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Implications of a tracked mathematics curriculum in middle school

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Implications of a tracked mathematics curriculum in middle school

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Implications of a Tracked Mathematics Curriculum in Middle School

by
Jason Scott Windust

Rossier School of Education
University of Southern California
A dissertation submitted to the faculty
in partial fulfillment of the requirements for the degree of
Doctor of Education

August 2022

© Copyright by Jason Scott Windust 2022

All Rights Reserved

The Committee for Jason Scott Windust certifies the approval of this Dissertation

Darline Robles
Morgan Polikoff
Lawrence Picus, Committee Chair

Rossier School of Education
University of Southern California
2022

iv

Abstract
The purpose of this study was to look at the relationship between tracking students in math into
two different tracks beginning in grade six in a private, international school in Singapore and if
there was an impact of course placement on (1) academic growth while in middle school, and (2)
academic outcomes as measured by completion of higher-level courses by the end of high
school. Students were grouped by the number of years they took advanced math while in grade
six through eight, then the academic growth they made on a school administered standardized
test was compared, as well as the number of high-level courses these students eventually
completed by the time they graduated from high school. The analysis showed that course
placement in middle school had a significant impact in both areas. With students placed on the
advanced math track, showing greater growth and better academic outcomes. The conclusion of
the researcher was that within the context of where this study was carried out, that delaying when
students are tracked, and creating more flexible pathways for students, may very well lead to
greater growth for students that currently have limited access to some of these more advanced
courses.
Keywords: ability grouping, acceleration, tracking, differentiation, de-tracking,
mathematical mindset, mathematical self-efficacy, mathematics

v

Dedication
To my late father-in-law, Dr. David Snell, who showed me what it meant to be a lifelong learner,
and was a model for living a faith-centered life.

vi

Acknowledgements
There are many people I would like to acknowledge who have helped me on my
educational journey, culminating in this research study. First to my committee chair, Dr. Larry
Picus, who has provided valuable guidance and reassurance through writing this dissertation.
Also, to my other committee members, Dr. Darline Robles and Dr. Morgan Polikoff. I have been
the beneficiary of your expertise and feedback over the past several months and for that, I am
very grateful. Finally, to Andrew Sheu and Meg Guerriero for helping this middle school math
teacher with his rusty data analysis skills.
On a personal level, I would be remiss if I didn’t thank my own parents, John and Sandra
Windust. From a young age, you both taught me the importance of hard work hard and
perseverance, through both your words and actions. These values have helped me immensely
during these past three years. To my daughter Indéa, who was always there to help me keep
things in perspective, helping me to laugh in stressful times. A willing procrastination partner
who would rarely say no to a game of cribbage when I should have been working. Last, and
certainly not least, to my amazing wife, Wendy. Starting a doctorate was a daunting decision, but
deciding to do this together made it so much easier. As has always been the case, you push me
out of my comfort zone in all the best ways, and I’ve loved learning together with you. Your
desire to do work that matters continues to inspire me. The past three years have been filled with
challenges, and at times sadness, but I will look back with pride knowing we were able to
accomplish this together.

vii

Table of Contents

Abstract ........................................................................................................................................... iv
Dedication ........................................................................................................................................ v
Acknowledgements ........................................................................................................................ vi
List of Tables .................................................................................................................................. ix
List of Figures .................................................................................................................................. x
List of Abbreviations ...................................................................................................................... xi
Chapter One: Overview of the Study .............................................................................................. 1
Statement of the Problem .................................................................................................... 2
Purpose of the Study ............................................................................................................ 3
Conceptual Framework ....................................................................................................... 4
Researcher Positionality ...................................................................................................... 6
Significance of the Study ..................................................................................................... 7
Limitation and Delimitations ............................................................................................... 9
Definition of Terms ........................................................................................................... 10
Organization of the Study .................................................................................................. 12
Chapter Two: Review of the Literature ......................................................................................... 13
The History of Tracking and Its Implications ................................................................... 13
Equity Issues to Consider in a Tracked Curriculum …. .................................................... 17
Impact on Academic Performance and Outcomes ............................................................ 20
Impact on Mindset and Self-Efficacy ................................................................................ 24
Alternatives to a Tracked Math Curriculum ...................................................................... 27
Conclusion ......................................................................................................................... 44

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Chapter Three: Methodology ........................................................................................................ 47
Context .............................................................................................................................. 48
Sample and Population ...................................................................................................... 52
Research Design………………………………………………………………………… 52
Instrumentation .................................................................................................................. 53
Validity & Reliability of Instrument ................................................................................. 55
Data Collection .................................................................................................................. 56
Data Analysis ..................................................................................................................... 57
Summary ............................................................................................................................ 63
Chapter Four: Results .................................................................................................................... 64
Results Research Question One, Academic Growth ......................................................... 64
Results Research Question Two, Academic Outcomes .................................................... 74
Summary ............................................................................................................................ 79
Chapter Five: Discussion ............................................................................................................... 81
Findings ............................................................................................................................. 81
Limitations ......................................................................................................................... 88
Implications for Practice .................................................................................................... 89
Future Research ................................................................................................................. 92
Conclusions ....................................................................................................................... 95
References ..................................................................................................................................... 97

ix

List of Tables
Table 1: Distribution of MAP Scores and Math Placement 49
Table 2: Index Variables for Regression Analysis 58
Table 3: Outcome Variables for Academic Outcomes 60
Table 4: Outcome Variable for Calculus Completion 61
Table 5: Academic Growth for Middle School Students 65
Table 6: Annual Growth for Students Taking At least One Year of Advanced Math 66
Table 7: Annual Growth in MAP Scores by Grade in Regular and Advanced Math 66
Table 8: Total Growth for Students taking At Least One Year of Advanced Math 68
Table 9: Total Growth Based on Number of Years of Advanced Math 68
Table 10: Regression Analysis for Number of Years of Advanced Math 69
Table 11: Academic Growth for Students Academically in the Middle 71
Table 12: Annual Growth for Middle Achieving Students 72
Table 13: Annual Growth by Middle Achieving Students 72
Table 14: Total Growth of Middle Achieving Students 73
Table 15: Total Growth for Middle Achieving Based on Years in Advanced Math 74
Table 16: Regression Analysis, Number of Years of Advanced Math for Middle 74
Table 17: High School Academic Outcomes 77
Table 18: Higher Level Courses 77
Table 19: Logistic Regression Completion of Calculus 78
Table 20: Higher-Level Courses Excluded High and Low Achievers 80
Table 21: Logistic Regression, Completion of Calculus Excluded High/Low Achievers 80

x

List of Figures
Figure 1: Factors Contributing to Academic Outcomes 5
Figure 2: Middle School Math Pathways at SIS 48
Figure 3: High School Math Pathways at SIS 51
Figure 4: Academic Growth in Middle School 67
Figure 5: Range of Grade Five MAP Scores Versus Placement (Growth) 70
Figure 6: Academic Growth for Middle Achieving Students 73
Figure 7: Range of Grade Five Map Scores Versus Placement (Outcomes) 79

