle school math department in 2014.
This tool comprehensively assesses the proficiency level of students’ mathematical
knowledge with the on-level sixth grade curricular skills as determined by PA Core
Curriculum Framework. If students are accelerated in sixth grade, they would be placed
beyond the on-level sixth grade course and enrolled into Pre-Algebra, which would be a
single-subject advancement. The other assessment is the Test of Mathematical AbilityThird Edition (TOMA-3). Although the district only uses the raw data from two subtests,
the TOMA-3 contains four core subtests: Mathematical Symbols and Concepts,
Computation, Mathematics in Everyday Life, and Word Problems. It is a “tool used to
identify, describe, and quantify mathematical deficits in school age children” (TOMA-3:
Test of Mathematical Ability, 2012, para. 1). The third element used for the
determination of acceleration in sixth grade mathematics is the cumulative average of a

PLACEMENT CRITERIA FOR MATHEMATICS ACCELERATION

26

student’s grade performance in fifth grade mathematics from each of the grading
quarters.
Regardless of grade level for entry or criteria used to determine entry, there are
disparities in the composition of students who are enrolled in accelerated courses, even
after the passing of the Civils Right Act in 1964. As Webel and Dwiggins (2019) describe
it, “a student’s placement in a track is likely not determined solely by mathematical
ability but also influenced by societal and economic factors” (p. 6). Utilizing data from
the National Assessment of Education Progress (NAEP), it was determined that nearly
50% of White, non-Hispanic eighth graders, along with 67% of Asian eighth graders, are
enrolled in advanced mathematics courses in comparison to 16% of Black eighth graders
and 38% of Hispanic eighth graders (Braddock, 1989; Domina, 2014; Gutiérrez, 2008;
Lubienski & Gutiérrez, 2008). Similarly, another study conducted by Kelly (2004)
revealed that there is a Black-White course-taking gap in which data shows that the
likelihood of White students being enrolled in advanced mathematics courses is double
that of Black students. However, Kelly (2009) in continued research, found that this
course-taking gap for Black students being enrolled in advanced mathematics courses is
significantly less in predominantly all Black schools in comparison to non-Black or
integrated schools. Race also seems to be a factor in the presence of tracking
opportunities in schools. Based on the data from the National Assessment of Educational
Progress (NAEP) 8th Grade Mathematics Assessment, which was a national collection of
data, a correlation between the existence of tracking in schools and the school’s racial
composition, particularly the percentage of White students was evident. The NAEP data

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27

revealed that tracking is most common when about half of the student population in a
school is made up of White students (VanderHart, 2006). When White students make up
either the majority or the minority of a school’s population, leveled coursework or
tracking by ability happens less frequently (VanderHart, 2006). Figure 6 displays the
likelihood of tracking or not based on the percentage of White students.
Figure 6
Percentage of White Students and the Prevalence of Tracking
Percentage of White Student Population

Tracked

Not Tracked

Less than 20%

0.294

0.706

20% to 40%

0.589

0.411

40% to 60%

0.761

0.239

60% to 80%

0.613

0.387

80% to 90%

0.582

0.418

More than 90%

0.267

0.733

Total

0.545

0.455

Note. Adapted from “Why Do Some Schools Group by Ability?: Some Evidence from
the NAEP,” by P. VanderHart, 2006, American Journal of Economics and Sociology,
65(2), p. 450. Copyright 2006 by American Journal of Economics and Sociology, Inc.
Similar to determining that tracking has racial implications, Walston and
McCarroll (2010) found that students from higher economic status are twice as likely to

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28

take an advanced mathematics course in eighth grade in comparison to a peer from a low
economic status. Loveless (1998) claims that socio-economic status is even more
significant in fostering the segregation of students in different tracks than race. Data from
Loveless (1998) states that when students’ prior achievement is equally considered, there
is no evidence of race disparities between high and low track students; however, even
with identical achievement scores as their wealthier counterparts, students from poor
families are more frequently placed in low tracks due to potential reasons such as status
discrimination or lower levels of parental influence. Oakes (1985), a fervent opponent of
tracking, believes that tracking intensifies social inequalities. Oakes (1985) illuminates
racial and socio-economic disparities by making the claim that there are
disproportionately higher numbers of poor and underrepresented students in low track
courses compared to the number of affluent, White students in high tracks. As shown
with race from the NAEP data, there was also a correlation between the existence of
tracked courses in a school relative to the school’s population of students from low socioeconomic backgrounds (VanderHart, 2006). Figure 7 shows the propensity for tracking in
schools when the percentage of students receiving a subsidized lunch due to low socioeconomic status is a low percentage.

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29

Figure 7
Percentage of Low-Income Students Receiving Free or Reduced Lunch and the
Prevalence of Tracking
Percentage of Low-Income Student

Tracked

Population

Not
Tracked

None

0.542

0.458

1% to 10%

0.625

0.375

10% to 25%

0.529

0.471

25% to 50%

0.435

0.565

50% to 75%

0.413

0.587

More than 75%

0.221

0.779

Total

0.508

0.492

Note. Adapted from “Why Do Some Schools Group by Ability: Some Evidence from the
NAEP,” by P. VanderHart, 2006, American Journal of Economics and Sociology, 65(2),
p. 450. Copyright 2006 by American Journal of Economics and Sociology, Inc.
Accelerative practices in mathematics are not without consequence. Although the
process of determining students for acceleration may not be uniformly defined,
enrollment in advanced level classes by students of color and of low socio-economic
status has been found to generally be less in comparison to these students’ White and
wealthier counterparts.

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30

Outcomes of Mathematics Acceleration
There is an abundance of research that has been conducted regarding the impact,
or lack thereof, on students as a result of accelerative practices including students’
achievement levels, self-beliefs, social comparisons, racial and socio-economic
disparities, secondary and post-secondary success, and job acquisition. Of these areas, the
one outcome of acceleration with the greatest abundance of research is if a correlation
exists between mathematics acceleration and academic achievement. However, not all of
the research is in consensus regarding the presence and type of correlation between
acceleration and achievement. A conclusion from a study conducted by Fuligni and
Stevenson (1995) found that accelerating medium and high ability students positively
correlated with these students’ achievement at the 10th grade level. Hallinan and
Kubistchek (1999) similarly concluded that students in advanced mathematics classes at
the end of middle school made significantly more achievement gains by their second year
in high school as opposed to their peers in on-level courses. In contradiction, not all
studies support positive achievement gains for higher level students. For example, in a
meta-analysis conducted by Kulik and Kulik (1992), they concluded, after having
conducted fifty-one studies, that acceleration has negative effects on higher level
students’ achievement. Furthermore, Slavin (1993) conducted a meta-analysis of 27
studies and found that accelerative (tracking) practices had no effect on student
achievement. Although there has been evidence established on both sides of the
achievement argument, several studies conducted at the secondary level have come to the
conclusion that any increased achievement impacts due to acceleration have occurred

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31

explicitly because students take different courses and, thus, are exposed to more specific
mathematical content (Gamoran, 1987; Gamoran et al., 1997; Ma, 2000; Rock & Pollack,
1995; Schneider et al., 1998; Stevenson et al., 1994; Schmidt, 2009).
While disparities exist in the potential influence acceleration has on positive
achievement gains, there is less dispute over the inequities that exist in the demographics
of the population of students who are accelerated. Studies reveal that students from low
socio-economic status families and underrepresented ethnic backgrounds are
underrepresented in acceleration programs (Wyner et al., 2007). Moreover, Baker and
Stevenson (1986) posits that there is a research-based link between high socio-economic
status and a child’s placement in advanced coursework. The researchers believe that this
is the case because higher socio-economic status of parents leads to these parents being
more well-educated adults who then are influential managers of the trajectories of their
children’s school career (Baker & Stevenson, 1986; Useem, 1992). Upper-class, White
mothers have been shown to utilize their personal educational experiences, along with
their involvement in their children’s school, as well as social networks, in order to gain
advantageous knowledge so that they can best prepare and influence the track of
mathematics courses for their children (Lareau & Shumar, 1996; McGrath & Kuriloff,
1998; Useem, 1992). Agreeing with the powerful role of parents, Kifer (1986) identifies
the transition from sixth to seventh grade as a pivotal point in which tracking impacts
most students, and, since the students are too far away from cementing their future career
plans, their parents impact the likelihood of and decisions related to accelerated
placement.

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32

Numerous studies have also been conducted relating a student’s self-concept to
acceleration in coursework. The results have considerably different findings, spanning
from students that are accelerated having higher self-esteem (DeLacy 2000; Gross, 1992;
Olszewski-Kubilius, 1995, 1998; Rogers, 1991; Sayler, 1992) to acceleration having no
positive impact on students’ self-esteem (Gross, 1994; Swiatek, 1994) to students who
are accelerated having a decline in their self-esteem over time due to social comparisons
(Lupkowski, 1992). Oakes (1985, 1990) arrived at the same positive relationship on
students’ self-concept when they were advanced into the high track. Oakes’s (1985,
1990) research found that low-track and high-track students displayed and maintained
opposite self-esteem levels and self-images, relative to their tracking placements. Kulik
(1992), much like Lupkowski (1992), found impacts to students’ self-concept that
contradicted those of Oakes’s findings. These researchers, along with others, found that
tracking actually lowers the self-concepts of students in high tracks, while it increases
self-concepts of students in low tracks (Hallam & Ireson, 2008; Kulik, 1992; Lupkowski,
1992; Wigfield et al., 1998; Zeleke, 2004). The support to these claims is that when
students are homogeneously grouped in tracked classes, they are more realistically able to
assess their ability in comparison to their peers, and, those in higher tracks have their own
self-concepts challenged more by their peers of similar high abilities (Goldberg et al.,
1966; Lupkowski, 1992; Nicholson, 1998). Challenging these results, Ma (2002)
conducted a study examining the self-esteem levels of accelerated and non-accelerated
students in three distinct categories: gifted students, honors students, and regular
students. Ma’s (2002) raw data concluded that in all three populations accelerated

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33

students had only slightly higher self-esteem than their non-accelerated counterparts at
the high school level. The difference was the least significant in the gifted population and
the most significant in the regular population (Ma, 2002).
Tracking not only has lasting impacts on individuals’ self-concepts, but also on
their high school and post-secondary success. The synthesis of numerous studies has
indicated that successful completion of Algebra I by eighth grade will lead to a greater
likelihood of high school completion, as well as an employable future (Finkelstein et al.,
2012). For students who matriculated to a college or university, Hoyt and Sorensen
(1999, 2001) found through an analysis of students’ transcripts at Utah Valley State
College that students who took higher levels of mathematics in high school were less
likely to need remedial courses in college. Another set of studies found that when
students successfully complete advanced mathematics courses in high school beyond
Geometry, they will be more likely to attain a college degree (Trusty & Niles, 2003).
Reviewing several studies that interviewed accelerated and non-accelerated students in
regards to post-secondary plans, Kulik (2004) concluded that accelerative practices
increase students’ educational ambition and positively impact their long-term educational
plans. Furthermore, Adelman’s (1999) analysis of the High School and Beyond data
illuminated that the strongest correlation to college degree completion could be drawn
directly back to the highest level of mathematics completed by a student in high school.
As a result, it could also be concluded that since middle school mathematics
coursework and sequencing influence the opportunities for course selection in high
school, accelerating students in middle school has a significant impact on students’

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34

futures. This impact is quantifiably described in two different national studies. Rose and
Betts (2004), utilizing the High School and Beyond data, discovered that students who
took advanced level mathematics courses in high school earned higher salaries in their
respective careers, ten years after graduation, regardless of their demographics, family,
school characteristics, or even their highest degree earned, college major, or occupation.
Using the same data, as well as the National Longitudinal Survey of Youth, Levine and
Zimmerman (1995) found that not only are the salaries higher for individuals based on
their enrollment in advanced high school mathematics courses, but when females are
accelerated, their wages increased in comparison to other non-accelerated females.
Conducting a separate study to corroborate these findings relative to how middle school
mathematics coursework is an indicator for future success in high school and beyond,
Finkelstein et al. (2012) gathered a dataset that spanned six school years and included
over 24,000 students from school districts in California. With a strong correlation to the
previously conducted research, Finkelstein et al. (2012) concluded that “course
performance as early as grade seven is a strong predictor of future high-school course
enrollment” (Finkelstein et al., 2012, p. 9). Figure 8 displays the percentage of students
enrolled in each type of mathematics course in grade 12 based on their cumulative letter
grade performance from their mathematics course in grade seven, according to
Finkelstein et al.’s (2012) study. The greatest percentages of enrollment in advanced
level courses correlates more significantly to students with higher grade averages in grade
seven based on the data in this display.

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35

Figure 8
Distribution of Grade 12 Mathematics Course Taking Based on Grade Seven
Mathematics Grade Averages

Note. Reprinted from “College Bound in Middle School and High School: How Math
Course Sequences Matter,” by N. Finkelstein, A. Fong, J. Tiffany-Morales, P. Shields,
and M. Huang, 2012, The Center for the Future of Teaching and Learning, p.13.
Copyright 2012 by WestEd.
Similar to the results of Finkelstein et al.’s study, the dataset of the HSTS in 2009
also found a strong association between a student’s placement in ninth grade mathematics
and the highest level of mathematics completed by graduation (National Assessment of
Educational Progress, 2009/2018). The percentage of students taking the highest level of
mathematics, Calculus, was greatest in the population of students who were enrolled in
an accelerated course of Geometry or higher in ninth grade (National Assessment of

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36

Educational Progress, 2009/2018). This percentage was over quadruple in comparison to
students who were enrolled in the general, on-level course of Algebra I in ninth grade
(National Assessment of Educational Progress, 2009/2018). However, the HSTS did
show that successfully completing Algebra I in ninth grade still provided high school
students with the opportunity to complete advanced level mathematics courses above
Algebra II (National Assessment of Educational Progress, 2009/2018).
In summary, both the studies by Finkelstein et al. (2012) and the data from the
HSTS of 2009 conclude that a successful completion of Algebra I, whether by eighth or
ninth grade, is a significant determination of students’ future mathematics coursework.
Recognizing the importance of Algebra I, Finkelstein et al. (2012) argue that acceleration
is a critical decision and that students should not be enrolled into Algebra I unless they
have developed the necessary foundation for learning algebraic concepts and skills.
Regardless of the grade level or if students are accelerated or advanced by natural
sequencing to Algebra I, data shows that without readiness for the concepts and skills,
there are negative consequences for the students’ placements and performances in higher
level high school mathematics courses, as well as post-secondary mathematics courses
(Finkelstein et al., 2012).
Mathematics Teachers’ Perceptions of Acceleration
Although the outcomes based on the utilization of accelerative practices vary,
there are common beliefs as to the reasons that teachers support the existence of tracking
in mathematics. One of these beliefs is that the curriculum and/or pace offered to each
level in a tracking system is more appropriate and suitable for the students’ ability levels

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37

within each track (Levine, 1983; Reed, 2008; Reuman, 1989; Richer, 1976; Schmidt,
2009). Another shared belief is that mathematics, unlike other subjects, is hierarchical in
nature (Ruthven, 1987) and, as a result, students who have different sets of mastered
concepts cannot optimally work in the same environment on the same task (Zevenbergen,
2003). Although teachers profess that their philosophy is to prefer a diverse group of
learners in their classrooms, researchers have ultimately found that they believe
homogeneous groups established through tracking are simply easier to teach (Loveless,
2013; Spear, 1994; Webel & Dwiggins, 2019).
Expanding on these beliefs, Robert C. Spear conducted a qualitative study with 31
seventh grade teachers regarding their perceptions on the advantages and disadvantages
of ability grouping. The general responses to advantages of ability grouping were that
separate groups increase student learning, and ability groups are easier to prepare for and
to teach (Spear, 1994). Furthermore, this group of teachers believed that ability groups
provide the students with the type of instruction and learning tasks that they need
intellectually (Spear, 1994). Through the study, Spear also found that those who favored
ability grouping or tracking communicated about their instruction in ways that were more
content-centered, whereas the dialogue of those who preferred the advantages of
heterogeneous, mixed-ability groups was more student-centered.
While research data provides its own disadvantages to accelerative practices, the
teachers who participated in Spear’s study claimed another disadvantage as the power
that parental influence has relative to the decisions made about ability grouping (Spear,
1994). This group of educators did not believe that the placement of students actually