xi

List of Abbreviations
AP Advanced Placement

AT Advanced Topic

BFLPE Big-Fish-Little-Pond-Effect

CCSS Common Core State Standards

CGI Conditional Growth Index

MAP Measures of Academic Proficiency

OFA Organized for Advancement

RIT Rasch Unit

SIS Singapore International School

1

Chapter One: Overview of the Study
At present, a majority of U.S. based and international schools use some form of ability
grouping or tracking in mathematics that sometimes starts as early as elementary school, and
continues throughout high school. Over time, secondary school tracks have developed, and
courses have been given labels such as basic, regular, honors, advanced, or accelerated (Hallinan,
1994). Many of these courses are aligned with pathways that have been laid out by the Common
Core State Standards (CCSS) for math so that students can “be college and career ready, and to
be prepared to study more advanced mathematics” (National Governors Association Center for
Best Practices & Council of Chief State School Officers, 2010, p. 1). Often schools define being
“college ready” as ensuring students have the opportunity to take calculus by their grade twelve
year. This definition of readiness has a trickle-down effect on the curriculum and various
pathways offered in secondary schools, and reinforces the perceived need to group students
homogeneously in order to accelerate high-achieving students through earlier grades, so that they
can complete school-defined pre-requisites for advanced courses, such as AP Calculus. By
creating homogenous groupings, many believe student achievement can be increased by
potentially providing targeted instruction to specific ability levels (Slavin, 1987).
Over the years there have been numerous studies and articles written about the tracking in
mathematics, looking at the potential benefits and drawbacks to this common practice. Much of
this work has understandably focused on answering the question whether this is an equitable
practice. As detailed in Chapter 2, numerous studies have shown that marginalized groups of
students are often underrepresented proportionally in higher-track classrooms. For example, after
examining data from the National Longitudinal Survey of Youth, with a sample size of 1922
2

high school graduates, Broussard and Joseph (1998) found that pathways for racial minority
groups were limited as a result of academic tracking.
One of the challenges identified in the findings of many of these studies is trying to
measure the degree to which tracking affects the current achievement gap, versus how much can
be attributed to other factors such as socioeconomic status, parental expectations, race and
gender to name a few. When doing research in the field, there is no way in which to control for
all variables, thus proving causation is difficult to do with any sort of reliability. This debate can
be traced back to the Coleman Report, a seminal work in which researchers tried to determine if
the primary cause of the achievement gap could be more greatly attributed to inequitable
structures in school or in society (Coleman, 1966). The Coleman Report concluded that societal
factors played a larger role, although the methodology of this study has its own critics (Downey
& Condron, 2016).
Statement of the Problem
The practice of tracking students, particularly in mathematics, has been debated in k-12
schools and academia for decades. Evidence that suggests there is a positive result or increased
learning for advanced students when grouped homogeneously is unclear. On one hand, several
recent studies have suggested that tracking students in mathematics has produced negative
results, especially for those on lower tracks, which often are disproportionally populated by
students from marginalized communities (Boaler & Selling, 2017; Mooney et al., 2021;
Schofield, 2010; Wells, 2018) however, not all researchers agree regarding the negative
consequences of tracking. In a meta-analysis of 261 studies, Hattie (2002), concluded that
tracking had little effect, either positive or negative, on academic achievement, finding instead
classroom practices and teacher pedagogy played a much more significant role. However, within
3

the conclusion of this meta-analysis, Hattie acknowledges the substantial amount of qualitative
research that suggests tracking may influence how these classes are taught. In another secondary
meta-analysis by Steenbergen-Hu et al. (2016) that was based on an analysis of 13 other meta-
analysis, and 172 studies, over a time period from the 1920s to the present, that tracking and
academic acceleration has in fact been shown to improve academic achievement. Even after this
substantial secondary meta-analysis, the researchers highlighted the need to continue studying
ability grouping and acceleration, recognizing that changing nature of education, and how
students might be affected differently depending on the domain, age, and background.
This study focused on two aspects of tracking. The first was the immediate impact it had
on academic growth in the first three years the students took part in a tracked mathematics
curriculum. This growth was measured by using standardized test scores from grades six, seven,
and eight. Second, the same cohort of students was followed into high school, and their academic
outcomes were measured by the number of higher-level math courses each completed by the
time they graduated.
Purpose of the Study
The purpose of this study was to examine the possible relationships between tracking in a
middle school math program, academic growth, and eventual student outcomes. This relationship
was explored by comparing students from two distinct math tracks. The data that was used for
this study came from students that attended Singapore International School (SIS) from grades six
through 12 over a four-year period.
Research Questions
1. What is the relationship between the level of math courses a student takes in middle
school and their academic growth from grade six through grade eight?
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2. What is the relationship between the level of math course assigned to a student in
middle school and the higher-level math classes completed by graduation?
To answer the first research question, standardized test data from Grades 6-8 was collected
and analyzed, comparing academic growth over this time period. Data from participants was
grouped based on which math track the student took part in. To answer the second research
question, academic outcomes for the same cohort of students were evaluated. To measure this,
each student’s high school academic records were used to determine how many higher-level
courses they completed by the time they graduated.
Conceptual Framework
The variables that impact student learning are a complex system. No single study,
regardless of the quality of its design, can hope to completely isolate any single variable and
determine its impact on student learning. As seen in the researcher’s conceptual framework in
Figure 1, there are many external and internal variables, which can affect academic growth and
outcomes. The focus of this study was tracking, examining the relationship between placement
on a particular mathematical track and a student’s academic growth and outcomes. There is a
wealth of literature about the effects of race, socioeconomic status, and gender in math
education. These variables are in the researcher’s conceptual framework to acknowledge their
importance, but because the focus of this study was on tracking, these variables will not be
considered. This is one reason this study does not aim to prove a causal relationship.

5

Figure 1
Factors Contributing to Academic Outcomes

Tracking students certainly is not the only variable of significance, but the literature shows
the far-reaching impact it can have on how a student experiences mathematics in school. One of
the foundational pieces of literature that has most influenced the researcher’s beliefs about
tracking stem from the work of Oakes (1980) and her seminal work Keeping Track: How
Schools Structure Inequality. In this study, Oakes looked specifically at how tracking affected
student outcomes and mindset, concluding that grouping students by ability widened the
achievement gap between students placed in different academic tracks. Another possible
explanation for a widening gap between students is known as the peer effect (Fitzpatrick &
Mustillo, 2020). This theory proposes that student mindset and achievement are affected by the
peers that they are surrounded by. Students placed in an advanced course are surrounded by other
high-achieving students, making it more likely to have a classroom of students that are engaged,
thus raising the level of a class. This peer effect can also have negative consequences for
6

students placed in a lower track classroom if surrounded by students that have less of a desire to
learn.
Variables have been separated within the framework as either external or internal. The
internal variables self-efficacy and mindset, have intentionally been placed closer to the students
moving through the curriculum as they are both constantly present and effecting students in
potentially positive and negative ways. In Figure 1, the higher a student is placed vertically,
represents higher levels of achievement and better outcomes. The rectangle to the far left
represents early education, prior to students being separated into different tracks of the
mathematics curriculum. In this box, students are still separated vertically, representing different
achievement levels caused by natural strengths, maturity, and other variables. The large arrow
pointing to the right lists the factors used within the context of this study to separate students into
different curriculum tracks. The first pair of diverging circles represents Grade 6 when students
are first separated into homogenous groupings in mathematics based on the teacher’s perceived
abilities and potential of each student. It should be noted that this paper will focus purely on the
potential effects of tracking students, while acknowledging here the multitude of other variables
that continue to have an influence on student learning.
Researcher Positionality
The researcher for this study is employed by SIS. For the past 5 years, he has held the role
of a grade seven math classroom teacher. In addition, his daughter, while not a participant in this
study, was part of the SIS middle school after the cohorts of students that were part of this study.
Both factors, and previous educational experience, have influenced how the researcher views
tracking. When doing research, it is nearly impossible to separate one’s personal goals; it is often
what motivates a researcher’s choice of topic. Of greater significance is the awareness of these
7

personal goals, and one’s ability to critically self-reflect in order to minimize any personal bias
they may introduce (Maxwell, 2013). The topic for this study was chosen by the researcher for
two reasons. First, he entered this study with a belief, based on personal experiences as a teacher,
that SIS separates students based on perceived ability at too young an age. Second, the
researcher’s daughter took part in the lower-curriculum track, which raised questions about the
potential harmful effects on students’ self-efficacy and mindset. The researcher entered this
study with an intention to use current research and data gathered on-site to ensure he did not
allow these preconceived beliefs to influence the analysis of results.
To limit potential bias, a quantitative approach was taken so that the researcher was not the
primary instrument used to collect data. In addition, the results will be shared with the institution
where it was carried out as part of a curriculum review cycle in which how and why tracking is
used is being questioned. The audience for this will undoubtably have diverse views and critique
any findings and/or conclusions drawn by the researcher.
Significance of the Study
This study aims to help fill a gap in the research in which the degree tracking impacts
learning and outcomes may be more easily identifiable because of the narrower context in which
it was carried out. This study was conducted in a private international school in Southeast Asia
that serves approximately 4,000 students from pre-kindergarten through grade 12, with
approximately 320 students per grade level in middle and high school. The school does not
collect data on race or ethnicity, but the student population is comprised of a diverse group of
students from all over the world. In the 2020/21 academic year, in the high school 48 different
nationalities were represented. For the 2020/21 graduating class about half of all students were
U.S. citizens, while the remaining were either citizens of other countries or held dual citizenship.
8

Similar to other international schools in the region, the high tuition fees mean that the student
population is very socioeconomically advantaged. Singapore International School is well-
resourced and class sizes are relatively small. The average high school class size this academic
year was 18.9 students. Students at SIS are high-achieving academically and the vast majority
attend universities upon graduation. For the class of 2019/20, 93% of graduating students began
university immediately, and six percent entered into a compulsory military national program. A
robust Advanced Placement (AP) program is offered. In total 1,619 AP tests were taken in 2020,
with an average score of 4.16 out of 5, compared to a worldwide average of 2.89. In this same
cohort of graduates, approximately 87% had a 3.35 (out of 4) grade point average or higher. This
recent data for student diversity and academic achievement is in line with historical data at SIS
and not out of the ordinary. All data was gathered from the organization’s website.
Tuition costs are prohibitively high, ensuring all students come from families of high
socioeconomic status. Despite the homogenous socioeconomic status of the student population,
not all the participants in this study enter middle school having had have equal access to a
rigorous curriculum, and cultural norms can vary significantly between different families. These
factors will be discussed in greater detail in the limitations section. With this being said, the SIS
student population shares more similarities than student populations that are the basis of most
other studies in current research. These similarities, and unique context, may have limited the
generalizability of any findings, but also it may have allowed for any difference found in
academic growth and outcomes to be more closely connected to tracking as a possible cause.
Besides contributing to the large body of research already done about tracking in
mathematics, the findings in this study will also be used by SIS to help inform decisions that
must be made within the school’s current curriculum review that is looking specifically at the k–
9

12 math curriculum at the school. If a decision is made to eliminate, or more likely to delay,
when students are tracked into different mathematics courses, it is likely to include a healthy
debate between various stakeholders in the community. Efforts to de-track math curriculums
have increased in frequency in recent history because of questions about equity and
effectiveness. Decisions to de-track are frequently resisted by parents, fearing their children may
be exposed to a watered-down curriculum (Oakes, 2005). Singapore International School
historically has had tremendous success producing students that score well on Advanced
Placement tests, and any change that may be perceived to be taking away opportunities from
students would likely be questioned. Singapore International School is in a competitive,
educational marketplace, substantial and possibly controversial program changes risk driving
families to other private international schools in the community. This is true if the parent
community perceives these changes as potentially disadvantaging their child’s chance of being
accepted into a more prestigious university. By grounding this study in a quantitative data, it may
help stakeholders to make more informed decisions about if change is needed, based on facts and
data, rather than perception and opinion.
Limitation and Delimitations
One significant limitation of the findings involves tutoring, or private tuition. In
Singapore, where this study took place, there is a culture of students attending private tutoring
(Jelita, 2017). Private tuition centers in Singapore are abundant and a sizable portion of students
that attend SIS also receive some sort of support out of school. The school does not have hard
data about the number of students receiving private tuition, defined as regular attendance to
classes outside of school, or private one-on-one tutoring, but estimates from teachers and
counsellors suggest a sizable number of students are receiving additional, private support outside
10

of the school. There was no way to determine if the participants in this study had received private
tutoring and how this may have affected their growth or outcomes. A second limitation is the
difficulty in accounting for the effects that COVID-19 may have had on the data collected from
the 2020/21 cohort of participants. Their choices of math course in the 2020/21 academic year
may have been affected by the school moving to remote learning for the last quarter of the
2019/20 school year.
One delimitation of this study was the generalizability of these findings. Singapore
International School is a private international school that caters to a community from a high
socioeconomic status. The school is well-resourced, class sizes in math are typically limited to
24 students, and students’ school education is frequently supplemented through private tutoring
outside of school. The findings for this study will only be able to be modestly extrapolated to
similar contexts.
Definition of Terms
Ability Grouping: Placing students in specific classes based on perceived ability, or grouped with
similar ability students in numerous classes (Oakes, 1987). Ability grouping can take different
forms including; between-class, within-class, cross-grade subject, and gifted and talented
groupings (Steenbergen-Hu et al., 2016).
Acceleration: Systematically progressing students through school, or a given curriculum, more
quickly than their peers. This may also include, or lead to, students taking courses at younger
ages than a typical student (Steenbergen-Hu et al., 2016).
Between-School Tracking: The process of tracking students into entirely different schools based
on their perceived ability (Trautwein et al., 2006).
11