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38

represented their ability levels but rather that of the involvement that their parents had in
their education (Spear, 1994). Through empirical research, Boaler (1997) supported an
additional negative perception of tracking by teachers. Although teachers believed that
placing students in different tracks was a successful method to increase achievement
levels, they actually found that this practice brought greater levels of stress to the students
in the low track and demotivated the students who were just shy of qualifying for the high
track (Boaler, 1997; Boaler et al., 2000).
Oakes (1985) also suggests that teachers have distinct perceptions related to their
classroom environments and relationships with the students in different tracks. Teachers
describe more positive relationships with students in high-track classrooms and more
peer conflicts and behavioral concerns in low-track classrooms (Oakes, 1985).
Pedagogical choices and instructional methods of teachers in tracked schools have also
yielded different results for students enrolled by ability group. Although most teachers
believe that ability grouping or tracking is beneficial to each student, research has found
that the level of expectation by the teacher within tracked groups varies. Teachers have
demonstrated to expect more from students in high track classes and have provided them
with more responsibilities, as well as more challenging work (Hallam & Ireson, 2005;
Oakes, 1992). In these high tracks, the teachers have also instructed at a faster pace and
required the students to complete a greater workload (Boaler et al., 2000). Oakes (1985)
noted that teachers of high track courses display more enthusiasm, are more organized,
have a greater set of instructional strategies, and elicit more student engagement.
Conversely, it has been found that teachers lower their expectations for students in the

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39

low track, often seeking behavioral compliance and completion of remedial tasks (Oakes,
1985, 1992; Reed, 2008). Finley (1984) claims that when teachers are assigned to instruct
low track classes for a period of time, both their instructional skills and efficacy decrease.
The students’ awareness of these different behaviors has also not gone undocumented.
Boaler et al. (2000), after interviewing numerous secondary students about their
experiences in tracked classes, summarized the students’ perceptions:
When students were divided into ability groups, students in high sets (tracks)
came to be regarded as “mini-mathematicians” who could work through highlevel work at a sustained fast pace, whereas students in low sets (tracks) came to
be regarded as failures who could cope only with low-level work- or worsecopying off the board. This suggests that students are constructed as successes or
failures by the set in which they are placed… (p. 643)
Comprehensively, the collective research suggests that the presence of tracking
influences teachers’ perceptions about the abilities, behaviors, and needs of their
students; Reed (2008) describes this as a practice of teachers creating a prototype of a
profile for students in each track before even working with individual students.
Conclusion
The approach to mathematics instruction in public schools has evolved from one
that reserved the taking of advanced level mathematics courses for a privileged few to
one that strives to provide most students with the opportunity to take advanced level
mathematics courses by the end of high school. Over the last 30 years, different

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40

organizations have attempted to create a national mathematics curricular framework in
order to improve mathematics instruction. Although a uniform, national curriculum still
does not exist, there are significant similarities in the mathematics course sequencing that
exist across the country. This common course sequencing, moving from rigid and tracked
in middle school, to more flexible and extensive in high school, has been found to play a
pivotal role in the likelihood of post-secondary success for students. Besides traditional
course promotion through the natural sequence, students have also been able to reach
advanced level mathematics courses in high school due to acceleration. However,
accelerative practices and the impact of such practices on students have been a debated
topic for much of the last 125 years, with researchers rarely arriving at a consensus.
Loveless (1998) summarizes years of studying tracking and the inability to truly quantify
its impact on education by stating that “research on tracking and ability grouping is
frequently summarized in one word: inconclusive” (p. 14).
The purest objective of accelerating students would be to provide each student
with the opportunity to enroll in the level of curriculum that is needed based on
previously mastered skills, regardless of age or grade level. Furthermore, advocates of
acceleration would offer that it allows students to reap more beneficial outcomes,
especially due to the multiple studies linking advanced level course taking with collegiate
and career success. However, critics of such practices would assert that the practices are
discriminatory, damaging to certain demographics, and lacking strong, positive benefits.
Teachers’ perspectives on acceleration and tracking are also often found to be
contradictory. Although teachers’ narratives often describe an intent to teach

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mathematics through differentiation in heterogeneously-grouped classes, their actions
often demonstrate a preference towards the ease of homogeneously, tracked groups.
Evidence has been established that there are positive achievement gains for
students who are accelerated, and contradictory evidence has also indicated that there is
not a significant impact for this population of students. Collections of studies have
indicated less beneficial outcomes, when particular factors of tracking for students of
underrepresented races, as well as low socio-economic statuses are considered. Debate
has also ensued about how accelerative practices impact students’ self-concepts.
However, the greatest culminating take-away, supported by data from multiple recent
studies, is the impact that acceleration in middle school has on positioning students for
high school and post-secondary course taking options. Following those advanced
trajectories, it has been shown that students who are accelerated and then subsequently
and successfully complete the advanced coursework will have a greater likelihood of
career attainment and success (Adelman, 1999).

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CHAPTER III
Methodology
The researcher has played a significant role in the placement process for
mathematics acceleration since the inception of the process, and, in seeking evidenced
research for this study, the researcher also conducted a review of many literature sources.
As a result of these actions, the researcher was better able to construct a comprehensive
methodology for research. This methodology takes into account the validity of the
screening process that is currently used for identifying students for mathematics
acceleration, the teachers’ perception of the placement of students in advanced
mathematics courses, and the possibility of a certain criterion having a greater likelihood
of predicting success for students in advanced mathematics courses over time. This
chapter’s purpose is to provide a detailed account of the actions taken in order to
complete this action research project.
The objective of this section is to fulfill the need for research to be conducted
relative to the screening process for placement of students in South Fayette Township
School District’s advanced mathematics coursework pathway. Guiding the purpose of
this section and action research project are three research questions which will be listed
and further explained in detail. Additionally, an explanation of the methodology and how
it was utilized for this action research project will be outlined in order to provide an
understanding of the outcomes.

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As previously explained, this research was prompted as a result of the academic
implications that the screening process for advanced mathematics has on a student at
early grade levels. This exclusionary process, which is completed prior to a student
entering middle school, has never undergone an audit or been thoroughly researched
regarding its accuracy in identifying the proper mathematics placement of students. With
research indicating that mathematics is the subject area in which accelerative practices
are used most frequently, it is necessary for a district to know the lasting impacts the
screening process has on its students (Mulkey et al., 2005). To assist the reader in having
a better understanding of the value of this study, it is critical that additional district
demographics and details are provided, as well as the justification for the use of the
targeted compilation of student data and teacher perspectives. For the use of student data
and teacher perspectives, the approved process and use of proper consent will be further
described.
Purpose
The purpose of this research is to validate or recommend revisions to the district’s
screening process for placement of students in the advanced mathematics coursework
pathway at South Fayette Township School District. Additionally, this research will
validate recommended revisions related to the accuracy of screening through the use of
collected data and evidence. Since the screening process occurs before students enter
middle school and because students who are placed in advanced math classes in sixth
grade are able to reach higher levels of mathematics over the course of grades six through
twelve,, it is critically necessary to determine if this process displays accuracy over

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multiple cohorts of students. Sharing the evolution of the district’s advanced mathematics
screening process from its original design and impact to its current form and impact will
help the reader gain a better understanding of the value of this research.
Prior to the transition to the PA Core Standards and redesigning the screening
process for advanced mathematics in 2014, South Fayette Township School District had
utilized two criteria in order to determine students’ placement in their mathematics
coursework pathway prior to sixth grade. Those two criteria were teacher
recommendation and a cumulative fifth grade math average of greater than 90%.
Teachers reviewed the students who reached the minimum grade average and then
identified the students who they believed were best qualified for advanced coursework.
This subjective process led to noticeable outcomes which, along with the need to align
the mathematics curriculum with the PA Core Standards, drove the overhauling of the
screening process criteria into its current form. These outcomes included the placement
of over 50% of the entire student population in advanced mathematics as a result of
inflated fifth grade cumulative mathematics averages, parental influence on teachers’
decision making, and a large number of students struggling in advanced mathematics
coursework at the high school level. Regarding the parental influence in the process, the
district agreed with Spear’s research that accelerated placement was being unfairly
affected by parents, particularly of our predominantly upper and middle class White
families (1994). The fifth grade cumulative mathematics averages for three years were
also reviewed at this time, and it was determined that, on average, over 86% of fifth
grade students had a cumulative yearly mathematics average greater than 90%. As a

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result of these outcomes and the timing of the adoption of the PA Core Standards at the
state level, the screening process was redesigned by a team of mathematics educators
from the district, representing grades six through eight. The objectives of the new process
were to remove subjectivity and parental influence, base the qualification on evidence of
students’ preparedness and mathematical ability, and assess the prerequisite skills needed
by a student who would be entering an advanced mathematics course in sixth grade.
Knowing that whatever screening process would be developed and utilized would make
academic determinations for students that could impact them beyond high school,
including their trajectory for career and college success, the mathematics educators
wanted to identify criteria that would accurately place students both in the short-term and
long-term.
As a former sixth grade mathematics educator and a member of the educational
team that redesigned the screening process for advanced mathematics placement, the
researcher had firsthand involvement with the district-level decisions that led to the
selection of the three criteria in 2014. Due to a change in professional roles, the
researcher then had the perspective of reviewing the results of the screening process and
scheduling students for their sixth grade mathematics courses. Throughout this time and
in both roles, the researcher has wondered if the district’s process leads students to the
most appropriate placement, not only for middle school, but as the literature supports, for
success in students’ future college and career-related decisions and outcomes.
In order to validate the accuracy of the criteria for the screening process, three
questions were constructed. The data and feedback collected through the guidance of

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these research questions will impact the mathematics course trajectory of future students
attending South Fayette Township School District. These three questions are:


Is the screening process for advanced mathematics coursework accurately
identifying students for acceleration based on the criteria?



Do teachers perceive that students are accurately placed in advanced
mathematics courses based on the qualification process that occurs prior to the
start of sixth grade?



Of the three criteria used in the screening process, does a pattern exist as to a
certain criterion indicating a greater likelihood of success in mathematics
advancement?

Not only have these questions been developed to guide the research, but they were also
paramount in identifying the targeted principles for the literature review and the
construction of the questions utilized in the mathematics teacher questionnaire.
As illuminated in the literature review from Chapter Two, there are positive
outcomes for students related to taking advanced level mathematics courses in high
school which include higher assessment scores, a higher likelihood of securing a
bachelor’s degree, higher salaries in the labor market, and increased levels of career
satisfaction (Altonji et al, 2012; Bozick & Lauff, 2007; Chen, 2009; Nord et al., 2011;
Pellegrino & Hilton, 2012). With the weight of those potential outcomes for students
and the decision for acceleration in mathematics being made at the end of fifth grade in
the South Fayette Township School District, the study, which is being guided by the
three research questions, is intended to determine if the district’s process is valid.

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Although there are secondary outcomes that the researcher hopes to glean from
this research, the primary outcome is to determine if the criteria used in the process for
accelerating students in mathematics in sixth grade is accurately fulfilling their
intention. From this outcome, the district will know whether or not a change in the
process is necessary in order to better identify students for acceleration. Additional
outcomes include determining how teacher perspectives align with the student
placement data, as well as with the research found in the literature review, and whether
one of the criteria is a stronger indicator at predicting long term success in advanced
level mathematics courses than the others. Regardless of what all of the outcomes
indicate, the criteria used for the screening process impacts the educational pathway
for each student who comes through the district for several years, and the outcomes
will provide the district with information needed to make informed decisions.
Therefore, it is critical that the process is comprehensively reviewed and the district is
provided with thorough research in order to be informed about such an impactful set of
criteria.
Setting
The setting for this study is the South Fayette Township School District. The
South Fayette Township School District is a fast-growing suburban, public school that
is located in Allegheny County in southwestern Pennsylvania. The district educates the
residents of eight different communities within the township, and both the township
municipality and district share the same borders within its 21-square mile region. The
population of nearly 16,000 has rapidly increased by approximately 11% since 2010

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and 48% since 1980, with the quality of education provided by the school district and
available new housing as the top attractions. Consistently, the township has had at least
100 new homes built each year for the last 10 years. Conveniently located equidistantly
from the city of Pittsburgh as well as the Pittsburgh International Airport, South
Fayette Township School District is a bedroom community with a median household
income of over $86,000 and a median household property value of approximately
$154,000. Primarily residential, the district contains over 7,000 households and relies
heavily on local taxes. About half of the township is still underdeveloped and only
recently has there been an increase in commercial development.
Community Demographics
The most recent census data revealed that the per capita income is $45,733, and
the median income for a household in the township is $86,858. Individuals had a
median income of $47,378, with males having a median income of $48,750 and
females earning a median income of $33,534. The unemployment rate is 1.5%, and the
poverty rate is 4.2%. Within the township, 88.4% of the residents are White, 9.5% are
Asian, 1.4% are Hispanic or Latino, 1.0% are African American, 0.6% are two or more
races, and 0.3% are American Indian and Alaskan Native. Between 2015-2019, the
percentage of township residents that identify as a “foreign born person” was 7.4%.
which is slightly higher than the state average of 6.8%. This is important to note
because there has been an increase in the Asian population in the township, particularly
from South Eastern Asia, within the last ten years. This is also reflected in the statistic

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that 9.7% of the households in the township speak a language other than English in the
home.
Of the approximately 7,000 households in the township, the most significant
household type is families with children at 65%. The average household size consists
of 2.50 persons, and the average family size was about 3.20 persons. About 32% of the
households have children under 18 years of age residing in the house. Approximately
56% of the households had married couples living together; whereas, 8.1% of the
households contained a female householder with no spouse present, and 4% of the
households contained a male householder with no spouse present. The make-up of
households that had someone residing alone who was 65 years or older was 10.7%.
The following distribution breaks down the residential population by age: 26.8%
of the population is under 20 years old, 8.3% of the population is between 20-29, 13%
of the population is between 30-39, 16.5% of the population is between 40-49, 13.7%
of the population is between 50-59, 8.9% of the population is between 60-69, and
12.9% of the population is 70 or older. The median age of males is 40.9 and the
median age of females is 40.8. About 47.3% of the township’s population is male, and
52.7% of the population is female.
School District History and Demographics
South Fayette Township School District was formed in 1928 with the opening of
the LaFayette High School. Fast forward to the next century, and the district is still
considered by many residents to be the “jewel of the community,” uniquely defined by

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existing on a single campus environment. There are four schools: an elementary
school, grades K-2; an intermediate school, grades 3-5; a middle school, grades 6-8;
and a high school, grades 9-12. Additionally, on the campus, there are separate
administrative and student services offices. Adjacent to the campus, the district houses
its transportation office and fleet of buses. Considering the district from a fiscal
perspective, the annual budget is approximately $64 million with 73% coming from
local revenue sources, 23.7% coming from state revenue sources, and 3.3% coming
from federal revenue sources. South Fayette Township School District is a district
within the services of the Allegheny Intermediate Unit.
The district prides itself on its rich tradition of school colors, mascot, and motto.
The colors are Kelly green and white, the mascot is the lion, and the motto is
“Tradition, Pride, and Excellence.” The district’s mission statement is, “The mission
of the South Fayette Township School District, in partnership with the community, is
to cultivate academic, artistic, and athletic excellence of the whole child by fostering
the skills to be confident, ethical, empathetic, and responsible global citizens.” The
district employs 239 teaching professionals, 188 part-time and full-time staff members,
and 26 administrators or supervisors.
Currently, there are 3,438 students enrolled in the district with the racial
composition of the student population being 72.8% White, 18.9% Asian, 4.3% Two or
More Races, 1.9% Black, 1.9% Hispanic, 0.1% Native American, and 0%
Hawaiian/Pacific Islander. The gender make-up is 48.6% female students and 51.4%
male students. Among the over 3,400 students, 10.7% are economically disadvantaged,

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8.7% are in special education, 0.007% are identified as gifted, 1.4% are English
Language Learners, 0.1% are in foster care, 0.5% are deemed homeless, and 0.8% are
military connected. It is important to note that the percentage of students who are
identified as gifted is considerably low in the district due to its approach of providing a
spectrum of enrichment services that meet the needs of all learners. By meeting the
needs of the learners, fewer students are identified as in need of gifted services. The
district has 43 students attending charter schools, 30 enrolled in approved therapeutic
schools, and 81 attending Parkway West Career and Technical Center in grades nine
through twelve. Upon graduation from South Fayette Township High School, 92% of
the student population attends a college or university, 4% enrolls in a trade/technical
school, 3% enters the workforce directly, and 1% enlists in the armed forces.
In 2001, the footprint of the campus consisted of two school buildings on the
campus: the elementary school, housing kindergarten through sixth grades, and the
junior-senior high school, housing seventh through twelfth grades as well as the
administrative offices. Due to rapid enrollment increases, the campus has grown by
two buildings in the last twenty years with three additional renovation projects. The
South Fayette Township High School was built in 2002 but required a $30 million
expansion project in 2017. During the same time as the high school building
construction, the middle school underwent a multimillion-dollar renovation project in
order to house grades five through eight. In 2013-2014, a brand new intermediate
school opened for students in third through fifth grades. This was a $24.5 million
development. Again in the summer of 2020, the middle school had a minor renovation