Curriculum Tracking/Differentiation: Placing students in a track with the expectation they are to
complete a sequence of courses designed to lead to a specific outcome e.g., college preparatory,
vocational, or general education (Oakes, 1980, 1987).
De-tracking: System that intentionally groups mixed-ability, heterogeneous students together
with the goal of maximizing equitable access to mathematical content for all students through a
common curriculum, effective instruction, appropriate supports, and high expectations for all
students (Mooney et al., 2021; Rubin, 2006; Rubin & Noguera, 2004).
Fixed Mindset: Belief that traits such as character, intelligence, and creative ability are static and
cannot be changed in any meaningful way (Dweck, 2016).
Growth Mindset: Belief that traits such as character, intelligence, and creative ability are not
static, and that these traits can be developed over time (Dweck, 2016).
Heterogenous Grouping: Groups that are made up of individuals that have a variety of interests,
attitudes, backgrounds, and/or abilities (Rubin, 2006).
Homogenous Grouping: Groups that are made up of individuals that have similar interests,
attitudes, backgrounds, and/or abilities (Rubin, 2006).
Mathematical Mindset (self-concept): A person’s self-perception about their ability in math, that
has been formed through experiences.
Mathematical Self-Efficacy: A person’s belief in what they are capable of doing or achieving in
mathematics (Schunk, 2020).
Overmatching: Placing a student within a grouping for which they may be unprepared, or
deemed to have lower ability than the majority of group members (Fitzpatrick & Mustillo, 2020).
12

Peer effect: Extent to which the composition of classes, which may be based on such factors as;
ability, ethnicity, gender, multigrade and multi-age classes, and class size affects learning
outcomes (Hattie, 2002).
Within-School Tracking: System where students, within a heterogenous school, are divided into
homogeneous groups and assigned to different classes based on any number of variables (Oakes, 2005;
Trautwein et al., 2006).
Organization of the Study
The second chapter of this study reviews some of the extensive literature centered on
tracking. This will include looking at the origins of tracking, issues around equity, effects on
academic achievement, mindset, and self-efficacy. Lastly, examples will be provided of schools
that have effectively de-tracked their math curriculums and what structures should be in place
when de-tracking. Chapter 3 will provide an outline of the methodology of the study. Within this
chapter, more details will be given about the context of where this study takes place, the
participants, and the statistical tools and tests which will analyze the quantitative data gathered.
In chapter four, the data will be analyzed by research question. Finally, chapter five will include
a discussion of the findings and their implications for schools that are considering different
options for how they group students and the pathways available to them.

13

Chapter Two: Review of the Literature
The literature around tracking is extensive. Within this review of the literature, the focus
will be organized into five major sections; the history of tracking, issues around equity, effects
on academic performance and student outcomes, effects on student’s mindset and self-efficacy,
and efforts to de-track math curriculums within schools. Despite many studies and meta-analysis,
researchers still have not reached a conclusive answer to whether tracking students in
mathematics is a net negative, net positive, or neutral. As seen in the researcher’s conceptual
framework, trouble reaching consensus on this issue can likely be attributed to the many
variables at play that influence student achievement. Most recent research supports efforts to de-
track, and the negative impact it has academically on students placed in lower-level math
courses. There is a growing body of work that strongly suggests that tracking can amplify
existing inequities in school between historically marginalized groups of students and their white
and Asian peers. The research around the impact on mindset and self-efficacy is more varied.
The History of Tracking and its Implications
Public education in the United States. started in the 1860s with a common curriculum. At
that point in time, very few students, less than 10%, moved on to secondary education, thus there
was little need to differentiate curriculum or track students into different levels of classes (Oakes,
2005). Around 1900, the number of students that attended secondary schools and later
universities increased, which led to a perceived need to differentiate the school curriculum for
those students that were more likely to further their education, versus students more likely to
move directly into the workforce. It was an efficient mechanism to prepare, and then channel
students into the roles of society they would most likely fill (Ansalone, 2010; Guiton & Oakes,
1995; Oakes, 1980). The initial goal of tracking was to act as a sorting mechanism for students
14

based on their most likely ultimate place in society, not their innate ability. Over time, the
practice of tracking has become more refined.
There are different forms of tracking that have developed, complicating the research as to
its effectiveness. Two of the most common forms of tracking are within-school and between-
school tracking. Within-school tracking is the most common form of tracking in the United
States today and is what is most applicable to the purpose of this study. Within-school tracking is
when a heterogenous group of students is separated into distinct classes based on many variables.
Different educational systems may do this at different ages for different subjects, the focus of
this study being mathematics. This should not be confused with ability groupings. Ability
grouping is a more general term, which may include within-school tracking, but can take place in
a heterogenous classroom, where different students are given different tasks or presented with
different content based on their perceived readiness. A second type of tracking is between-school
tracking. Between-school tracking involves tracking students into entirely different schools
based on their perceived ability. Between-school tracking is more common in some European
countries, such as the Netherlands, Germany, Austria, and Switzerland, as well as some Asian
countries like South Korea and Singapore (Anderson, 2015; Ho & Li-Ching, 2014; Reed et al.,
2015; Ryan, 2004; Trautwein et al., 2006).
Even when school populations are considered heterogeneously grouped, it is not
guaranteed that students will receive equivalent educational experiences. In Social Class and the
Hidden Curriculum of Work, Anyon (1980, 1981) further identified “hidden curriculums” within
schools that correlated to the socioeconomic status of most of the students they served. Anyon
classified these schools, ranging from working-class schools that serve the lowest socioeconomic
populations, to executive elite schools whose students came from the most privileged families.
15

Working-class schools as institutions had an ethos and practiced instructional strategies more in
line with what research has shown to be common in lower track classrooms; a focus on student
compliance and rote-learning. Whereas students that attended executive elite schools were given
more freedom to express their own voice and opinions, similar to what is often found in upper
track classrooms.
Assumptions Made When Tracking
Despite a large body of contradictory research, grouping students by ability in one form
or another continues to be widespread in educational systems throughout the world. This leads to
the questions, what are the reasons for the persistence of tracking, and why are so many people
opposed to de-tracking school curriculums, particularly in mathematics? To answer these
questions, one must first consider the long-held beliefs and assumptions upon which tracking is
based (Ansalone, 2010; Guiton & Oakes, 1995; Mooney et al., 2021; Oakes, 1987; Oakes, 2005;
Werblow et al., 2013).
The first assumption is the belief that bright, high-achieving students will be held back by
their slower, lower-achieving peers (Oakes, 2005). This assumption is often held by parents of
high-achieving students that are more familiar with a tracked math curriculum because of their
own educational experiences (Oakes, 2005). Often, this group of parents is resistant to suggested
changes to potentially de-track the math curriculum, as they believe it will disadvantage their
own children (Burris & Welner, 2005; Rubin, 2006). The second assumption is that low-track
student’s mindset would be negatively affected being with peers that are stronger math students
(Ansalone, 2010; Marsh et al., 2008). The third assumption is that student placement into tracks
can be done subjectively and fairly (Ansalone, 2010).
16