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project in order to accommodate increased enrollment. The addition of seven new
classrooms occurred by reconfiguring existing interior spaces for the cost of just under
$1 million. The district has plans to add an additional primary center in the future and
to redistribute the grade levels across the buildings in order to absorb the projected
enrollment numbers.
The current enrollment at South Fayette Township High School is 1,071 students.
The racial demographics are 77.7% White, 15.1% Asian, 2.8% Two or More Races,
2.5% Black, 1.8% Hispanic, 0.2% American Indian/Native Alaskan, and 0.0%
Hawaiian/Pacific Islander. 53.3% of the student population is male and 46.7% of the
population is female. Of the approximately 1,000 high school students, 10.2% are
economically disadvantaged, 9.0% receive special education services, 0.1% are
English Language Learners, 0.2% are in foster care, 1.0% are deemed homeless, and
1.2% are military connected. South Fayette Township High School was ranked 1,253
among high schools in the nation, 45 among high schools in the state of Pennsylvania,
and 12 among high schools in the greater Pittsburgh area.
South Fayette Middle School, which now consists of grades six through eight, has
an enrollment of 835 students. The racial composition of the student body is 72.9%
White, 18.7% Asian, 4.6% Two or More Races, 2.3% Hispanic, 1.3% Black, 0.3%
American Indian/Native Alaskan, and 0.0% Hawaiian/Pacific Islander. 50.3% of the
student population is male and 49.7% of the population is female. With similar
statistics in student groups compared to the high school, the middle school has 11.7%
of the population identified as economically disadvantaged, 9.1% receiving special

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education services, 1.3% English Language Learners, 0.2% in foster care, 0.3%
deemed as homeless, and 1.3% with military connections. South Fayette Middle
School was ranked as the number one middle school in Allegheny County for four
consecutive years, from 2014-2018, based on state assessment achievement levels.
South Fayette Intermediate School, grades three through five, houses 801 students
comprising the following racial groupings: 71.2% White, 21.2% Asian, 4.8% Two or
More Races, 1.8% Black, 1.0% Hispanic, 0.0% American Indian/Native Alaskan, and
0.0% Hawaiian/Pacific Islander. The gender make-up is 49.2% males and 50.8%
females. Student groups have the following compositions: 10.1% economically
disadvantaged, 9.4% special education, 3.0% English Language Learners, 0.0% foster
care, 0.4% homeless, and 0.4% military connected. The Intermediate School also had
the top-ranking position in Allegheny County for three years, from 2014-2017, based
on the school’s academic achievement on state assessments.
South Fayette Township Elementary School, the only building not to have any
recent renovations or expansions, currently has 731 students enrolled in kindergarten,
first, and second grades. Within those 731 students, 68.1% are White, 21.7% are Asian,
5.4% are Two or More Races, 2.5% are Hispanic, 2.0% are Black, 0.1% are
Hawaiian/Pacific Islander, and 0.0% are American Indian/Native Alaskan. 52.2% of
the students are male and 48.8% of the students are female. Students who are identified
as economically disadvantaged are 10.9% of the population, 6.8% of the students
receive special education services, 1.9% are English Language Learners, 0.3% are in
foster care, 0.1% are deemed homeless; and 0.4% are connected to the military. South

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Fayette Elementary School has been recognized on the national level as a Blue Ribbon
School for its levels of achievement and academic programming. The elementary
school is the building that is projected to be the next construction project. It is
proposed to include third grade in the future.
Participants
The teacher participants in this research project were identified due to their
teaching of a mathematics course in grades six through twelve. This group of 23
educators was asked to complete a questionnaire through the means of a Google Form.
Of the 23 educators who were invited, 17 completed the questionnaire and provided
consent by submitting the form anonymously. In order to gain their consent, the
participants were informed in a written disclaimer prior to submitting the
questionnaire. They were made aware that they could exit the questionnaire at any time
before submitting and elect not to participate. Appendix A provides a view of the
survey that includes the written disclaimer. Participants were also informed in the
written disclaimer, prior to starting the survey, that minimal risk of identification
existed through triangulation based on identifying data including gender, years of
experience, and response to teaching accelerated/advanced courses.
The completion rate of the questionnaire by the teacher participants was 73.9%.
This majority level of completion could be due to the minimal amount of time that was
required on each participant’s behalf to complete the questions. Additionally, since the
questionnaire was electronic, the participants could complete it at their convenience,
both in terms of time and location. One possible participant communicated with the

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researcher directly that he/she would not be completing the questionnaire because
his/her teaching assignment never includes advanced level students, although this was
not a requisite to participate. The researcher was satisfied with the level of completion
and could only make conjectures related to the five remaining participants who did not
complete the questionnaire. Some of those considerations include distrust in the
process, lack of interest in the action research project, or lack of prioritization of time
to complete the questionnaire.
The data collection in this mixed methods research project also included student
data. In order to utilize the student data and fulfill the requirements of the Institutional
Review Board, no actual students nor any identifying information related to students
were utilized in the data collection. Additionally, the researcher had a district-level
administrator codify every student who would have data used in this study so that,
prior to the researcher using any of the students’ data, the compilation would be
independent of any identifying information. Based on not having nor using identifying
information related to the student data, neither parental consent nor student assent was
necessary; therefore, the use of student data was approved by the Institutional Review
Board.
Researcher
The researcher has worked in education for twenty years and has been exclusively
employed by the South Fayette Township School District. In those twenty years, the
researcher has held the roles of sixth grade science teacher, sixth grade mathematics
teacher, assistant principal of the middle school, principal of the middle school, and

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assistant to the superintendent for secondary education. The researcher holds a
Bachelor of Science degree in Elementary Education and a Master of Education degree
in Educational Leadership. The researcher has gained knowledge and experience
related to school operations, curriculum and instruction, data analysis, and districtlevel financial planning based on the multiple roles held over two decades. As
previously mentioned, it is the researcher’s intent to provide the district, particularly in
the role of assistant to the superintendent for secondary education who is responsible
for overseeing secondary level curriculum, with the outcomes from this action research
in order to validate or revise the screening process for advanced mathematics
placement.
Research Plan
Before any literature, participant, or data research began, a timeline outlining the
process was developed. This timeline helped to structure the scope of the entire
project into manageable, incremental, and logical steps. The literature review, which
occurred prior to completion of the data collection and participant submissions, was
vital in assisting the researcher in honing the focus of the action research, as well as
connecting it to or juxtaposing it against other research studies. This review provided a
detailed evolution of mathematics education in the nation’s public school system,
identified different approaches to how schools and districts sequence their mathematics
courses, and provided multiple explanations of the ways that schools commonly
choose to accelerate students. Additionally, the literature review presented the impact

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that acceleration can have on the future of all students, on different subgroups of
students throughout schooling, and teachers’ instructional perspectives.
After the conclusion of the review of literature, the researcher shared a
questionnaire with secondary level mathematics teachers in the South Fayette
Township School District to gain their perspectives on the accuracy of the screening
process used to determine placement of students in advanced mathematics courses.
Occurring parallel to both the review of literature and participant questionnaire, the
researcher gathered anonymously coded data about the five cohorts of students who
qualified for the advanced mathematics course pathway based on the current screening
process.
The first research question, “Is the screening process for advanced mathematics
coursework accurately identifying students for acceleration based on the criteria?” will
be answered through a quantitative analysis of the academic achievement data related
to the multiple cohorts of students who have been identified to be placed on the
accelerated mathematics course pathway. Not only will these students’ results on each
of the three criteria be critically reviewed for patterns and trends, but so will the
students’ grade achievements in each advanced mathematics course from sixth through
eleventh grades. Successful completion of these courses is determined by the district as
a cumulative average of 80% or greater. The students who have reached that minimum
average are permitted to ascend to the next advanced level course. However, a
cumulative average lower than 80% will remove students from the advanced level
pathway and require them to take the course again or take a lower level of the

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subsequent course. Eleventh grade is selected as the final year for examination in the
research because the district’s graduation requirement is a minimum of three years of
mathematics, and some students may elect not to take a mathematics course in their
senior year. In seeking answers to this particular research question, a spreadsheet was
created that contained thirteen columns of data related to each individual displaying
statistics ranging from the results on each of the criteria to the cumulative averages in
each of the advanced mathematics courses. The literature related to an ideal screening
process for determining the placement of students in accelerated courses is not
conclusive; however, the literature does concur that placement in middle school
mathematics, specifically Algebra I, should not be rushed for students because it is
paramount to their success in high school mathematics and post-secondary courses.
The second question, “Do teachers perceive that students are accurately placed in
advanced mathematics courses based on the qualification process that occurs prior to
the start of sixth grade?” utilizes a qualitative approach to discern the teachers’ beliefs
related to the accuracy of the advanced mathematics placement screening process. The
survey not only asks the teachers about their perceptions of the characteristics of the
students who were placed in advanced courses and the process’s accuracy, but also
asks about the consideration of students who were not accelerated and the
appropriateness of their placement in general level courses.
The final question, “Of the three criteria used in the screening process, does a
pattern exist as to a certain criterion indicating a greater likelihood of success in
mathematics advancement?” returns to the examination of the data collected from the

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multiple student cohorts. Since the collection of data includes the achievement levels
of each student in the advanced mathematics courses from sixth to eleventh grade,
there is a possibility that there will be a drop off point in which the greatest number of
students identified for acceleration exit the advanced course pathway. Connecting that
potential drop-off point back to the performance levels on each criterion, a certain
criterion may emerge as an indicator of long term success for students. Determining if
there is a pattern relative to a certain criterion would be beneficial for the district being
able to accurately identify students for acceleration since each criterion is currently
considered to be equally important. If a pattern emerges from the data, that individual
criterion could be considered with a greater weight or significance during the screening
process.
As previously stated, the literature review, accompanied by these guiding
questions, provides the researcher with the ability to assess the current criteria used in
the screening process for placement in the advanced mathematics course pathway.
Although the student data is significant to the research from the perspective of their
achievement outcomes in the accelerated courses, the questionnaire was just as
important because it considered the teachers’ perspectives. The teachers, experts in
their content, are a critical piece to the success of students in advanced courses, as also
revealed in the literature review. Therefore, their perspective about and familiarity with
the district’s current process, provides insights into how they value the criteria and
view the placement of students. Ultimately, with the synthesis of the literature and the
data, the district will have evidence to determine if the process that is being utilized to

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determine the mathematics course trajectory for students at the secondary level is
accurately placing students. Although the financial implications related to the
outcomes of this research are minimal, the impact on students and their futures is not.
The placement of students in advanced or general level courses prior to sixth grade is
one that can permit or prevent a student from achieving at the highest level in
mathematics. This is an educational decision made exceptionally early yet one that
yields significant, long term ramifications.
Fiscal Implications
The cost to conduct the screening process to determine students who qualify for
the accelerated mathematics sequence is very minimal. Of the existing three criteria,
two are data pieces that are generated at no cost. One, the students’ cumulative grade
averages in fifth grade math, comes from a query in our online grading system. The
second criteria, the comprehensive, summative assessment tool of the general sixth
grade mathematics course has already been designed by our math educators. It only
needs to be copied each year for the students to take. An expense to the district related
to the criteria is the purchasing of the Test of Mathematical Ability (T.O.M.A.) 3.
Copies of these exams are budgeted for annually by the assistant to the superintendent
for secondary education. Additionally, six teachers are paid an hourly rate each year to
score the two assessments. Three fifth grade teachers score the T.O.M.A. 3 tests, and
three sixth grade teachers score the comprehensive sixth grade assessments. These
teachers are paid an hourly rate and are not permitted to exceed five hours of grading.

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The hourly rate is $40.25 and is budgeted for within the district’s general budget for
extra professional responsibilities.
Should the data and research from this Capstone Project indicate that the criteria
are not accurately identifying students for the accelerated mathematics sequence, there
could be a financial impact on the district of selecting an alternate (or multiple
alternate) screening tool/s. The funds allotted to pay the six teachers may also be
impacted if new tools would require less or more time for scoring.
Research Design
A mixed-methods approach was selected as the research method due to the
utilization of both quantitative and qualitative data collections. The quantitative part,
relative to the first and third research questions, includes the collection of thirteen data
points for each student included in the five cohorts. As shown in Appendix B, these
data points include each student’s performance on the three criteria from the screening
process and the cumulative grade average for each accelerated mathematics course
taken from grades six through eleven. If a student ever exited the accelerated pathway
or moved from the district, data would not be available. The teacher questionnaire also
included certain quantitative aspects, as shown in Appendix A. These questions were
designed as a result of the thorough research conducted in the literature review process.
The quantitative questions included whether or not teachers could identify and name
the three criteria used to screen students for advanced mathematics placement and how
well they could describe this process from “very well” to “not at all.” Additionally, the
teacher participants rated, on a five-point Likert scale, how important they believed

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knowing the criteria was to them as a mathematics educator. The 1 on the scale
represented, “I do not need to be made aware or be familiar with the criteria,” while the
5 represented, “I should be fully aware and extremely familiar with the criteria.”
Another quantitative aspect of the teacher questionnaire asked the participants to
identify the most important indicator from the three criteria used in the screening
process. This question was followed by an open-ended response opportunity for which
qualitative data could be collected. The participants were provided the opportunity to
express their beliefs in the screening process and provide a rationale for a different tool
or assessment to use in place of one of the already-existing criteria. Similar to this pair
of mixed method questions, the next four questions were designed to yield quantitative
and qualitative data. The participants were asked, based on their perspectives from
teaching advanced level mathematics courses, how accurate the placement of students
in the accelerated courses was. The responses were assigned a rating of one (“Not
accurate”) to five (“Extremely accurate) on a Likert scale. Then, the teachers were
asked to respond, in their own words, about the characteristics of students who were in
accelerated courses but seemed to be inaccurately placed. The next question asked
participants to rate how often they felt that students in general mathematics courses
belonged in an advanced level course. This rating occurred from “Never” (1) to
“Always” (5) on the scale. Participants subsequently had the opportunity to describe
the characteristics of the students who they perceived should have been in accelerated
courses but were not. The final question was open-ended and allowed all participants
to describe any additional recommendations relative to the screening process that they

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believed may help to improve the accuracy of student placement and long-term
success. Overall, this mixed-methods approach will provide the researcher and,
ultimately, the district, with a comprehensive data set, both from the students’
outcomes and teachers’ perspectives, so that an informed decision can be made
regarding the accuracy of the screening process for accelerating students in
mathematics.
Data Collection
After making the necessary revisions required to receive approval by the
Institutional Review Board (IRB) to conduct the research, the researcher initiated the
process as it was outlined in the timeline provided for IRB approval. The first step of
the collection was to request the data related to the multiple student cohorts. In order to
be in compliance with the approved process, an independent, district level
administrator was needed in order to code the students in such a way that no student
could be identified by the researcher. Since the student data included multiple cohorts
and thirteen pieces of data for each student, requesting this data in September provided
enough time for it to be exported and compiled into a detailed spreadsheet. The
questionnaire that was utilized in the research was electronically shared with the 23
mathematics educators in January, and the participants were provided with two weeks
to submit their responses. Seventeen of the 23 educators participated.
The questionnaire was shared electronically since it was designed as a Google
Form. Participants' names and email addressed were not collected through the
submission of the form. Therefore, participation was anonymous and voluntary. On the

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Google Form, however, the participants were made aware of a minimal possibility of
data triangulation due to identifying data including gender, years of experience, and
response to teaching accelerated/advanced courses. Also included in the Google Form
was the pertinent information regarding informed consent. Respondents were also
given the opportunity to contact the researcher with any questions.
After the window closed for the teachers to participate in the research by
submitting the questionnaire, the responses were exported into a Google Sheet. Both
the Google Form and Sheet were chosen as the collection tools due to their ease of use
and access, by both the respondents and researcher, respectively. Within the Google
Sheet, the researcher had the ability to aggregate similar data, as well as disaggregate
individual data, based on the goal of a particular analysis or question within the
questionnaire. The Google Sheet also allowed for the search of repeated key terms or
phrases used in the participants’ open-ended responses.
The spreadsheet that was constructed with all of the students’ data was able to be
manipulated in multiple ways. The spreadsheet could be sorted by an individual
column (criterion or course) in ascending or descending patterns. Graphs could be
easily generated from data in order to determine the magnitudes of different selected
data points. The data could be analyzed for trends and patterns, both within the
spreadsheet, as well as through other exported graphical displays. Collectively, the
analysis of both data sets provided evidence related to the problem statement, as well
as in response to the three research questions, such that the researcher was to construct
recommendations for the district.

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Ethical Concerns and the Institutional Review Board
In order to conduct this mixed-methods research study, approval from the IRB
was necessary. Prior to submitting a proposal to the IRB, a research plan was
submitted to the researcher’s Doctoral Capstone Committee on August 2, 2020.
Additionally, the superintendent of the South Fayette Township School District
provided written endorsement and permission for the study to take place within the
district, utilizing student data and teacher feedback. This letter is provided as Appendix
C. With the approval of the researcher’s committee and district, a proposal with the
necessary IRB forms, which can be viewed in Appendix D, was submitted to the
Institutional Review Board for approval on August 14, 2020. The plan did not receive
initial approval, as stated in a letter from September 2, 2020; the Chair of the
Institutional Review Board requested that a statement about the possible triangulation
of identifying information be included in a written disclaimer for the questionnaire.
Additionally, there was questioning regarding the use of student data for individuals
under 18 and the need for parental consent. The notification of these requests can be
found in Appendix E. On September 4, 20202, the researcher responded to these
requests by amending the questionnaire to include the statement regarding possible risk
through triangulation, as well as an explanation that the student data would be coded
by a district level administrator and that no identifying information would be known to
the researcher. This response can be seen in Appendix F. The researcher received
formal approval (Appendix G) of the research plan on September 11, 2020.

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Validity of Research Plan
The researcher took multiple steps in order to increase the validity of the research.
The first step taken was a request to have the student data coded by a district level
administrator. The data codification removed any possibility that the researcher would
be able to identify a student or have subjectivity during the research based on current
or previous roles held in the district. Additionally, the questionnaire that was used was
first reviewed by the researcher’s Doctoral Capstone Committee in order to make sure
that the questions were not leading and did not contain bias. The survey was also
created so that the respondents were anonymous with a minimal risk of identification
through triangulation of certain data. The questions that were developed in the
questionnaire came as a result of the literature review and the guiding questions in
order to include the teachers’ perspectives into the research of the accuracy of the
criteria used for placement of students in advanced level mathematics courses.
The researcher utilized spreadsheets as the tools to gather and analyze the data.
The student data was inputted by an independent administrator, and the teacher data
was exported to the Google Sheet directly from the Google Forms. Therefore, the data
was unaltered by the researcher. Although the researcher has held roles that have been
close to the screening process, by having the anonymity of the students being
preserved through codification, potential subjectivity is removed. Additionally, the
researcher holds no further bias towards the study or its findings. The goal is to be able
to share data-based evidence with the district related to the accuracy of a critical
screening process.

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Summary
This chapter described the methodology process for how the researcher would
gather data in order to answer the three guiding questions while connecting the study
and its outcomes to the review of literature. This chapter revealed important aspects
that contributed to the methodology of the study. The details related to the setting and
participants, the research plan, the methods for collecting data, the fiscal implications,
and validity of the research were thoroughly explained. In order to arrive at meaningful
recommendations that may result from this research, it is important to have a
comprehensive understanding of how data was gathered and analyzed. Chapter 4 will
provide the results of this study as evidenced in the data in order to answer the guiding
questions and to provide the district with feedback regarding the criteria used in
determining the acceleration of students in mathematics courses.

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CHAPTER IV
Data Analysis and Results
In the following chapter, the analysis of the data related to the three research
questions that were previously described will be presented. The results include both
quantitative and qualitative datasets that were collected and synthesized from five cohorts
of students who qualified for acceleration in their mathematics course pathway, as well as
responses from middle school and high school mathematics teachers. The quantitative
data collected came from the results of a total of 150 students who were a part of the five
most recent cohorts of students. Additionally, these students were identified for
mathematics acceleration prior to entering sixth grade. Each student was anonymously
coded to maintain objectivity within the analysis, adhering to the requirement from the
Institutional Review Board. Results from the three pieces of criteria used in the screening
process, as well as cumulative grade averages from each student’s accelerated
mathematics courses from sixth grade to the most recently completed course were
utilized in the data analysis. Additionally, responses from six of the questions from the
teacher questionnaire yielded quantitative data. The qualitative data that was collected
resulted from the teacher questionnaire and included responses to five open-ended
questions.
Data from both the spreadsheet of the student results, as well as responses to the
teacher questionnaire, were utilized to determine if the screening process accurately
identifies students for mathematics acceleration. This concept was framed in the first
research question. The data related to the student cohorts was organized so that it could

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be determined if students were successfully promoted each year in order to remain on the
accelerated mathematics pathway. Also, a comparison was conducted for each cohort
based on the number of students in each cohort who began in the accelerated sixth grade
mathematics and how many of those students remained in the last mathematics course
that each respective cohort had completed. The teachers’ feedback from the questionnaire
relative to the criteria used in the screening process was also thoroughly analyzed. The
second research question was informed by teachers’ responses to open-ended questions
related to their perceptions of the accuracy of student placement in accelerated
mathematics courses based on the screening process. The final question was also
conducted as a mixed methods analysis. The spreadsheet that contained the students’
results was sorted in multiple ways and analyzed for patterns of whether or not a certain
criterion of the screening process indicated a greater likelihood of success for students in
the accelerated mathematics course pathway. Additionally, the teachers’ responses to the
items in the questionnaire that addressed their perception of the three criteria were
reviewed.
Data Analysis
A correlational analysis of the students' results during the screening process
compared to their long-term achievement in the accelerated mathematics coursework was
conducted to address the first research question, "Is the screening process for advanced
mathematics coursework accurately identifying students for acceleration based on the
criteria?" Separately, a comparison of the retention rates for each cohort in the
accelerated mathematics courses from sixth to eleventh grade was constructed.

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Additionally, the results from teachers’ ratings on two Likert scales from the
questionnaire were utilized in order to address this first question.
The student data, comprised of the five most recent cohorts of students who
qualified for the accelerated mathematics course pathway, consisted of 150 individual
students. These students were codified in a spreadsheet by being assigned an alphanumeric code. The first student of the first cohort was identified as “aa001.” As cohorts
changed, the second letter changed; however, the numeric assignment continued in
cardinal order throughout all cohorts. The last student in the fifth cohort was identified as
“ae150.” Each student had up to 13 pieces of data assigned, based on the last
mathematics course completed. Every student had the following data: raw cumulative
score on the two T.O.M.A. subtests, percentage earned on the curriculum-based
assessment, cumulative grade average from fifth grade mathematics, points earned from
the screening process rubric for the T.O.M.A. results, points earned from the screening
process rubric from the curriculum-based assessment results, points earned from the
screening process rubric from the cumulative grade average results, total points earned
from the three criteria, cumulative grade average from the sixth grade accelerated PreAlgebra course, and cumulative grade average from the seventh grade accelerated
Algebra I course. Then, depending on the last mathematics course that a specific cohort
finished, each student may have had the following data: cumulative grade average from
the eighth grade accelerated Geometry course, cumulative grade average from the ninth
grade accelerated Honors Algebra II course, cumulative grade average from the tenth

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grade Honors Pre-Calculus course, and cumulative grade average from the eleventh grade
accelerated Advanced Placement Calculus AB course.
A series of correlational analyses were conducted in which the students’ total
points earned from the screening process were compared to their cumulative grade
averages in each of the accelerated mathematics courses in order to explore whether or
not students who qualified for advanced placement were continually finding success in
subsequent accelerated mathematics courses after their initial placement. Students must
earn a minimum cumulative grade average of an 80% in order to be promoted to the next
accelerated course.
Addressing the second question, “Do teachers perceive that students are
accurately placed in advanced mathematics courses based on the qualification process
that occurs prior to the start of sixth grade?” both quantitative and qualitative data was
analyzed. The participants in the questionnaire not only rated their perceptions of the
accuracy of student placement in advanced mathematics courses on Likert scales, they
also responded to three open-ended questions. The ratings on the Likert scales were
examined for volume and consistency of responses by participants. The responses to the
open-ended questions were reviewed to identify if there were commonalities and
alignment with the other data related to the accuracy of placement of students in
advanced mathematics courses.
The analysis of the final research question, “Of the three criteria used in the
screening process, does a pattern exist as to a certain criterion indicating a greater
likelihood of success in mathematics advancement?” utilized both quantitative and

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qualitative data. Using the comprehensive spreadsheet of the scores that the students in
each of the cohorts received during the screening process, the data was sorted and
critically examined for the existence of a pattern. In addition to that, two items from the
questionnaire were examined for how the teacher participants perceived the accuracy of
the existing criteria or other potential screening tools.
Results
Comparing the total points earned by the students as a result of the three criteria
used in the screening process to each of their cumulative grade averages in the
accelerated math courses yielded a general trend. As shown in each of the six
scatterplots, Figures 9 to 14, students who earned a total of 10 or 11 points during the
screening process consistently had the lowest cumulative averages, even lower than
students who had earned a total of nine points. The group of scatterplots also revealed
that earning a higher total of points during the screening process did not increase a
student’s chance of getting the highest cumulative grade averages in each course.
Students who qualified with any of the point totals, 9-15 points, were able to achieve the
highest cumulative grade average.
As shown in Figure 9, all 150 students in the five cohorts successfully completed
the accelerated sixth grade Pre-Algebra course and were promoted to the accelerated
Algebra I course in seventh grade. The highest cumulative average earned by any student
was 99%, and the lowest average was 81%. The first average below a 90% (A range)
occurred for a student who earned a total of 13 points.

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Figure 9

Cumulative Grade Percentage

Correlational Analysis- Total Points from
Screening Compared to Pre-Algebra
Cumulative Grade
120
100
80
60
40
20
0
0

2

4

6

8

10

12

14

16

Screening Point Total (out of 15)

Note. The comparison of the total number of points students earned during the screening
process to their cumulative grade average in Pre-Algebra 6
A similar shape and trend with the data occurred with these five cohorts of
students as they advanced to the accelerated seventh grade course of Algebra I (Figure
10). However, there were two students, both earning ten points during the screening
process, who fell below the passing average. One had earned a cumulative average of
69%, and the other had earned a cumulative average of 70%. These two students were
exited from the accelerated program following their enrollment in Algebra I. The range of
cumulative grade averages was 100% to 69%. In this course, it was also the same student
with a screening point total of 13 who earned the first cumulative average below 90% (A
range).

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Figure 10

Cumulative Grade Percentage

Correlational Analysis- Total Points from
Screening Compared to Algebra I Cumulative
Grade
120
100
80
60
40
20
0
0

2

4

6

8

10

12

14

16

Screening Point Total (out of 15)

Note. The comparison of the total number of points students earned during the screening
process to their cumulative grade average in Algebra I
In Figure 11, the data began to show that more students who had earned a 10 or
11 during the screening process were not meeting the minimum passing average of 80%.
Two students with a screening point total of 11 were not successful and each had
cumulative grade averages of 79%. One student with this same screening process had a
point total of 11, although this individual had also met the grade minimum to be enrolled
in Honors Geometry, but had elected not to take the course and exited the accelerated
pathway. That student is indicated as a circle at the bottom of the graph above the number
11. There were three students who had a screening point total of 10 who did not reach the
minimum. They had cumulative grade averages of 79%, 76%, and 69%. Similar to the
formerly mentioned student, one student with this point total also opted not to continue

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on the accelerated pathway even though the previous grade average permitted this student
to do so. The two students who did not meet the grade average minimum exiting Algebra
I are also represented as circles at the bottom of Figure 11. The range of cumulative grade
averages were 100% to 69%. Yet again, the first cumulative grade average earned that
was less than 90% (A range) was the same previously mentioned student with a screening
point total of 13.
Figure 11

Cumulative Grade Percentage

Correlational Analysis- Total Points from
Screening Compared to Honors Geometry
Cumulative Grade
120
100
80
60
40
20
0
0

2

4

6

8

10

12

14

16

Screening Point Total (out of 15)

Note. The comparison of the total number of points students earned during the screening
process to their cumulative grade averages in Honors Geometry
Figure 12 displays the first occurrence in which a student who had earned above
an 11-point total during the screening process did not choose to enroll in the Honors
Algebra II course. This particular student had earned the requisite grade average in
Honors Geometry but had elected to exit the accelerated pathways. Additionally, a

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student who was on track to be enrolled in this accelerated course and had a screening
point total of 12 withdrew as a student in the district. All students who did not meet the
minimum cumulative grade average at the end of this course had 10 points as their
screening point total. There are other circles at the bottom of this graph representing
additional students who made the minimum cumulative grade average in the previous
course but who chose not to continue in the accelerated mathematics pathway. This group
consisted of four students with 11 points as their screening total and eight students with
10 points as their screening total. In addition, three other students who had an 11-point
screening total and met the cumulative grade average to maintain status in the accelerated
pathway withdrew from the district. The range in cumulative grade averages was 100% to
79%. Also, the trend continues with the first cumulative average under 90% being
associated with the aforementioned student who had a screening point total of 13.

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Figure 12

Cumulative Grade Percentage

Correlational Analysis- Total Points from
Screening Compared to Honors Algebra II
Cumulative Grade
120
100
80
60
40
20
0
0

2

4

6

8

10

12

14

16

Screening Point Total (out of 15)

Note. The comparison of the total number of points students earned during the screening
process to their cumulative grade average in Honors Algebra II
The data indicates that sizes of the cohorts continue to decrease at the close of
Honors Pre-Calculus based on more students earning a cumulative grade average below
the minimum of 80%. Additionally, there are more students, represented by circles at the
bottom of Figure 13, that chose not to enroll in this course. One student who had earned a
15-point screening total and met the grade requirement for enrolling in this course elected
not to continue on the accelerated pathway. This student has been previously mentioned
as the first student to score below a 90% in each of the previous courses. This student had
a cumulative grade average of a 73% in Honors Pre-Calculus and, thus, exited the
accelerated pathway. There was one student with a screening point total of 12 who did
not qualify to enroll in this course and one student with a 12-point total who fell below

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the minimum cumulative grade average of 80%. Of the students with a 10-point
screening total, there were five who did not qualify to enroll in this course, one who was
eligible to enroll but withdrew as a student in the district, and one who earned a 79%
cumulative grade average in this course, which would subsequently remove the student
from the accelerated pathway. The range in cumulative grade averages was 99% to 73%,
and the first average below a 90% (A range) occurred with a student who had a screening
point total of 14.
Figure 13

Cumulative Grade Percentages

Correlational Analysis- Total Points from
Screening Compared to Honors Pre-Calculus
Cumulative Grade
120
100
80
60
40
20
0
0

2

4

6

8

10

12

14

16

Screening Point Total (out of 15)

Note. The comparison of the total number of points students earned during the screening
process to their cumulative grade average in Honors Pre-Calculus
In Figure 14, no data is displayed for a screening point total of 9. This occurred
because only the first two cohorts had reached this level in the course sequence, and their
screening criteria had a minimum cut-off score of ten points. Their cohorts completed the

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former version of the Test of Mathematical Abilities, the T.O.M.A. 2. Therefore, the
scatterplot is fully representing the cohort of students who have begun and continued
through the entire series of accelerated mathematics courses through eleventh grade. In
addition to the students who had previously exited the accelerated pathway due to choice,
grade average, or withdrawal, there were additional students impacted in Advanced
Placement Calculus AB. One additional student within the 11-point screening total group
failed to meet the minimum grade average of 80% at the end of this course. This student
earned a 72% and subsequently exited the accelerated pathway. Nine additional students,
ranging in screening point totals from 13 to 10 points, elected not to take this course,
even though they had met the minimum cumulative grade average requirement.
Separately, one student who had a 14-point screening total withdrew as a student in the
district. In Advanced Placement Calculus AB, the range in cumulative grade averages
was between 99% and 72%. The first cumulative average that was below 90% occurred
for a student with a 14-point screening total.

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Figure 14

Cumulative Grade Percentage

Correlational Analysis- Total Points from
Screening Compared to Advanced
Placement Calculus AB Cumulative Grade
120
100
80
60
40
20
0
0

2

4

6

8

10

12

14

16

Screening Point Total

Note. The comparison of the total number of points students earned during the screening
process to their cumulative grade average in Advanced Placement Calculus AB
The previous scatterplots showed the cumulative grade performances by the
students in the five cohorts, and they also showed trends of when students began to exit
the accelerated pathway. The following chart, Figure 15, is another representation of such
student data in which each cohort’s retention rate of students remaining in the accelerated
mathematics pathway is displayed. The graphical display indicates that, for the three
courses of accelerated mathematics in the middle school, students are highly successful
and nearly all of the students are able to qualify for promotion to the next accelerated
course. Two cohorts, Cohort 1 and Cohort 5, each had a 96% enrollment rate compared to
the original group. Cohort 1 had two students, whereas Cohort 5 had one student, who did
not meet the minimum grade requirement upon exiting Algebra I. The greatest decrease

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in students remaining in the accelerated course pathway occurred after Honors Algebra
II. For the three cohorts that had already advanced to enroll in Honors Pre-Calculus, there
was a noticeable decline in the amount of students compared to the total amount who
began on the accelerated pathway in sixth grade. Cohort 1 saw a decrease of 12% after
Honors Algebra II, Cohort 2 saw an 8% decrease after Honors Algebra II, and Cohort 3
saw a decrease of 16% after Honors Algebra II. Another significant decrease in student
enrollment occurred after Honors Pre-Calculus. Both Cohorts 1 and 2 had their greatest
decreases, 16% and 24%, respectively. If you consider the entire group of 150 students,
40 exited the program in total, with 14 of those students exiting due to not meeting the
cumulative grade requirement. From start to finish with these cohorts, 90.7% of the
students who qualified for advanced placement through the screening process were able
to maintain their accelerated status through to the most recent course that they had
finished.