There is a large body of research that has shown that race and socioeconomic status often
play a significant role in determining whether students should or should not be placed within
advanced tracks in mathematics (Ehlers & Schwager, 2020; Guiton & Oakes, 1995; Mooney et
al., 2021; Werblow et al., 2013). Even when some argue school’s decisions are based purely on
academic achievement, often standardized test scores, there is no guarantee these tests are not
culturally biased. Current standardized tests in mathematics often reward students that are
proficient at doing calculations quickly, but don’t reward those students that are strong problem
solvers (Mooney et al., 2021). Ultimately, decisions on course placement are often made by
teachers, that whether they know it or not, have their own biases. Even when students perform
equally well on measures of academic achievement, African American and Latino students are
still more likely to be placed in lower track classrooms (Guiton & Oakes, 1995).
The fourth and final assumption is that teaching homogeneous classrooms is easier for
teachers because of less of a need to differentiate. Ultimately, this is probably an accurate
statement (Ansalone, 2010; Oakes, 2005). While some educators probably believe offering
heterogenous math classes is helpful to students, it presents a more challenging classroom
environment to manage, as well a more complex task to design curriculum and assessment tools
that are appropriate for a more diverse set of learners. Perhaps the most prevailing reason for the
persistence of tracking is stated by Anselone (2009), “Tracking seems to be the program of
choice primarily because it is embedded in the broader reality and culture of the school rather
than because of its overall effectiveness” (p. 150). Because of the cumulative effect of tracking
(Mooney et al., 2021; Oakes, 1980; Oakes, 1987) and the impact it has on student outcomes, it’s
difficult to imagine structures where it does not exist.