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Figure 15

ACCELERATED MATHEMATCIS COURSES

RETENTION RATES IN ACCELERATED COURSES
Pre-Algebra 6
Algebra I
Honors Geometry
Honors Algebra II
Honors Pre- Calculus
AP Calculus AB
0.00% 10.00% 20.00% 30.00% 40.00% 50.00% 60.00% 70.00% 80.00% 90.00%100.00%
STUDENTS REMAINING IN ACCELERATED COURSES
(% OF WHOLE GROUP ENTERING IN 6TH GRADE)
Cohort 5 ("ae")

Cohort 4 ("ad")

Cohort 3 ("ac")

Cohort 2 ("ab")

Cohort 1 ("aa")

Note. This bar graph displays the percentage of students in each cohort who remained in
the accelerated courses from entrance until the completion of the most recently finished
course.
Two questions from the teacher questionnaire specifically asked the teachers to
rate their perception of how accurate the placement of students was in accelerated
courses. Both of these questions asked the participants to respond on a Likert scale from
1 to 5. For the first question, “If you have taught the highest level mathematics course at
a respective grade level in last seven years, please describe how accurate the placement
of students seems to be, from your perspective, knowing that the nearly all of the students
were in that course because they qualified for advanced placement through the screening
process at the end of fifth grade,” 5 represented “Extremely accurate; all students seemed

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to be appropriately placed in advanced level courses” and 1 represented “Not accurate; all
students seemed to be inappropriately placed in advanced level courses.” Figure 16
shows the results of the nine participants that responded. Although 17 teachers completed
the questionnaire, this question may not have applied to all participants if they did not
teach an advanced level course within the last seven years. In response to the question,
67% of the participants expressed that the accuracy of student placement in advanced
mathematics courses was at a score of 4. No participants felt that the placements were
extremely accurate; nor did the participants express that the placement was completely
inaccurate.
Figure 16

Number of Participant Responses

Teacher Responses to Accurate Placement in
Advanced Courses
7
6
5
4
3
2
1
0
1

2

3
Likert Sale Ratings
Responses

4

5

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Note. This bar graph displays the total number of responses for each of the ratings on the
Likert scale, from Not Accurate (1) to Extremely Accurate (5)
The second question, which was similar in nature, asked the participants to
consider the following, “If you have taught mathematics courses that are not at the
highest level at each grade level, how often do you find that students in these classes
should have been placed in the advanced course sequence?” Again, the participants’
responses were rated on a Likert scale of 1 to 5 with 5 being “Always” and 1 being
“Never.” The results revealed that there was less consistency among the perceptions of
the 17 participants who responded to this question (Figure 17). Most teachers,
approximately 65%, rated the frequency in which non-accelerated students should have
been placed in accelerated courses between a 2 and 3. There were two individuals who
had opposite perceptions of how often they believed that non-accelerated students should
have been placed in advanced courses. One of these teachers felt that it never occurred;
whereas, another teacher felt that it always occurred.

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Figure 17

Number of Participant Responses

Teacher Responses to Accurate Placement of NonAccelerated Students
7
6
5
4
3
2
1
0
1

2

3

4

5

Likert Scale Ratings
Responses

Note. This bar graph displays the total number of responses for each of the ratings on the
Likert scale, from Never (1) to Always (5)
Following each Likert scale question, participants were given the opportunity to
respond to an open-ended question. The first open-ended question asked the teachers, “If
you felt that students had been inaccurately placed in the highest level mathematics
course, please describe the characteristics of such students.”. Nine participants chose to
respond to this question. As shown in Figure 18, there are some common themes of
characteristics that participants identified which included deficits in prior knowledge,
difficulty with higher level/application-based problems, lack of confidence, immaturity,
and lack of self-advocacy skills. The theme that was repeated the most frequently was

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that of students having deficits in their mathematical knowledge related to previously
taught skills.
Figure 18
Teacher Responses to Characteristics of Advanced Students Who May Not Have Been
Appropriately Placed
Characteristic

Teacher Responses

Occurrence
of
Theme

Deficits in Prior
Knowledge

“Deficits in prior knowledge”; “lack of necessary mental
math and basic calculation skills”; “Basic skills are weak.”;
“They do not have the appropriate prerequisite skills”;
“Reliant on calculators”; “Basic Algebra skills are weak.”

6

Difficulty with
higher
level/applicationbased problems

“Difficulty with problems that went beyond basic skills and
concepts”; “Weak completing application problems”; “Not
able to think beyond a procedure”

3

Lack of Confidence

“Lack of confidence in completing individual tasks or when
participating in class.”; “Lack of experience of what to do
when a concept is not understood”; “Struggled with work
ethic/organization/study habits/independence”

3

Immaturity

“It is typically immaturity. These students are very good at
school but are not quite ready to take on the rigor and the
work load.”; “Work ethic is not mature.”

2

Lack of SelfAdvocacy Skills

“Difficulty or hesitation in formulating questions to ask when
struggling with a concept”; “Failure to self-advocate”

2

Other

“Expectation of Extra Credit to achieve a grade”

1

Similar to the open-ended question, participants of the questionnaire were given
the opportunity to respond to another question after they rated the frequency with which
they found students in non-accelerated courses who they believed should have been

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placed in the advanced course pathway. The teachers were asked to expand by
responding to, “Please describe the characteristics of students that you perceive should
have been placed in the advanced course sequence but were not.” With a response rate of
88%, there was minimal repetition in themes. Some overlap did exist with the themes of
onset of later maturity, completion of work, level of motivation, level of understanding of
mathematics concepts, and performance above peers. The most commonly mentioned
characteristic for students who were not in the advanced level mathematics courses but
were perceived by their teachers as being capable was the level of understanding of
mathematics concepts with four occurrences (Figure 19).

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Figure 19
Teacher Responses to Characteristics of Non-Advanced Students Who May Not Have
Been Appropriately Placed in General Mathematics Courses
Characteristic

Teacher Responses

Occurrence
of
Theme

Level of
Understanding of
Mathematics
Concepts

“Exhibit complete or nearly complete understanding of
certain topics prior to discussing them in class”; “Lesser
challenging problems seem trivial”; “Questions show interest
in the ‘why’ instead of just the ‘how’”; “If a student has a
deep conceptual understanding on how things work and the
‘why’ behind how the problem works.”

4

Performance
Above Peers

“Students that score 98% or above in my class.”; “Performs
well-above peers on majority of assessments and learning
tasks”; “Performance above peers in class performance and
mastery of content”

3

Completion of
work

“Consistent effort in and out of class”; “Homework
completion is consistent and accurate”; “Strives to complete
all assignments and extra learning opportunities with
motivated work ethic”

3

Level of
Motivation

“Driven to succeed at the highest level”; “Seeks out
additional opportunities for enrichment/instruction”

2

Onset of Later
Maturity

“Students who have matured over the summer.”; “Maturity
also plays a factor at later level.”

2

Other

“Students have a focus on learning and treat grades as a
reflection of learning.”; “They were too intimated to take the
advanced level course.”

2

In sorting the comprehensive spreadsheet that contained the results of the
students’ performances on the three criteria during the screening process, as well as their
success in each of the accelerated courses for which they were enrolled, the following
information was ascertained. The first way in which the data was sorted was to take each
of the columns that contained the points earned from the screening rubric associated with

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each of the screening criterion and order them from greatest to fewest points earned.
Then, how many students successfully remained in the accelerated pathway in each
column was determined. Once the first student exited, it was noted. For example, after
ranking all 150 students based on their T.O.M.A. scores, the student who was 31st on the
list was determined to be the first to exit the accelerated course pathway. Students beyond
this student may have been more successful and continued on the accelerated pathway,
but, as one measure of each criterion being an indicator of long-term success, that first
student to exit was identified (Figure 20).
Figure 20
Each Criterion’s Exiting Student Report Based on Scores

Number of students in
spreadsheet before a
student exited the
accelerated course
pathway

T.O.M.A. Score

Curriculum-based
Assessment

Cumulative Fifth
Grade Math
Average

30 students

42 students

18 students

The next analysis of data occurred to determine if a pattern existed related to a
certain criterion indicating a greater likelihood of success in mathematics advancement.
The analysis included examining the magnitude of students who exited the accelerated
pathway according to the points they earned for each criterion from the screening process
(Figure 21). Of note, students that fell in the 4-point range on the rubric based on their
T.O.M.A. results constituted the greatest population of students that, at some point on the

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pathway, did not meet the qualifications to remain in the accelerated series of courses.
This population was significantly higher than any other point range associated with the
T.O.M.A. results. The variation in the students who were required to exit the accelerated
pathway due to not meeting the grade minimum based on their curriculum-based
assessment results was not as significant. The greatest amount was nearly 17% of
students who had earned 1 point. Three points was the fewest amount of rubric points that
could be earned in the screening process based on students' cumulative fifth grade math
averages; this point total yielded the largest population of students who did not remain in
accelerated coursed due to not meeting the requisite cumulative average. That data also
revealed that almost the same amount of students who had earned the maximum rubric
score of 5 points, as well as the score of 3 and 2 points, based on their T.O.M.A. results,
had elected to exit the accelerated pathway at some point. Even though these students
were identified as being appropriately prepared for advanced level work, they made a
choice not to continue in the sequence of courses. The greatest portion of students who
chose to leave the accelerated pathway based on the points earned from their curriculumbased assessment results were those who earned 1 point. There is also an overall trend
with this category that, as the points increase, the percentage of students who chose to
leave the accelerated pathway decreases. For this same type of comparison relative to the
cumulative fifth grade math average, those students who had earned 3 points comprised
the greatest number of students who opted to exit the accelerated pathway.

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Figure 21
Amount for Each Type of Exited Student Based on Rubric Scores for Each Criterion

T.O.M.A.

Curriculumbased
Assessment

Cumulative
Fifth Grade
Math
Average

5 points

4 points

3 points

2 points

1 point

0 points

Did not
qualify to be
promoted

Did not
qualify to be
promoted

Did not
qualify to be
promoted

Did not
qualify to be
promoted

Did not
qualify to be
promoted

Did not
qualify to be
promoted

8.3%

31.3%

5.9%

5%

0%

0%

Elected to
exit
accelerated
pathway

Elected to
exit
accelerated
pathway

Elected to
exit
accelerated
pathway

21.7%

14.3%

23.5%

20%

0%

0%

Did not
qualify to be
promoted

Did not
qualify to be
promoted

Did not
qualify to be
promoted

Did not
qualify to be
promoted

Did not
qualify to be
promoted

Did not
qualify to be
promoted

6.7%

2.7%

10.6%

6%

16.7%

N/A

Elected to
exit
accelerated
pathway

Elected to
exit
accelerated
pathway

Elected to
exit
accelerated
pathway

Elected to
exit
accelerated
pathway

Elected to
exit
accelerated
pathway

Elected to
exit
accelerated
pathway

Elected to
exit
accelerated
pathway

Elected to
exit
accelerated
pathway

Elected to
exit
accelerated
pathway

0%

5.4%

14.9%

27.3%

44.4%

N/A

Did not
qualify to be
promoted

Did not
qualify to be
promoted

Did not
qualify to be
promoted

Did not
qualify to be
promoted

Did not
qualify to be
promoted

Did not
qualify to be
promoted

3.8%

13.4%

0%

N/A

N/A

N/A

Elected to
exit
accelerated
pathway

Elected to
exit
accelerated
pathway

Elected to
exit
accelerated
pathway

17.5%

20.9%

33.3%

Elected to
exit
accelerated
pathway

N/A

Elected to
exit
accelerated
pathway

N/A

Elected to
exit
accelerated
pathway

N/A

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Utilizing only the first two cohorts, since they are the only cohorts who had
progressed from the first accelerated course in sixth grade to the final course considered
in the study, Advanced Placement Calculus AB, a chart was constructed to analyze how
many students in each criterion and rubric point total remained in the accelerated
pathway. These students’ data revealed the following for each criterion of the screening
process. For rubric points earned based on T.O.M.A. results, the greatest number of
students who remained in the pathway, as well as who exited, earned the maximum point
total of 5. For the curriculum-based assessment, similar to all of the cohorts, the greatest
number of students remaining in the accelerated pathway had a 4-point total, and the
most significant number of students who exited had a 1-point total. Finally, the trend for
the students who completed the full pathway based on their cumulative fifth grade math
average followed the point values. The greatest number of students who completed the
pathway scored 5 points, whereas, the fewest number scored 2 points. It is notable that
both of the students from these two cohorts who qualified for accelerated placement with
only 2 points finished the entire sequence (Figure 22).

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Figure 22
Comparison of Students Who Exited and Did Not Exit Accelerated Pathway Based on
Rubric Scores for Each Criterion
5 points

T.O.M.A.

Curriculumbased
Assessment

Cumulative
Fifth Grade
Math
Average

4 points

3 points

29

5

6

students
completed
the pathway

students
completed
the pathway

students
completed
the pathway

18

5

6

students
exited the
pathway at
some point

students
exited the
pathway at
some point

3

2 points

1 point

0 points

N/A

N/A

N/A

students
exited the
pathway at
some point

N/A

N/A

N/A

13

7

13

3

students
completed
the pathway

students
completed
the pathway

students
completed
the pathway

students
completed
the pathway

students
completed
the pathway

1

1

7

10

11

students
exited the
pathway at
some point

students
exited the
pathway at
some point

students
exited the
pathway at
some point

students
exited the
pathway at
some point

students
exited the
pathway at
some point

N/A

20

17

2

students
completed
the pathway

students
completed
the pathway

students
completed
the pathway

N/A

N/A

N/A

13

17

0

students
exited the
pathway at
some point

students
exited the
pathway at
some point

students
exited the
pathway at
some point

N/A

N/A

N/A

N/A

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The following three histograms (Figures 23, 24, and 25) display the distribution of
students who exited the accelerated pathways based on their performances on each of the
three criteria in a different manner. Exited students include students who did not meet the
minimum cumulative grade average needed to maintain accelerated status, as well as
students who elected to exit the advanced mathematics course pathway at any point on
their own. Of the 150 students comprising the five cohorts, 40 students, or 26.7%, exited
the program. Fourteen out of the 40 students who exited were placed off of the
accelerated pathway due to not earning the required cumulative grade average. Analysis
of the T.O.M.A. results showed that all 40 students who exited the program, regardless of
reason, earned less than a 52 raw score point total on the assessment (Figure 23). Of the
three criteria, this display has the clearest separation of the total scores for students who
have or have not exited. The greatest number of these students, nearly half of those who
exited, had a raw score between 42 and 46 on the T.O.M.A. In regards to the curriculumbased assessment results, the standard deviation was the greatest of the three graphical
displays, meaning that there was a greater range of students who exited the accelerated
pathway in comparison to the average percentage earned (Figure 24). Most significantly,
though, students who earned an 80% or less on the curriculum-based assessment were
more likely to exit the accelerated course pathway. Those who scored 80% or less on the
curriculum-based assessment comprised 87.5% of the 40 students who exited the
program. When analyzing the criterion of the fifth grade mathematics cumulative grade
averages, little deviation was found in the data between the students who exited and did
not exit the accelerated course pathway. The greatest frequency of those exiting the

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pathway occurred with students earning a 98% cumulative fifth grade mathematics
average (Figure 25). Notably, with the exception of averages of 100% and 92%, students
who exited had every other cumulative average. It is also worthwhile to acknowledge
that, according to the screening rubric and process, students earned zero points for any
cumulative average below a 90%. Therefore, it would require high scores for the other
two criteria for a student to qualify for advanced mathematics placement with a
cumulative fifth grade math average of less than a 90%.
Figure 23

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Note. This histogram displays the frequency of students who exited the accelerated
course pathway within the stated ranges of raw scores from the T.O.M.A.
Figure 24

Note. The histogram displays the frequency of students who exited the accelerated course
pathway within the stated ranges of percentages earned on the curriculum-based
assessment.

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Figure 25

Note. The histogram displays the frequency of students who exited the accelerated course
pathway for each of the cumulative fifth grade mathematics averages between 90% and
100%.
On the teacher questionnaire, the participants were asked to identify which of the
three existing criteria of the screening process they perceived to be the most useful in
accurately placing students in accelerated mathematics coursework. As shown in Figure
26, the majority of the teachers responded with the “comprehensive curriculum-based

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assessment of sixth-grade content.” Over 58% of all of the questionnaire participants
identified this assessment as the most accurate criterion in the process.
Figure 26

Most Accurate Criterion from Teachers' Perspective
0

Curriculum-based Assessment
7
T.O.M.A. 3
10

Cumulative Fifth Grade Math
Average

Note. This circle graph displays the responses by questionnaire participants regarding
which of the three criteria they perceived to be the most accurate in placement of students
in accelerated coursework.
The teachers who participated in completing the questionnaire were also given an
open-ended question in which they could state a different tool or specific assessment that
they believed should be used in place of one of the existing criteria. Here are the
responses from the seven participants who chose to respond:

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● “An interest questionnaire to assess students’ interest in participating in
accelerated/advanced math courses”
● “I don’t have a specific tool; however, I currently have a student that has 100% in
both Q1 and Q2 that should be in Pre-Algebra but didn’t meet the criteria prior to
6th grade.”
● “I believe that students need to be advanced on their PSSA Math 5th grade test
before they can even be considered for advanced math placement, but that can’t
be done because the results are released too late.”
● “I believe that students’ scores on the 5th Grade PSSAs should also be factored
into the placement.”
● “Not sure how much the cumulative math average assists”
● “If they were basic, proficient, or advanced in math on the PSSA”
● “Teacher recommendation”
The participants most frequently recommended to include the achievement levels of the
fifth grade mathematics Pennsylvania State School Assessment in the screening process
for each student.
Triangulation of data occurred in this study because multiple, different points
were analyzed regarding students who qualified for advanced placement in mathematics.
The student data included results for each of the three criteria used in the screening
process, those scores converted to points from the screening process’s rubric, and
cumulative mathematics averages for each of the accelerated courses that every student
has completed. The student data encompasses these data points for the past eight years

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100

because that is when the first cohort would have entered its first accelerated mathematics
course. Additionally, data was gathered from teachers in regards to their years of teaching
experience, certification, perceptions of the screening process, and perceptions of the
qualities of both the accelerated and non-accelerated students in mathematics.
Information and statistics gathered from the review of literature was examined and
considered when looking at the results of the student data and teacher questions.
Discussion
1. Is the screening process for advanced mathematics coursework accurately identifying
students for acceleration based on the criteria?
Based on the analysis of the data, the most significant indicator that the current
screening process is accurately identifying students for acceleration is that 90.7% of the
students who qualified were able to maintain their status in advanced placement based on
achievement. This statistic does not include the students who qualified to remain in the
accelerated pathway but made the choice to exit. Including those students, the overall rate
that would represent the population of students who have maintained their status in the
accelerated pathway is 73.3%. Although the reasons for students exiting on their own
accord were not identified in this study, their cumulative averages indicated that they
were performing at a level that would be considered successful for accelerated courses.
Additionally, the data showed that the students who had qualified for accelerated
placement prior to entering sixth grade were most successful in the middle school
accelerated courses: Pre-Algebra 6, Algebra I, and Honors Geometry. In each of the
recorded cohorts, very few students exited the program. These results may also be

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attributed to the phenomenon described by other researchers who have found through
multiple studies that course sequencing in middle school is less flexible than in high
school, which causes students to move less from the accelerated pathway during sixth,
seventh, and eighth grades (Loveless, 1998, 2013; Lucas, 1999; Mulkey et al., 2005). The
most significant loss of students in the accelerated pathway occurs after Honors Algebra
II. This loss includes both students who do not qualify to remain in the accelerated
pathway, as well as those who self-select to exit. This is not unlike the trend that occurred
in the data from the High School Transcript Study (HSTS) of 2009 in which the most
noticeable decline in the population of students advancing to the next accelerated course
happened after Algebra II and before Pre-Calculus (National Assessment of Educational
Progress, 2009/2018). If the intention and measure of success of the screening process is
for the entire cohort that began as accelerated students in sixth grade to remain
accelerated through Advanced Placement Calculus AB, then that has not been achieved at
a rate of 100%. In the two cohorts that have completed the full sequence of courses from
sixth to eleventh grade, fewer than 60% of the students from their respective original
groups still enrolled in Advanced Placement Calculus AB. However, those statistics
again include students who could have qualified to make it to that course but chose to exit
at one point on their own.
2. Do teachers perceive that students are accurately placed in advanced mathematics
courses based on the qualification process that occurs prior to the start of sixth grade?
The data indicates that the majority of teachers do believe that students are
accurately placed in advanced mathematics courses based on the qualification process.

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Unlike existing research that indicates the use of standardized tests, teacher subjectivity,
and parental influence, the criteria that is currently employed and preferred by the group
of teachers who participated in the questionnaire represents two assessments, neither
which are standardized (Bitter & O’Day, 2010; Hallinan, 2003; Kelly, 2007; Loveless,
1998; Meehl, 1954; Oakes, 1985; Useem, 1992). Although one teacher in the survey
suggested “teacher recommendation” as an added component to the screening process, it
was not a tool that the majority of teacher participants expressed. However, and in
alignment with the research, three of the teacher participants believed Pennsylvania's
annual standardized math assessment should be added as a measure to the screening
process.
3. Of the three criteria used in the screening process, does a pattern exist as to a certain
criterion indicating a greater likelihood of success in mathematics advancement?
After analyzing the data in multiple ways, there was not a conclusive pattern that
existed relative to one criterion indicating a greater likelihood of success in the district’s
accelerated course sequence. However, there were outcomes from this collection of data
that did reveal certain indicators about each criterion. First, the fifth grade cumulative
mathematics average revealed no correlation to a student’s long-term success in the
accelerated course pathway. Students of every grade average, from 90% to 100%, exited
the program, with the greatest number having a cumulative average of 98%. In addition
to what the quantitative statistics revealed, the teachers also reinforced the notion that this
criterion was not perceived as valuable. None of the 17 participants chose this as the most
useful tool in accurate placement of students in accelerated mathematics coursework. One

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teacher went as far to say, “Not sure how the cumulative math average assists” in an
open-ended response. In regards to the other two criteria, the T.O.M.A. and the
curriculum-based assessment, a variety of analyses pointed to different recommendations
based on the results of the five most recent cohorts. First, all 40 students who exited the
program earned a 51 or lower raw point total on the T.O.M.A. Although there was not a
pattern revealing that the T.O.M.A. was the strongest indicator, there was a clear
distinction between the scores of students who exited and did not exit the pathway. There
was a similar pattern for the curriculum-based assessment results for which there was a
threshold separating those who were more and less successful in advanced courses.
Students who earned an 80% or lower on the curriculum-based assessment exited the
accelerated pathway at a higher volume than those who scored 81% and higher.
Compounding these two patterns and based on the data, the profile of a student who
would have a greater likelihood of success in advanced mathematics would consist of
results greater than 51 as a raw point total on the T.O.M.A. and greater than 80% on the
curriculum-based assessment. Reviewing the data associated with the only two cohorts
who had finished the entire sequence of accelerated mathematics courses defined in this
study, no pattern was found indicating one criterion as stronger than another for
predicting long-term success for students. The only criterion’s data pointing towards a
slight pattern for these two cohorts is the curriculum-based assessment. For students who
earned 4 or 5 points, only one with each point value exited the accelerated pathway. That
number of students was the fewest of any category and point value. Similarly, but on the

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low end of the point values, the greatest number of students who exited the program had
earned 1 point based on results from the curriculum-based assessment.
Summary
The collection of quantitative and qualitative data was used to inform the three
research questions. It was important to analyze the multiple data sets in order to be
informed about the current screening process that is used to place students in advanced
mathematics coursework prior to entering middle school. Since this process, which
happens early in a student’s career, has far reaching implications, including
postsecondary and career aspirations, it was critical to gather evidence that would speak
to the efficacy of the screening process.
Accuracy of the current screening process was determined to be true for
approximately 90% of the 150 students enrolled in the five most recent cohorts. The
teachers’ perceptions indicated that two of the three current criteria are valued as accurate
tools to determine placement. Those perceptions were further supported by the student
data indicating that there was no correlation between the third criteria of the students’
cumulative fifth grade math average and success in the accelerated pathway. Although
none of the three criteria were shown to be the single best indicator for a student’s
likelihood of success in the accelerated mathematics pathway, certain analyses provided
thresholds of scores on the T.O.M.A. and curriculum-based assessment that would point
to a greater accuracy of advanced placement of students, as well as their long-term
success in the accelerated pathway.

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The considerations taken from all of this analyzed data will convert into
recommendations for the district, as well as additional, recommended research topics that
may need to be considered or conducted before any changes are implemented. These
recommendations and potential research will be discussed in the next chapter.

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CHAPTER V
Conclusions and Recommendations
The purpose of this study was to validate or recommend revisions to the district’s
screening process for placement of students in the advanced mathematics coursework
pathway at South Fayette Township School District. This process, as previously
described, is exclusionary and only permits students who qualify to have access to the
highest level of mathematics courses as a result of a screening process that occurs prior to
students entering sixth grade. As research has indicated, taking advanced level
mathematics courses in high school leads to higher assessment scores, a higher likelihood
of enrolling in college and completing a bachelor’s degree, greater career earnings, and
increased career satisfaction (Altonji et al., 2012; Bozick & Lauff, 2007; Chen, 2009;
Nord et al., 2011; Pellegrino & Hilton, 2012). Therefore, this decision of accurately
accelerating students is of paramount importance and can drastically impact the students’
futures. In order to determine if the screening process was effectively identifying students
for acceleration in mathematics, multiple sets of data were reviewed. Those sets included
data related to 150 students’ performances on the three criteria of the screening process,
the cumulative grade averages for all of the students in the advanced mathematics courses
they had finished, and teachers’ responses to a questionnaire.
Conclusions
In order to inform the South Fayette Township School District if its screening
process for placing students in advanced mathematics courses at the secondary level was

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accurate, five cohorts of student data were reviewed. These cohorts contained 150
students who had already qualified for acceleration due to the screening process that has
existed for the past eight years. With a 90.7% retention rate of students who had qualified
originally and remained in the accelerated course pathway, the screening process seems
to be accurate for the majority of students. However, there was a discrepancy with the
total population of students who remained as accelerated when those who chose to exit
the pathway on their own were also included. The accuracy rate then dropped to 73.3% of
students remaining in the accelerated pathway. Therefore, the screening process may be
accurately identifying students based on their mathematical knowledge and skills;
however, the process may account less for students who may not be interested in
pursuing advanced mathematics courses through high school.
An additional conclusion drawn about the retention of students in the accelerated
pathway was that students remain in the advanced level courses in middle school at a
higher rate than in high school. The retention of all five cohorts for Pre-Algebra and
Algebra I was 100% and three of the five cohorts dropped to approximately 95% with
each of them losing one student each from the original cohort for Geometry in eighth
grade. The rate starts to decrease more significantly as the students progress through high
school. The data showed that there was between a 3% to 12% decline in enrollment after
Honors Geometry in eighth grade. That was followed by between a 9% to 16% decrease
in enrollment after Algebra II in ninth grade. Lastly, there was between a 15% to 23%
decrease in student enrollment after Honors Pre-Calculus in tenth grade. Related to the
two cohorts that finished the entire sequence of courses, there was a loss of only one

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student in one of the two cohorts by the end of middle school compared to a loss of 40%
(20 students) of the students in the first cohort and 37.5% (9 students) of students in the
second cohort during high school.
Considering the decline of students throughout the sequence of accelerated
courses in high school, the first significant number of students exiting the accelerated
cohort consistently occurred after Honors Algebra II and prior to Honors Pre-Calculus.
The three cohorts that have gotten this far in the sequence showed a drop in retaining the
full group of students who qualified in each cohort by a loss ranging between 9% and
16%. This is in alignment with a reported national trend, but the underlying reasons were
not investigated nor revealed in this study (National Assessment of Educational Progress,
2009/20018). The drop-off that occurs within these cohorts of students is a result of both
students not qualifying due to their cumulative grade average, as well as students who
elected to exit the accelerated pathway. Since Honors Algebra II is taken in ninth grade as
the highest level course, those who opted to exit on their own still had to take two more
mathematics courses in order to fulfill local graduation requirements. The data shows,
however, that they are not taking the highest level course that is expected in the
accelerated sequence and for which they demonstrated requisite knowledge and skills.
There is another significant decrease in the population of students in the accelerated
pathway after Honors Pre-Calculus. For the two cohorts who have completed the full
sequence of this study by finishing Advanced Placement Calculus AB, only
approximately 56% and 59%, respectively, of the students remained compared to their
original cohort population. This decrease, however, does not align with the results from

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the High School Transcript Study of 2009. That study saw a slight increase in students
taking Pre-Calculus in tenth grade when compared to those taking Calculus in eleventh
grade (National Assessment of Educational Progress, 2009/2018).
Based on the results that 67%, or four out of the seven, teachers who completed
the item of the question rated their perception of the accuracy of the screening process a 4
out of 5 rating on a Likert scale, it can be concluded that their perceptions of the accuracy
of the screening process are more favorable than not. No teacher gave a complete
endorsement of a 5 rating, or “extremely accurate,” nor did any teacher rate the accuracy
level lower than a 3. Considering the sample size, seven teachers is not a large
population, however, that group is inclusive of every mathematics educator that teaches
the highest level of mathematics between grades six and twelve.
Based on the review of the teachers’ responses to which criterion they believed to
be the most accurate for advanced placement, it was concluded that the teachers only
validated two of the three criteria. The use of the cumulative fifth grade mathematics
average was not identified by any of the participants as being the most accurate tool.
Overall, 58.8% of the teachers endorsed the T.O.M.A. 3 as the most accurate criterion
used in the screening process for advanced placement; whereas, 41.2% selected the
curriculum-based assessment. The cumulative fifth grade averages from the student data
were also determined to have the least impact on determining a student’s likelihood for
success in the accelerated mathematics pathway. With the exception of students who had
a cumulative fifth grade math average of 100% and 92%, all other grade averages led to
students who exited the accelerated pathway, either by performance or choice. Thus,

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there shows to be no correlation with this grade average and a student’s likelihood to
remain on the accelerated pathway. Conversely, data related to the other two criteria, the
T.O.M.A. and the curriculum-based assessment, presented enough evidence to indicate
their usefulness and accuracy as tools in the screening process. Specifically, for students’
raw scores from the T.O.M.A. assessment, if they earned a 52 or greater, their likelihood
of remaining on the accelerated pathway was significant. From the five cohorts, 100% of
the students scoring in this range remained on the pathway. The curriculum-based
assessment had a comparable indication, although not as strong. For students who scored
an 81% or greater on the curriculum-based assessment, there was a high likelihood that
they were accurately placed and will remain in accelerated courses through high school.
This was determined to be true for 91.2% of the students who scored in this range from
the five cohorts.
As previously proposed, if the use of the fifth grade cumulative mathematics
average was to be replaced by another tool, there could be financial implications for the
district, depending on what would be utilized as the new criterion. If standardized tests
were selected, as some teacher participants suggested, there would be no cost to the
district to incorporate the results as a measure in the screening process. The assessments
are mandated and funded by the state, and comprehensive results are provided to the
district on an annual basis. Similar to collecting and utilizing the cumulative fifth grade
mathematics averages, an administrator would have to sort and organize the standardized
test data to incorporate into the screening process spreadsheet for each cohort. The
challenge with incorporating the state standardized test results is the time in which the

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scores are received compared to when the screening process concludes. Scores have
historically come in after the existing screening process has already finished and students
have been scheduled for the accelerated courses in sixth grade. If these results were to be
included, there may need to be a shift in the completion of the process, communication of
results to parents, and the scheduling of these students for sixth grade mathematics. This
recommendation of utilizing the standardized test results, however, should not be
implemented until further research is conducted. Using the same cohorts of students that
were included in this study, a correlational analysis could be completed to compare their
achievement levels on the state standardized assessments with their success of remaining
in the accelerated pathway. In addition, the district may want to seek out and consider
other tools to replace the cumulative fifth grade mathematics average that were not
mentioned by the teacher participants.
One measure that should not be used as a potential replacement for the cumulative
fifth grade mathematics average, based on research from literature and the teachers’
responses to the questionnaire, would be parental input. Parental influence has
demonstrated to be an inequitable consideration because it favors students of families
from higher economic status (Baker & Stevenson, 1986; Lareau & Shumar, 1996;
McGrath & Kuriloff, 1998; Useem, 1992). Additionally, teachers have expressed that
when parents have influence in the accelerated placement process, it is not advantageous
and the placements do not accurately represent the students’ true ability levels (Spear,
1994). None of the 13 teacher participants in this study’s questionnaire who provided
recommendations on improving the screening process included parent input. Actually, the

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input that respondents valued the most, which is currently not a part of the process and
was mentioned three times, was teacher recommendation. One teacher described this
sentiment in an open-ended response by stating, “The teacher ultimately knows what the
students are capable of and shouldn’t feel pressured by parent concerns.”
A particular conclusion, not specifically related to a research question, yet one
that can be compared to research from literature was how the cumulative average of the
students from the first two cohorts in seventh grade compared to their subsequent courses
taken in mathematics. Finkelstein et al. (2012) had concluded that a student’s
performance in grade seven mathematics is a strong predictor for high school
mathematics course selection. When looking at South Fayette’s accelerated population
related to the two cohorts that have completed the sequence of courses in this study, a
trend similar to that of Finkelstein et al. (2012) was found. Specifically, the results from
these two cohorts indicated that students with cumulative averages in the A and A+
ranges in Algebra I are most likely to enroll in the highest level eleventh grade
mathematics course of Advanced Placement Calculus AB (Figure 27). Conversely, the
number of students who elect to take non-accelerated mathematics courses increases as
the cumulative averages in seventh grade occur in the B+, B, and B- ranges. It should be
noted that the two students in the A+ range who did not enroll in Advanced Placement
Calculus AB were qualified to do so but had elected on their own accord to exit the
accelerated pathway.