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Equity Issues to Consider in a Tracked Curriculum
Schools have long been touted as a great equalizer in meritocratic societies. Students who
work hard can use education to better their standing in society. In reality, like most systems,
schools and education benefit those in power. Schools get to define what knowledge is valued,
and reward student that conform to the status quo (Oakes, 1980). Despite efforts to classify
mathematics as a neutral subject, one does not need to dig deep to see issues of power and
subjectivity in what many consider the most objective of subjects, in which tracking is just one
component (Battey, 2013). Examples of power and subjectivity in the math classroom begin with
acknowledging it to is a cultural space where normative practices typically favor the privileged
(Hand, 2010). How math has been taught has changed little over time. The dominant culture has
created a narrow definition of what math is, and how we measure success. Louie (2017)
describes this as a “culture of exclusion” for those students that don’t conform to these narrow
definitions.
The merits of tracking have been long debated, and often those debates center on issues
of equity. Wells (2018) describes. The question that arises is does tracking amplify existing
inequity within schools, and if so, to what degree? Well-intentioned educators have long viewed
it as a powerful tool for differentiation, a goal that is typically applauded in education. Studies
have shown five factors that can lead to iniquity in a tracked curriculum (Guiton & Oakes, 1995;
Mooney et al., 2021; Oakes, 1980). First is the quantity of leaning that occurs. Students in higher
track classrooms are simply exposed to more content. Making different knowledge available to
different students, or curricular differentiation (Oakes, 1980), is often the key factor that bars
students in lower tracks from being allowed access to advanced level courses in high school. It
may also mean that students that move on to college either cannot pursue degrees that require
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advanced levels of math, or are required to take multiple levels of remedial math courses to get
them up to a satisfactory level for university studies.
Civil rights activist, Robert “Bob” Moses in his book Radical Equations: Civil Rights
from Mississippi to the Algebra Project, has compared giving students of color early access to,
and the ability to be successful in algebra, as analogous to the voting rights movements of the
1960s. In response to the need to create more access to algebra for all students, he created an
organization called The Algebra Project (Moses, 2020; Moses & Cobb, 2002). Moses recognized
the role algebra was playing as a gatekeeper, that students placed in lower mathematical tracks
did not have sufficient access to algebra and were then less likely to qualify for advanced courses
later in high school, meet minimum university requirements, or develop the skills involving
symbolic representation that are needed in working in careers around technology. Moses’
conclusions are reinforced when examining national statistics in the United States that show
students, disproportionally white, that take higher-level math courses in school are more likely to
earn substantially more money over the course of their lives (Battey, 2013). The second factor to
consider is the cognitive demand placed on students (Fitzpatrick & Mustillo, 2020). Even when
exposed to the same content, students in high track classrooms are often given more demanding
tasks that require the use of higher-level thinking skills, as opposed to the rote-learning that is
often associated with lower-track classes. A third factor is the experience and enthusiasm
provided by teachers (Fitzpatrick & Mustillo, 2020; NCTM, 2020; Oakes, 1980; Guiton &
Oakes, 1995). In many schools, new and unexperienced teachers are more commonly assigned to
lower track classrooms. Teachers that have greater pedagogical knowledge, accumulated through
more time in the classroom, are “awarded” with more advanced classes to teach, creating another
layer of inequity (Mooney et al., 2021). Fourth is the impact of peer interactions on the learning
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of students. Studies show overwhelmingly that students in high track classrooms have students
that have more positive interactions with their teachers and with one another (Guiton & Oakes,
1995). Finally, are the expectations teachers bring with them into the classroom. The act of
labeling students has the effect of creating a different set of expectation by teachers, which can
then lead to different outcomes (Boaler, 1997).
Another potential source of inequity is the implications of overmatching. Overmatching
refers to the practice of placing students in a class they may seem not to be prepared for
(Fitzpatrick & Mustillo, 2020). In math, this has shown to have a positive effect on student
learning. Students from privileged families are more likely to be overmatched in their classes
(Burris et al., 2006; Fitzpatrick & Mustillo, 2020; Oakes, 1980) but the reasons for this are not
clear. Parents that are more familiar with the structures in place that track students may be more
likely to exert their influence to make sure their children have opportunities to take these courses,
as compared to a student from a lower socioeconomic background. The benefits from
overmatching can carry into university admissions (Fitzpatrick & Mustillo, 2020). Privileged
families use seemingly meritocratic educational structures to help their child gain access to
universities they may not otherwise be considered for. This can be done through test preparation
courses, private tutoring, or working with school counsellors or other individuals in the
admissions process. These practices help to effectively maintain inequality in education.
When students are tracked at a young age, it does impact when they are first exposed to
algebra. Placement into different classes can be subjective, particularly with students that are on
the borderline of whatever criteria is being used to group them. The choice to overmatch these
students or not may very well determine when they take their first formal algebra course, one of
the most significant markers in a student’s math journey. Algebra in Grade 8 is considered the
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“gateway” to advanced courses in high school. In a school that tracks its students, being able to
take algebra in Grade 8 is not guaranteed. First-generation students, those whose parents didn’t
attend college, are much less likely to take algebra in Grade 8, thus less likely to go to college
(Horn & Nuñez, 2000). The same can be said of African American and Latino students (Moses
& Cobb, 2002). Students that don’t take algebra by Grade 8 are 60% more likely to eventually
drop out of school (Werblow et al., 2013).
Impact on Academic Performance and Outcomes
The variables that impact the academic achievement of students are a complex system in
which tracking is just one part. Despite the multitudes of studies centered on tracking, it is still
difficult, if not impossible, to fully separate the effects of tracking versus other variables, such as
a student’s socioeconomic status, race, and gender (Oakes, 2005). Regardless of this complexity,
tracking has a direct impact on four key areas: content, instructional strategies, peer-to-peer
interactions and outcomes that have been well documented over the years.
Often, a key difference between high and low tracks is the content that is made available
to students in those respective classes (Oakes, 2005), referred to as curricular differentiation. To
cater to students with a stronger background in math, curriculums have developed that will
accelerate students in higher track classrooms, this becomes reasons for lower-track students to
lose access to advance level math courses because they have not had the same access to the pre-
requisite skills needed for these courses. This acceleration in advanced level courses can be
achieved because of a higher percentage of time devoted to instruction in high track classrooms
(Oakes, 2005), increased participation among students (Hand, 2010), and more demanding
pacing. Accelerating students does not come without a cost. As a counterpoint to the benefits of
overmatching (Fitzpatrick & Mustillo, 2020), Boaler (1997) found that sometime the strongest
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students in advanced math classes experiences negative consequences of rushing through
content, such as increased anxiety, lower levels of retention, and not being able to connect what
they are learning to real-life situations.
Besides differences in the amount of content students are exposed to are differences that
develop in instructional strategies. Lower track classes have shown to have a greater reliance on
didactic instruction, providing fewer opportunities for students to take part in sense-making and
collaborative work. Instead, the emphasis increasingly focuses on classroom management and
compliance (Hand, 2010). One reason for this difference in instructional strategies is the teachers
that are more likely to be teaching advanced courses. Teachers of higher tracked classes are more
experienced than those of their colleagues that teach lower track classes, thus having stronger
pedagogical knowledge than their less experienced peers (Mooney et al., 2021). However, as
with much of the research, there is not consensus about the impact tracking has on instructional
strategies employed in the classroom. In a metanalysis Hattie (2002) concluded that tracking had
little impact on academic achievement and the classroom environment, suggesting that
differences in student performance are more likely related to the quality of instruction, regardless
of whether students are grouped heterogeneously or homogeneously by perceived skill.
Another difference is the type of peer interactions students have in math class, which
some studies have shown to affect academic achievement. These studies have shown that lower
track classrooms have shown to have more defiant students, which causes teachers to provide
fewer opportunities for deeper learning and limiting student participation to control classes
(Hand, 2010). The participation that occurs becomes more polarizing, with groups of students
that are involved, and those that choose to remove themselves from valuable learning
opportunities (Hand, 2010). When surrounded by peers that are stronger academically than
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themselves, lower ability students often benefit from a “positive peer group effect” (Ehlers &
Schwager, 2020; Fitzpatrick & Mustillo, 2020). Those in favor of tracking will point to its
positive effect math self-concept, a student’s perceived competence in mathematics, or
commonly referred to as the Big-Fish-Little-Pond-Effect (BFLPE) (Marsh et al., 2008). The
logic underlying the BFLPE is that students, when placed with students of similar ability, will
have a greater math self-concept, an important predictor to future achievement.
Different content exposure, instructional strategies, and peer interactions all can lead to
differing student outcomes. It has been shown non-tracked schools consistently improve
performance of low-ability students, and possibly improve overall averages (Boaler & Staples,
2008; Ehlers & Schwager, 2020; NCTM, 2019). There is more debate on the benefits and
drawbacks to high-ability students, but rarely have heterogenous classrooms been shown to have
a significant negative impact for higher ability students (Boaler & Selling, 2017; Burris et al.,
2006). For example, in a study, looking at schools in New York over a 6-year period, 3 before
de-tracking and then 3 post-tracking, all groups of students were shown to have performed better
once the schools had de-tracked their math curriculum, including high-level students (Burris et
al., 2006). In a second study in England, schools that offered a project-based math curriculum
and heterogenous math classes were compared to schools with a more traditional structure that
included tracking. The project-based, non-tracked schools ended up with students that had better
test scores, a deeper understanding of the content, and eventually better long-term job prospects
(Boaler & Selling, 2017).
De-tracking is not without its detractors. Despite a growing body of research that
suggests tracking may be an ineffective practice, there are counter studies that have shown the
opposite to be true, that tracking may in fact improve student performance. Two significant meta-
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analysis of different grouping programs (Kulik & Kulik, 1992; Steenbergen-Hu et al., 2016)
found clear and consistent academic benefits from grouping programs, particularly for those
students in the higher ability groups, and that students in the lower-tracked groups were not
harmed academically by grouping, sometimes even gaining academic ground. In contrast to
Oakes’ (2005) findings, Kulik and Kulik (1992) found that ability grouping did not have
devastating effects on student self-esteem, as Oakes has charged, believing instead that their
findings showed the effects of grouping on self-esteem to be near-zero overall. A second meta-
analysis (Steenbergen-Hu, et al., 2016) also reached the same conclusion, that tracking benefited
students, stating “The preponderance of existing evidence accumulated over the past century
suggests that academic acceleration and most forms of ability grouping, such as cross-grade
subject grouping and special grouping for gifted students can improve K–12 students’ academic
achievement (p.893).” Because of the vast number of studies that were considered in these meta-
analyses, their findings should not be easily discounted.
Despite apparently conflicting research, a host of educational organizations and
governing bodies have endorsed reducing or eliminating tracking within the school curriculum.
Some of these include; The National Governors Association, Carnegie Council for Adolescent
Development, College board, National Council of Teachers of English, The California
Department of Education, Massachusetts State Legislature (Welner, 2001), National Research
Council (National Academies Press, 2003), and more recently the National Council of Teaching
Mathematics (NCTM, 2020). One of the key recommendations by the National Council of
Teaching Mathematics was that “Middle school mathematics should dismantle inequitable
structures, including the tracking of teachers and the practice of ability grouping and tracking
students into qualitatively different courses” (NCTM, 2020, p.19). Clearly stating that even when
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the same content is taught, different experiences caused by intentionally grouping students based
on perceived ability into different classes, leads to different outcomes. This conflicting research
is one reason that the debate about how and if to group students is unlikely to end soon.
Impact on Mindset and Self-Efficacy
Interrelated with student’s academic achievement, is their attitudes about themselves and
mathematics as a discipline, specifically their mindset and self-efficacy as it relates to
mathematics. In her seminal work Keeping Track: How Schools Structure Inequality, Oakes
(1980) found that the attitudes of students in different tracks shared common attitudes. She found
that participation within a track affects how students viewed themselves and plans for their
educational futures. Students enter school and individual classrooms at different places, in terms
of their readiness to learn, for a wide variety of reasons. Dweck (2016) estimated that 40% of
students have a growth mindset, 40% fixed, and 20% mixed. However, rather than a supposed
meritocratic school structure acting as an equalizer, the differences in attitudes between students
in high and low track classroom increases over time, showing that tracking students may amplify
differences in students’ mathematical mindset and self-efficacy.
The research doesn’t fully agree on the impact tracking may have on mindset and self-
efficacy. The primary justification for tracking having a beneficial impact on mindset is that low-
achieving students taught in homogenous math classes will learn alongside those of similar
ability, thus avoiding potentially damaging comparisons with classmates that may be stronger
students (Marsh et al., 2008; Trautwein et al., 2006). The argument against this is that the action
of labelling students by placing them in a lower math class creates in them a fixed mindset about
their abilities and decreases their self-efficacy (Boaler, 2013).
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Students’ self-reported feelings about mindset and self-efficacy often act as an argument
for the potentially damaging act of tracking students (Hand, 2010; Reed et al., 2015). Hand
(2010) found that when students in lower tracks were interviewed and compared themselves to
students enrolled in higher track classes, they often self-identified as being “slow” and “lazy” in
comparison. They attributed their lack of success as simply not being a math person, showing a
negative, fixed mindset around their abilities. When examining math self-efficacy and self-
concept in an educational system with early tracking in the Netherlands, researchers found the
track played a large effect on both (Reed et al., 2015).
Other studies have shown that tracking may actually help mindset. Marsh et al. (2008)
found evidence supporting the Big-Fish-Little-Pond-Effect (BFLPE) that is based on social
comparison theory. The BFLPE predicts that when students attend schools, or are placed in
classes with other high achieving peers, their academic self-concept suffers due to comparing
oneself to the group, as opposed to students that are surrounded by similar, or lower-ability peers.
Here, BFLPE predicts an improved academic self-concept when students are grouped
homogeneously. Although this study and the concept of the BFLPE is not without critique. For
example, when students were put in a gifted program, academic self-concept decreased but only
temporarily, and this was not even true in all students studied (Dai, 2004; Dai & Rinn, 2008).
However, in this study, researchers found evidence that supported BFLPE, providing an
argument that tracking may improve academic self-concept.
A similar study carried out in Germany looked at the impact of tracking in between-
school and within-school tracking (Trautwein et al., 2006). The study explored two concepts, the
contrast effect and assimilation effect. The contrast effect looks at the impact of comparing
oneself to their peers. The thought of being in a homogenous classroom lower ability students
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would have improved mindset because they were comparing themselves to similar ability
classmates. The second idea is that of an assimilation effect, which says that self-concept is
based on the group they are placed in. This would be an argument against tracking because of the
belief that the mindset of students placed on a lower track would be negatively affected.
Ultimately, they found that tracking had minimal impact on student’s math self-concept. The
difference found in self-concept were instead attributed to past performance and teacher assigned
grades.
Tracking is only one variable that may affect mathematical mindset and self-efficacy, and
tracking can affect different students in different ways. Several of these studies identified gender
as playing a significant role in how students responded to being tracked. Boaler (1997) showed
that girls were most negatively affected by being in high track classrooms because of the
emphasis on speed and moving quickly through the curriculum. These girls felt as though they
weren’t given sufficient time to make sense of the content, which damaged their mathematical
self-concept. This did not seem to affect the boys in the study as much. Tracking students may
also cause fixed ability beliefs, but especially for high achieving girls. This group was frequently
praised at an early age for being smart, usually for arriving at correct answers. Initially, this may
not seem to be a problem, but later as the difficulty level of math increases, these same girls were
more likely to opt out of more challenging courses because of the message they had previously
received, equating struggling with a lack of intelligence. Dweck’s (2016) work showed the
cumulative and corrosive repercussions of tracking over time, finding heterogenous groupings
more conducive to developing a growth mindset when we avoid labelling children as gifted (or
not gifted), and instead label educational programming descriptively by level of difficulty, or just
by using a grade level descriptor. This is one possible explanation why there continues to be a
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gap between the number of boys that ultimately take advanced level math course and the number
of girls, despite girls achieving academically just as well, if not better, when younger.
Alternatives to a Tracked Math Curriculum
This section will discuss the process of de-tracking. It will begin with looking at
impediments to this process. Second, what are the actual steps or processes involved if a school
plans to de-track their math program? Third, will be an examination of the structures that can be
put into place to make it more likely that schools can implement these changes, and that once
they do, students will be given the best opportunity to succeed. Fourth, will look at changes that
need to be made to instructional practices and philosophy when transitioning to a de-tracked
structure. Fifth, examples of school systems that have successfully de-tracked will be examined.,
This section will conclude with things to avoid when going through the de-tracking process and
potential benefits beyond academic achievement, mindset, and self-efficacy.
Impediments to De-Tracking
As discussed previously, tracking in schools, particularly in mathematics, has been the
status quo for a long time. Despite growing evidence that suggests tracking is ineffective and
inequitable, the practice persists. According to the Brookings Institution, 75% of all ability-based
classes are mathematics classes, making it the most tracked discipline in schools today
(Loveless, 2013). We have outlined the assumptions that have led to educators and other
stakeholders to resist de-tracking math classrooms; the belief that high-achieving students will be
held back by peers that have a perceived lower ability (Ansalone, 2010; Oakes, 1987; Oakes,
2005), that placing students that would traditionally be in a lower-track math classroom with
higher achieving peers will negatively affect their mindset (Oakes, 1987; Oakes, 2005), that