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Figure 27

Distribution of Grade 11 Mathematics Course
Taking Based on Grade Seven Mathematics Grade
Averages
(Cohorts 1 and 2 only)
Number of Students

20
15
10
5
0
D

C-

C

C+

B-

B

B+

A-

A

A+

Cumulative 7th Grade Math Grades
AP Calculus AB

Other Course

Note. The comparison of the seventh grade cumulative mathematics averages for students
in cohorts 1 and 2 and their course enrollment for eleventh grade mathematics
Limitations
A particular limitation that may have impacted the interpretation of the findings
would be that all of the cumulative mathematics grades from the 2019-2020 school year
had the potential to be skewed. Due to the global pandemic and the shutdown of schools,
South Fayette Township School District elected to implement a “Pass/Fail” grading
system for the fourth nine weeks of that school year. If a student earned a “Pass,” that
would be equivalent to a 100% average for that grading period. If a student received a
“Fail,” that would be equivalent to a 50% average for that grading period. The fourth
quarter grade was factored into each student’s overall cumulative average. Therefore, the

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alignment of the students’ cumulative averages to what their actual achievement would
have been if the grading system was not impacted could be different.
Another limitation to the study was that all of the open-ended questions were
optional for participants to complete. Therefore, some of the results contained less than a
50% response rate from the total participant population. Due to this, the conclusions that
were drawn may not be fully representational of all possible responses and perceptions.
The final limitation that existed would be that not all of the cohorts of students
involved had progressed far enough to finish the full sequence of advanced mathematics
courses. Although two of the cohorts did complete the sequence, the remaining three
cohorts only had cumulative averages up to the last course that they finished. Without the
completion of all courses by the remaining three cohorts, some of the data related to
remaining in the accelerated pathway through Advanced Placement Calculus AB may not
be fully accurate and representational of all students. This study, however, was inclusive
of all of the cohorts from when this screening process was officially implemented.
The summation of these conclusions will be used to inform the South Fayette
Township School District of next steps for their screening process for the placement of
students in advanced mathematics. The district should feel validated that, based on
mathematical knowledge and skills, the criteria of the screening process was accurately
identifying students at a 90.7% rate for advanced placement in courses. However, based
on the outcome of student data and teacher feedback, the district may want to consider
exploring the replacement of one of the criteria, the cumulative fifth grade mathematics
average. Although teachers expressed that they are mostly in agreement with the

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accuracy of the placement of students, this particular measure was not valued by the
teachers, nor did the averages show correlation to the populations of students who
remained in the accelerated pathway versus those who exited. Also, these fifth grade
mathematics averages did not correlate with the cumulative grade performances students
earned in the sequence of accelerated courses following the screening process. The
district now also has evidence to inform families about a student’s likelihood for success
in the accelerated course sequence. The profile of a student with the most success would
earn a score greater than a 52 on the T.O.M.A. 3 assessment and greater than 80% on the
curriculum-based assessment. Also, the district now has local data that parallels national
studies that show a student’s performance in Algebra I serves as a predictor for high
school mathematics course taking. This can be a useful statistic as students select courses
for high school and consider their plans for post-secondary and goal aspirations.
Recommendations for Future Research
With any meaningful research comes the potential for more unanswered questions
that are worth investigating as an outcome. That experience occurred with this study, and
those unanswered questions led to the following recommendations for future research or
considerations. Each of these could be rich, comprehensive studies on their own;
however, collectively, they all have potential implications, not only for the success
students can experience in the accelerated pathway, but for their likelihood of qualifying
for acceleration.
A consideration for additional research would be to analyze the demographics of
the students who qualified for the accelerated pathway, as well as those that subsequently

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exited. Existing research acknowledges that the make-up of students in accelerated
courses is imbalanced based on race (Braddock, 1989; Domina, 2014; Gutiérrez, 2008;
Lubienski & Gutiérrez, 2008; Webel & Dwiggins, 2019). The races and demographics of
students were not considered in this study. However, it would be beneficial to determine
if racial divides exist in the composition of the cohorts of students who have qualified for
acceleration. Also associated with race, further research could be done relative to the
Asian population enrolled in the accelerated course pathway compared to other
underrepresented races and to the race representations for the whole student population.
The district’s Asian population is approaching 20% and has significantly increased over
the last ten years, which makes it a particular demographic worthy of studying.
Not only has research indicated that students of color are underrepresented in
accelerated courses, it has also been determined that students from higher economic
status are two times as likely to take advanced mathematics in middle school than peers
from a low economic status (Walston & McCarroll, 2010). South Fayette Township
School District has a population of approximately 10.7% of students who are identified as
economically disadvantaged. Although this study did not analyze the economic status of
the students in the cohorts, there is potential to study how the overall economically
disadvantaged population aligns to those of the students who are accelerated.
The focus of the research for this study only incorporated the results of the
students who had qualified for acceleration as a result of the screening process. Another
potential study could be looking at the data and course sequences for the students who
did not originally qualify and how they fared in regular and advanced level courses. This

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117

consideration leads to questions such as: How many students who were not originally
accelerated take additional measures in order to end up in the highest level of
mathematics in high school? How many students move into advanced level courses by
the time they complete high school but were in the general mathematics pathway in
middle school? Is there need to provide students the opportunity to enter the accelerated
pathway after the screening process occurs at the end of fifth grade? This last question is
one that is of particular interest as a result of comments in the teacher questionnaire. Four
of the teachers expressed in open-ended responses that, through their observations, there
have been students who did not qualify for acceleration through the screening process but
later demonstrated the maturity and ability to be a part of this exclusive group.
The teachers of the accelerated mathematics courses have not changed
significantly over the past eight years. However, there is always potential for changing
the assignments of educators, especially at the high school level. Therefore, a possible
further study related to students’ maintaining their accelerated course status would be to
do a comparative study of outcomes related to different teachers. With some courses, this
may be difficult because there is only one section and one teacher. The only comparative
study that could occur would be if that single teacher changed over time. Research
found that, when teachers are assigned to high level courses, they display more
enthusiasm and employ a greater set of instructional strategies. Therefore, it may be
worthwhile to investigate the impact that different teachers could have as the teacher of
the most advanced course in various grade levels (Oakes, 1985, 1992).

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Lastly, a recommendation based on this research would be to implement an exit
interview or survey for students who elect to exit the accelerated pathway even when they
qualify to be promoted to the next advanced level course. Multiple data analyses
illuminated the existence of these students, but their reasons for exiting were not
incorporated in this study. Should there be a way to gather this information in the future,
the results could potentially further inform the screening process. One of the participants
in the questionnaire even recommended that the screening process include “an interest
questionnaire to assess students’ interest in participating in accelerated/advanced math
courses, their willingness to put forth their greatest amount of effort in order to be
successful, and their goals as math students.” Research found in literature indicated a
variety of outcomes linking students’ self-concept to being accelerated, some that showed
positive outcomes for accelerated students while others found negative ramifications
(DeLacy, 2000; Gross, 1992, 1994; Olszewski-Kubilius, 1995, 1998; Rogers, 1991;
Sayler, 1992; Swiatek, 1992). Self-concept has a potential outcome for the students in
this study, even when their grades deem them eligible to advance to the next accelerated
course. Additionally, within this potential future research, there is space to examine a
possible link between students who exit the accelerated pathway and that decision being
based on their plans for postsecondary or career aspirations. Ultimately, are these
students leaving the accelerated mathematics pathway because they have determined that
they do not need the highest level of mathematics in order to pursue their post-secondary
and/or career goals?

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Another factor that could contribute to students originally qualifying for
accelerated placement but not remaining in the full sequence of courses could be parental
influence. The magnitude of parental influence on the screening process and subsequent
selection of courses by students could be an additional area for research. Literature
indicates that parental involvement, especially by parents of higher economic status,
often contributes to students’ placement in advanced level courses (Bitter & O’Day,
2010; Hallinan, 2003; Kelly, 2007; Loveless, 1998; Meehl, 1954; Oakes, 1985; Useem,
1992). The South Fayette Township School District’s screening process does not
currently quantify parental influence, nor does it include parental input as a tool in the
screening process. However, based on the researcher’s former involvement with the
process, the researcher observed parental demands for students to be considered and reconsidered for the acceleration when students did not qualify for advanced placement
prior to sixth grade.
Summary
The purpose of this study was to examine the screening process the South Fayette
Township School District uses to determine advanced mathematics placement of students
prior to entering middle school. The process, adopted eight years ago, has impacted the
mathematics course pathway of over 2000 students, with 150 students qualifying for
accelerated placement. This is the first study that has been conducted to determine the
accuracy of the screening process, as well as to determine if the students who originally
qualify for advancement remain in the program through eleventh grade. Although the
results indicated that almost 91% of the students who qualified for acceleration

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successfully met the annual requirements to be advanced, the retention rate dropped to
about 73% when the students who opted to exit the pathway were added into statistics.
This result led to a recommendation for a future study involving the reasons why students
choose to exit the accelerated pathway even when they demonstrate the aptitude and
grade requirement to continue. Teachers expressed their endorsement of two of the three
criteria, excluding the cumulative fifth grade mathematics average. In addition to their
feedback, the data did not support this criterion as a strong tool for predicting student
success. Therefore, another recommendation is for the district to consider replacing this
measure in the screening process. The other two measures, the T.O.M.A. 3 and the
curriculum-based assessment, were revealed through this study to be useful as indicators
for accurate placement and long-term success based on student data. Overall, and with
the consideration that multiple researchers have drawn the conclusion that advanced
mathematics placement in middle and high school is linked to college and career
readiness and success, coupled with the fact that students cannot reenter the accelerated
pathway without exceptional efforts in high school, the district could continue to use its
existing screening process. However, the two considerations relative to students choosing
to exit the pathway and the lack of correlational value to the cumulative fifth grade
mathematics average should be addressed.

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APPENDICES

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Appendix A
Teacher Perception Survey
(This is the Action Research Survey that was developed to gather mathematics teachers’
perspectives regarding the placement criteria used in identifying students for accelerated
mathematics courses.)

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Appendix B
Student Data Spreadsheet Template
(This provides the headings for each of the data points that were identified in the full
spreadsheet for the five cohorts of students that were accelerated. There were 150
students assigned an ID from the five cohorts.)

ID

Raw
Point
TotalTOMA

Cumulati
Percenta ve 5th
ge
Grade
TOMA
earned Math
Rubric
on CBA Average Points

CBA
Rubric
Points

Total
Earned
Cumulati Points
ve
from
Average Screening
Points
Rubric

Pre
Algebra
Cumulati
ve
Average

Honors
Geometr
Algebra I y
Cumulati Cumulati
ve
ve
Honors
Average Average Alg II

Honors
Pre Calc

AP Calc
AB

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Appendix C
South Fayette Township School District Approval Letter
(This is the letter of approval to conduct the study within the South Fayette Township
School District. This letter was issued by the superintendent of schools.)

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Appendix D
Institutional Review Board Approval Application
(This contains the application materials that were submitted to the IRB for approval to
conduct the action research study as outlined.)

IRB Review Request

Institutional Review Board (IRB) approval is required before beginning any research and/or
data collection involving human subjects
Submit this form to instreviewboard@calu.edu or Campus Box #109

Project Title:

The Accuracy of the Advanced Mathematics Placement Criteria in Identifying Students for

Mathematics Course Acceleration
Researcher/Project Director
Phone #

Kristin M. Deichler

412.478.5936

Faculty Sponsor (if researcher is a student)
Department Education

E-mail Address
Dr. Kevin Lordon

dei1175@calu.edu

PLACEMENT CRITERIA FOR MATHEMATICS ACCELERATION
Anticipated Project Dates

August 2020

to

148

August 2021

Sponsoring Agent (if applicable)
Project to be Conducted at
Project Purpose:

South Fayette Township School District

Thesis

Research

Class Project

Other

Keep a copy of this form for your records.

Required IRB Training
All researchers must complete an approved Human Participants Protection training course. The training requirement can
be satisfied by completing the CITI (Collaborative Institutional Training Initiative) online course at
http://www.citiprogram.org New users should affiliate with “California University of Pennsylvania” and select the “All
Researchers Applying for IRB Approval”course option. A copy of your certification of training must be attached to this IRB
Protocol. If you have completed the training within the past 3 years and have already provided documentation to the IRB,
please provide the following:

Previous Project Title
Date of Previous Project IRB Approval

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Please attach a typed, detailed summary of your project AND complete items 2
through 6.
1. Provide an overview of your project-proposal describing what you plan to do and how you
will go about doing it. Include any hypothesis(ses)or research questions that might be
involved and explain how the information you gather will be analyzed. All items in the
Review Request Checklist, (see below) must be addressed.
2. Section 46.11 of the Federal Regulations state that research proposals involving human
subjects must satisfy certain requirements before the IRB can grant approval. You should
describe in detail how the following requirements will be satisfied. Be sure to address each
area separately.
(text boxes will expand to fit responses)
a.
How will you ensure that any risks to subjects are minimized? If there are
potential risks, describe what will be done to minimize these risks. If there are risks,
describe why the risks to participants are reasonable in relation to the anticipated
benefits.
Risks will be minimized by actions that are taken by the researcher. Data that is
utilized in this research study will be maintained in secure, electronic files that are
password-protected, as well as contain no identifying information. All screening
process data points will be confidential and anonymous. No children under 18 will be
involved in this research. Additionally, for the teacher survey, participation is
voluntary and all responses will be confidential and anonymous. The voluntary survey
will not collect any email addresses, names, or identifying information.
The confidentiality of the both the data sets and the survey results will minimize any
risk presented in this study. The beneficial outcome of validating or making
improvements to a screening process used by the South Fayette Township School
district that impacts the educational program for all students will outweigh these
minimal risks.
Since there will be no in-person or face-to-face encounters or interviews involved in
this study, no risk will be present related to COVID-19. The surveys will be electronic
and can be completed in a safe environment of the willing participants’ choosing.

b.
How will you ensure that the selection of subjects is equitable? Take into
account your purpose(s). Be sure you address research problems involving vulnerable
populations such as children, prisoners, pregnant women, mentally disabled persons, and
economically or educationally disadvantaged persons. If this is an in-class project
describe how you will minimize the possibility that students will feel coerced.
Since the research is targeted at the criteria used in the screening process for
advanced mathematics coursework at the South Fayette Township School District, all
teachers of mathematics from the point of screening on will be able to participate in a
voluntary survey. No teachers will be excluded or eliminated from having the option
to complete the survey, maintaining equity of participants.
Additionally, the data sets that represent multiple cohorts of student performances
during the screening process, as well as in academic courses subsequently, will

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150

include all results, anonymously coded, applicable to the problem statement and
research questions. Although the research takes into account the data from student
performance during the screening process and in academic courses, no children will
be involved in this research study. Only data points will be included, which will be
gathered, utilized, and analyzed, in a coded, non-identifiable way.
c.
How will you obtain informed consent from each participant or the subject’s
legally authorized representative and ensure that all consent forms are appropriately
documented? Be sure to attach a copy of your consent form to the project summary.
Consent would only be necessary for the voluntary survey that will be distributed to
the South Fayette Township School District’s mathematics teachers in grades six
through twelfth. Consent will be obtained from each participant by their submission of
responses to the electronic survey.
The consent documentation will be embedded into the electronic survey for
documentation purposes before any questions are asked of the participants.

d.
Show that the research plan makes provisions to monitor the data collected to
ensure the safety of all subjects. This includes the privacy of subjects’ responses and
provisions for maintaining the security and confidentiality of the data.
Participants of the electronic survey will have anonymity. The survey will not collect
email addresses, names, or identifying information. Also, the results of the survey will
be secured through password-protected means in which only the researcher has
knowledge and can access. The data sets from the screening process and subsequent
math courses will be organized in a confidential, secure spreadsheet. It will be
similarly housed electronically in which only the researcher will have the knowledge
of a complex password in order to access. All of the data points in the multiple sets
will be coded in the spreadsheet.

3. Check the appropriate box(es) that describe the subjects you plan to target.

Adult volunteers

Mentally Disabled People

CAL University Students

Economically Disadvantaged People

Other Students

Educationally Disadvantaged People

Prisoners

Fetuses or fetal material

Pregnant Women

Children Under 18

Physically Handicapped People

Neonates

4. Is remuneration involved in your project?

Yes or

No. If yes, Explain here.

PLACEMENT CRITERIA FOR MATHEMATICS ACCELERATION

5. Is this project part of a grant?

Yes or

No

151

If yes, provide the following information:

Title of the Grant Proposal
Name of the Funding Agency
Dates of the Project Period
6.

Does your project involve the debriefing of those who participated?

Yes or

No

If Yes, explain the debriefing process here.

7. If your project involves a questionnaire or interview, ensure that it meets the requirements
indicated in the Survey/Interview/Questionnaire checklist.

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152

California University of Pennsylvania Institutional Review Board
Survey/Interview/Questionnaire Consent Checklist (v021209)
This form MUST accompany all IRB review requests
Does your research involve ONLY a survey, interview or questionnaire?
YES—Complete this form
NO—You MUST complete the “Informed Consent Checklist”—skip the remainder of this form
Does your survey/interview/questionnaire cover letter or explanatory statement include:
[X_] (1) Statement about the general nature of the survey and how the data will be used?

[X_] (2) Statement as to who the primary researcher is, including name, phone, and email
address?

[X_] (3) FOR ALL STUDENTS: Is the faculty advisor’s name and contact information provided?

[X_] (4) Statement that participation is voluntary?

[X_] (5) Statement that participation may be discontinued at any time without penalty and
all data discarded?

[X_] (6) Statement that the results are confidential?

[X_] (7) Statement that results are anonymous?

[X_] (8) Statement as to level of risk anticipated or that minimal risk is anticipated? (NOTE: If
more than minimal risk is anticipated, a full consent form is required—and the Informed
Consent Checklist must be completed)

[X_] (9) Statement that returning the survey is an indication of consent to use the data?

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153

[X_] (10) Who to contact regarding the project and how to contact this person?

[X_] (11) Statement as to where the results will be housed and how maintained? (unless
otherwise approved by the IRB, must be a secure location on University premises)
[X_] (12) Is there text equivalent to: “Approved by the California University of

Pennsylvania Institutional Review Board. This approval is effective nn/nn/nn and
expires mm/mm/mm”? (the actual dates will be specified in the approval notice from
the IRB)?
[X_] (13) FOR ELECTRONIC/WEBSITE SURVEYS: Does the text of the cover letter or
explanatory statement appear before any data is requested from the participant?

[X_] (14) FOR ELECTONIC/WEBSITE SURVEYS: Can the participant discontinue participation
at any point in the process and all data is immediately discarded?

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California University of Pennsylvania Institutional Review Board
Review Request Checklist

(v021209)

This form MUST accompany all IRB review requests.
Unless otherwise specified, ALL items must be present in your review request.
Have you:
[X_] (1.0) FOR ALL STUDIES: Completed ALL items on the Review Request Form?
Pay particular attention to:
[X_] (1.1) Names and email addresses of all investigators
[X_] (1.1.1) FOR ALL STUDENTS: use only your CalU email address)
[X_] (1.1.2) FOR ALL STUDENTS: Name and email address of your faculty
research advisor
[X_] (1.2) Project dates (must be in the future—no studies will be approved which
have already begun or scheduled to begin before final IRB approval—NO
EXCEPTIONS)
[X_] (1.3) Answered completely and in detail, the questions in items 2a through 2d?
[X_] 2a: NOTE: No studies can have zero risk, the lowest risk is “minimal
risk”. If more than minimal risk is involved you MUST:
[_] i. Delineate all anticipated risks in detail;
[_] ii. Explain in detail how these risks will be minimized;
[_] iii. Detail the procedures for dealing with adverse outcomes due
to these risks.
[_] iv. Cite peer reviewed references in support of your explanation.
[X_] 2b. Complete all items.
[X_] 2c. Describe informed consent procedures in detail.
[X] 2d. NOTE: to maintain security and confidentiality of data, all study
records must be housed in a secure (locked) location ON UNIVERSITY
PREMISES. The actual location (department, office, etc.) must be specified
in your explanation and be listed on any consent forms or cover letters.
[X_] (1.4) Checked all appropriate boxes in Section 3? If participants under the age
of 18 years are to be included (regardless of what the study involves) you MUST:

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155

[_NA] (1.4.1) Obtain informed consent from the parent or guardian—
consent forms must be written so that it is clear that the parent/guardian
is giving permission for their child to participate.
[NA_] (1.4.2) Document how you will obtain assent from the child—This
must be done in an age-appropriate manner. Regardless of whether the
parent/guardian has given permission, a child is completely free to refuse
to participate, so the investigator must document how the child indicated
agreement to participate (“assent”).
[X_] (1.5) Included all grant information in section 5?
[X_] (1.6) Included ALL signatures?

[NA_] (2.0) FOR STUDIES INVOLVING MORE THAN JUST SURVEYS, INTERVIEWS, OR
QUESTIONNAIRES:
[NA] (2.1) Attached a copy of all consent form(s)?
[NA_] (2.2) FOR STUDIES INVOLVING INDIVIDUALS LESS THAN 18 YEARS OF AGE:
attached a copy of all assent forms (if such a form is used)?
[NA_] (2.3) Completed and attached a copy of the Consent Form Checklist? (as
appropriate—see that checklist for instructions)
[X_] (3.0) FOR STUDIES INVOLVING ONLY SURVEYS, INTERVIEWS, OR QUESTIONNAIRES:
[X] (3.1) Attached a copy of the cover letter/information sheet?
[X] (3.2) Completed and attached a copy of the Survey/Interview/Questionnaire
Consent Checklist? (see that checklist for instructions)
[X_] (3.3) Attached a copy of the actual survey, interview, or questionnaire
questions in their final form?

[X_] (4.0) FOR ALL STUDENTS: Has your faculty research advisor:
[X_] (4.1) Thoroughly reviewed and approved your study?
[X] (4.2) Thoroughly reviewed and approved your IRB paperwork? including:
[X] (4.2.1) Review request form,
[X] (4.2.2) All consent forms, (if used)
[NA_] (4.2.3) All assent forms (if used)

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[X] (4.2.4) All Survey/Interview/Questionnaire cover letters (if used)
[X] (4.2.5) All checklists
[X_] (4.3) IMPORTANT NOTE: Your advisor’s signature on the review request form
indicates that they have thoroughly reviewed your proposal and verified that it
meets all IRB and University requirements.
[X] (5.0) Have you retained a copy of all submitted documentation for your records?

The Doctoral Capstone Project Proposal

Identifying Information
a.
b.
c.

d.

Doctoral Student Name: Kristin M. Deichler (email: dei1175@calu.edu)
Proposed Doctoral Capstone Project Title: The Accuracy of the Advanced Mathematics
Placement Criteria in Identifying Students for Mathematics Course Acceleration
Doctoral Capstone Project Committee
Faculty Capstone Committee Advisor: Dr. Kevin Lordon (email:
lordon@calu.edu)
External Capstone Committee Member: Dr. Jeffrey Evancho
Anticipated Doctoral Capstone Project Dates: __August 2020________ to _August
2021______

Description of what you plan to research (problem statement)
The mathematics course trajectory for all students entering middle school is
determined by a set of three pieces of criteria at the end of fifth grade. The
outcome from this set of criteria can permit or prevent students from enrolling
in the highest level of mathematics coursework. The intention of this criteria is
to accurately identify students for placement in accelerated mathematics
courses yet there is no researched evidence supporting such accuracy.
Description of why this problem is in need of research.
This problem needs to be researched because there has never been a study done
to determine if the placement criteria accurately identifies students at an early
grade level for math advancement. Since the qualification process is an
exclusionary academic decision prior to middle school, the research will be

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157

informative to determine if the criteria are accurately identifying the
mathematics placement of students.
The three pieces of criteria include: cumulative grade average for fifth grade
mathematics, the raw cumulative score from two subtests from the Test of
Mathematical Ability (T.O.M.A. 3), and the percentage correct on a
comprehensive, summative assessment based on the general sixth grade
mathematical course. Since these three criteria are from different sources and
are on different scales, the performances on each are converted to points
ranging from 0 – 5, and students earn a total out of 15 points.
Description of how you plan to go about doing your action research (research method)
This research will be conducted through a mixed methods approach. One of the
ways in which both descriptive and quantitative research will be conducted is
through survey research of the mathematics teachers. These district teachers,
from grades five to twelve, will be asked to complete a survey on their
perceptions of the accuracy of the accelerated placement criteria based on their
observations of student achievement in their courses. The remaining research
will quantitative in nature and will be done to determine the accuracy of the
advanced mathematics placement criteria, as well as the potential pattern
between three criteria and likelihood for predicting a student’s success in math
advancement. The research will occur for multiple cohorts of students who were
screened through identical accelerated placement criteria prior to entering sixth
grade.
A thorough Literature Review will occur in the fall of 2020 as the researcher
completes a course (EAL 706) which is focused on this comprehensive process.
The collection of literary evidence will be included in this research project.
Research Questions
Is the screening process for advanced mathematics coursework accurately
identifying students for acceleration based on the criteria?
Do teachers perceive that students are accurately placed in advanced
mathematics courses based on the qualification process that occurs prior to the
start of sixth grade?

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Of the three criteria used in the screening process, does a pattern exist as to a
certain criterion indicating a greater likelihood of success in math advancement?
Explanation of how you plan to collect your data for each of the research questions.
(Include attachments of data collection instruments i.e. surveys, interview questions)
1) Use of collected data from multiple past cohorts of students relative to their
placement in the advanced mathematics course in sixth grade, as well as
their successful completion (earned at least an 80% cumulative average) in
the subsequent advanced course sequence through junior year. The
collected data will stop at the junior year in order to be valid because the
district’s graduation requirement is a minimum of three years of
mathematics courses. The data may also reveal if there is a “drop off” point
in which the greatest number of students identified for acceleration do not
continue on the advanced course pathway. (Research Question 1 and 3).
A spreadsheet will be built for each cohort of students included in the study. The
spreadsheet will have columns that have their results from each of the three
criteria, their overall score in the screening process, and each of their subsequent
math courses with the cumulative grades for each.

2) Use the results of a survey given to district mathematics teachers regarding
their perception of the accuracy of the current placement indicators based
on their observations of students in advanced mathematics courses
(Research Question 2 and 3).

Develop a timeline for data collection.
July – August
2020



August- October
2020








Seek necessary approvals (IRB, School District)
related to developed survey and use of district data
Gather data from past years related to students’
math acceleration and long-term math coursework
Gather multiple literary sources and pieces to
review and include in a literature review related to
the topic and research
Organize literature and execute a thorough review
Distribute survey for mathematics teachers to
complete

PLACEMENT CRITERIA FOR MATHEMATICS ACCELERATION
OctoberDecember 2020
January – April
2021



May – July 2021



July-August 2021






159

Organize, sort, and outline the collection of various
data points in preparation for data analysis
Analyze and interpret data from both spreadsheet
and teacher surveys
Begin to develop tables, charts, and graphs related
to analyzed data
Embed data into written portion of research
project, articulate findings, and develop
recommendations
Share and present research and findings

Explanation of how you plan to analyze your data
This action research plan is data-heavy and will consist of numerous spreadsheets,
charts, tables, and/or graphs. There will be multiple cohorts of students on detailed
spreadsheets revealing their placement scores, as well as subsequent math courses and
cumulative averages. This data will be analyzed for correlations as proposed in
Research Q1 and Q3. Also, the data collected from the surveys completed by district
mathematics teachers will be analyzed for magnitudes of similar responses and how
their responses may also correspond with Research Q2 and Q3 and triangulate with the
data from the screening criteria.

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Appendix E
First Response from the Institutional Review Board
(This was the first response received from the IRB indicating that there were issues that
needed to be addressed in order for approval to be obtained.)
Institutional Review Board
California University of Pennsylvania
Morgan Hall, 310
250 University Avenue
California, PA 15419
instreviewboard@calu.edu
Melissa Sovak, Ph.D.

Dear Kristin,
The IRB is in the process of reviewing your proposal titled “The Accuracy
of the Advanced Mathematics Placement Criteria in Identifying Students for
Mathematics Course Acceleration.” (Proposal #19-079) the following issues
have arisen:
Please resubmit the following:

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Also:
-The teacher survey portion of the study basically appears to meet OHRP and CALU IRB
requirements, However, the consent form should be edited to include a statement that
although responses are kept confidential, there is a risk that individuals could be identified by
triangulation of identifying data—e.g. gender, certifications held, grades taught, courses taught.
-The researcher states that no children under 18 will be involved in the study. However-mention
is made of accessing student performance data sets which is contradictory to that statement.
There is no explanation of what data is being obtained and how detailed it is, etc. No mention is
made of obtaining parental consent or student assent for student data to be used in this study.
These items need to be clarified.

Please respond to these issues so the Board may continue its review.
Email responses (with attachments as needed) are preferred. If hard copies
are submitted in response, they must be sent to Campus Box #109.

PLACEMENT CRITERIA FOR MATHEMATICS ACCELERATION
If you have any questions or comments, do not hesitate to contact me.
Melissa Sovak, Ph.D.
Chair, Institutional Review Board

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Appendix F
Researcher’s Response to Institutional Board Review Request
(After issues were outlined in the first response from the IRB Chair, this is the
researcher’s response, including revisions and further clarification to appeal to IRB
Chair’s request.)
Dear Dr. Sovak,

Thank you for reviewing my research proposal. In response to the issues that have arisen, I am
responding with and providing the following:

In regards to the use of student data, the district has granted permission to use the data
identified for this study. The reason that it was stated that no children under 18 would be
involved is because no students will be interviewed, questioned, or actively participating in the
study. The researcher will strictly be using data points only for which no identifying information
relative to any student will ever be revealed or used in the study, nor included by the
researcher. Additionally, the researcher will have a district-level administrator code every
student who would have data used in this study so that, prior to the researcher using any of the
students’ data, the researcher will not have names and will be independent of any identifying
information. Based on not having nor using identifying information related to the student data,
the researcher does not believe that parent consent or student assent is necessary.
The data that is proposed to be obtained is the accelerated students’ scores on the qualifying
criteria (3 pieces of criteria and 3 scores), as well as their cumulative grade average in each of
the subsequent accelerated math courses after qualifying for this advanced mathematics track.
The details that are included would be a coded identification for each student (no name), the
numeric score for each of the three criteria, the equivalent points earned for each of these
criteria based on the district’s point system for qualification, and the cumulative grade
percentage for each subsequent accelerated math course for each student.

Relative to the consent form for the teachers including a statement about triangulation, it has
been added to the Google Form. This is evident in the highlighted statement that was added to
the consent form: (Link t Full Consent Form and Survey:
https://forms.gle/aEvgAPFXWfaVpCmR9 )

PLACEMENT CRITERIA FOR MATHEMATICS ACCELERATION

Thank you for your consideration,
Kristin M. Deichler

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Appendix G
Approval from Institutional Review Board
(This is the letter of approval that was received after the researcher made revisions to
address the initial issues with the proposal.)

Institutional Review Board
California University of Pennsylvania
Morgan Hall, 310
250 University Avenue
California, PA 15419
instreviewboard@calu.edu
Melissa Sovak, Ph.D.

Dear Kristin,

Please consider this email as official notification that your proposal
titled “The Accuracy of the Advanced Mathematics Placement Criteria
in Identifying Students for Mathematics Course Acceleration”
(Proposal #19-079) has been approved by the California University of
Pennsylvania Institutional Review Board as submitted.

The effective date of approval is 9/11/20 and the expiration date is
9/10/21. These dates must appear on the consent form.

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Please note that Federal Policy requires that you notify the IRB
promptly regarding any of the following:

(1) Any additions or changes in procedures you might wish for your
study (additions or changes must be approved by the IRB before they
are implemented)

(2) Any events that affect the safety or well-being of subjects

(3) Any modifications of your study or other responses that are
necessitated by any events reported in (2).

(4) To continue your research beyond the approval expiration date of
9/10/21 you must file additional information to be considered for
continuing review. Please contact instreviewboard@calu.edu

Please notify the Board when data collection is complete.

Regards,

Melissa Sovak, PhD.
Chair, Institutional Review Board

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Appendix H
Current Screening Process Rubric
(This is the current rubric used for the screening process of accelerated mathematics
placement in the South Fayette Township School District. This rubric applies to the raw
scores associated with the Third Edition of the Test of Mathematical Ability (T.O.M.A.
3). This rubric applied to the three most recent cohorts of accelerated students used in this
study.)

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Appendix I
Initial Screening Process Rubric
(This rubric was used for the first two cohorts and included the raw scores from the
Second Edition of the Test of Mathematical Ability (T.O.M.A. 2). The distribution of
rubric points based on raw scores earned are different in this rubric than the current
version that is used. This rubric was used for the first two student cohorts that were a part
of this study.)