placing students into tracks can be done objectively and fairly (Ansalone, 2010; Mooney et al.,
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2021; Oakes, 1987; Oakes, 2005), and that teaching homogenous groups of students is easier for
teachers because of a lessened need to differentiate the curriculum (Ansalone, 2010; Oakes,
1987; Oakes, 2005). However, once these issues have been addressed, and a decision has been
made to move to a de-tracked mathematics curriculum, there are still a host of challenges to
overcome. This should not be surprising, as any systemic changes that challenge long held
practices and beliefs will undoubtably be met with resistance.
One of the first areas that needs to be examined are stakeholders’ views on what students
are capable of. These stakeholders include community members, administration, teachers, and
the students themselves. (Louie, 2017; Mooney et al., 2021; NCTM, 2020; Rubin, 2006). Over
time, deficit views of what students are capable of have been formed for a variety of reason, one
of which may be tracking and labelling students. Overcoming this type of deficit thinking can be
difficult because it is grounded in long-standing structures and practices. Schools can provide
freedom of choice for students to pursue more advanced studies, and put in place structures that
will support these students, but if a student does not believe in themselves, that they can do the
work, then they are unlikely to succeed or even pursue these opportunities (Rubin, 2006). The
same can be said for family and other community members that are part of each student’s support
system. The inequitable systems that many parents went through as students themselves may
lead to a lack of trust that their own child will be supported. Schools need to convince
community members that math classrooms are a safe and supportive place.
Another key stakeholder is the teacher. In a research study in which Louie (2017)
observed four geometry teachers while teaching, during department meetings, and in formal and
informal conversations. These teachers held a firm commitment to using equitable practices,
having previously taken part in professional development in this area. Despite a sincere desire to
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engage in inclusive teaching practices, three out of the four teachers quickly reverted to a more
exclusive framing, or what Louie referred to as a “culture of exclusion”. In meetings and
collegial conversations, they would frequently apply hierarchical labels like “low”, “strong”, or
“smart” when describing students. During class, each teacher made an attempt at inclusive
framing, using activities that were student-centered and open-ended, but most quickly reverted to
an exclusive frame, which included more rote-learning, routine and closed tasks, and repetitive
practice, with a lack of opportunities at sense-making. Often, the language that teachers, district
leaders, parents and families, policymakers, and other stakeholders use shapes beliefs about
student potential and reveals perceptions of students.
In their book Detracking for Excellence and Equality, Burris and Garrity (2008) note how
ingrained the culture of sorting students is in our educational systems, using as evidence the
frequency and nonchalance that teachers assign and use labels when discussing students. Making
changes, such as replacing the word ‘ability’ with ‘achievement’ when discussing students, can
have a meaningful impact. By labelling a student as ‘high ability’, it implies there is something
innate that makes them better at mathematics, as opposed to saying ‘high achieving’, which is
just a measurement of what a student knows at a moment in time. By changing the language that
is used, it can change how we view students and their potential.
There are many arguments that practitioners and researchers used to support the need for
tracking (Bannister, 2016; LaMar et al., 2020; Mooney et al., 2021; Wells, 2018). The first is that
low-achieving students need more direct instruction and routine practice than that of their peers.
To witness this type of pedagogy in many low-track classrooms, it is easy to believe that, in fact,
many of these low-achieving students may be the ones to benefit the most from less rote
learning. In his case study in which Bannister (2016) investigates whether differentiated
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instruction is ideologically opposed to goals for equitable classrooms, he found the student that
was the subject of his study, was hindered academically when given more direct instruction and
routine practice, particularly if it was at the cost of working with classmates in heterogenous
groupings. While the generalizability of a case study with one student is certainly limited, the
questions it raises are valid.
A second argument addressed in this case study was that creating classes that cater to
different learning styles is best practice, a viable argument for the benefits of tracking or
grouping students with peers of similar perceived ability. Recent research has shown little to no
evidence that crafting instruction or courses to match learning styles is effective (Kirschner &
van Merrienboer, 2013; Newton, 2015). Trying creating classes or tracks that cater to particular
learning styles is problematic for a variety of reasons. First, individuals do not fit neatly into one
learning style, let alone groups of students (Kirschner & van Merrienboer, 2013). Further
complicating the use of learning styles to group students is that different tasks or content might
be better suited to be taught using different styles, irrespective of the learner (Kirschner & van
Merrienboer, 2013; Riener & Willingham, 2010). For example, some students may learn well
kinesthetically, but not all concepts lend themselves to this type of learning.
One final impediment to de-tracking is the rise in importance of content standards, such
as the Common Core and standardized testing (LaMar et al., 2020; Mooney et al., 2021; Wells,
2018). When students, teachers, and schools are judged primarily on their performance on
standardized tests, it leads to a narrow focus on what mathematical learning is valued in the
curriculum. Following a strict set of narrowly defined standards prevents students from exploring
what they might find interesting within that subject. This evolution has led to math instruction
that teaches students to believe math is a discrete set of skills that needs to be progressed
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through, with little emphasis on sense-making or connections within and outside of the
mathematics classroom (Wells, 2018).
The current curricular structure in mathematics has emphasized certain types of
knowledge and skills, which de-values other important aspects of the discipline (Louie, 2017).
Students that can perform multi-digit, pencil-and-paper computations, quickly and efficiently,
are judged to be our best math students, those that are then most commonly accelerated and/or
placed into high-track classrooms (Mooney). This although technology has rapidly made many
of these overly valued skills obsolete. Louie (2017) warns that this narrow focus on
computations and speed may disadvantage students that may be excellent problems solvers and
deep thinkers, but may struggle with performing calculations quickly and with precision. By
devaluing this knowledge, it can create a system in which certain students are rewarded with
access to the most challenging curricular options, while the rest of the students are assigned to
classes that may leave them feeling unchallenged. As the technology students have access to
develops, there is a need to continue to change, or sometimes, overhaul, what is taught, how it is
taught, and how it is assessed (Li & Ma, 2010). The goal of a mathematics curriculum should
extend beyond just racing through content to ensure students can take calculus by the time they
graduate (LaMar et al., 2020; Mooney et al., 2021).
Process of Implementation
In this section, the process of how de-tracking may be implemented will be explored. As
with any substantial change, there should be a clear purpose, rationale, or goal that drives the
decision to de-track. De-tracking curriculums, particularly in mathematics, is not a recent
phenomenon. Often, the original goal of de-tracking has been to increase social integration
(Rubin & Noguera, 2004). This is true in schools that see a distinct separation of racial groups
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being placed in high-track versus lower-track math classrooms. There is often a clear
discrepancy in how various races are proportionally represented (Ehlers & Schwager, 2020;
Guiton & Oakes, 1995; Oakes, 1980; Oakes, 1987; Oakes, 2005; Wells, 2018; Werblow et al.,
2013). Thus, increasing the level of integration in these classes is a sensible and worthy goal.
With that being said, to de-track effectively, this should not be the sole purpose (Rubin &
Noguera, 2004).
The goal of schools is to create an environment that fosters learning in its students,
therefore the second goal of any de-tracking effort must also be to engage all students in as many
rich learning experiences as possible (Rubin & Noguera, 2004). By focusing solely on
integration, a school is likely setting up a system in which students are either being placed in
classes beyond their current capabilities, leading to an increased likelihood of failure, or you are
lowering your standards and not all students are being as challenged as they could be.
Once a school has established its purpose to de-track, there are certain questions that
should be asked to help guide what needs to be done in the process ahead (Burris & Garrity,
2008). Many of these questions are practical in nature, but will have major implications for what
steps need to be taken. At what grade level are students first grouped within homogenous
classes? What rationale is used for their grouping? Do these groupings affect long-term outcomes
for the students involved? When are students first placed into different courses or tracks? What is
the basis for these placements and is the accuracy and validity of those placements ever
assessed? The answers to these questions can help to guide how a school may choose to
implement de-tracking. Burris and Garrity (2008) offer other suggestions. Schools should focus
on data, grounding the conversation in fact rather than opinion. Tracking can be an emotional
issue, for parents of both high and low-achieving students. Besides looking out for the best
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interest of their own child, parents bring with them their own educational experiences, many of
which will influence their perceptions about the best course of action. One valuable piece of
information is student transcripts. Examining and sharing how placing students within a track
can affect long-term outcomes can be a powerful argument to de-track mathematics classrooms.
As an example, using a nationally representative set of data, it was found that students taking
part in a low-track mathematics curriculum were 60% more likely to drop out of school
(Werblow et al., 2013). When examining the student populations in each track, one is likely to
find an overlap in achievement, described as “hidden heterogeneity”. This overlap can be caused
by a variety of reasons, and is further evidence of the subjectivity of course placements at the
point when tracking begins. Last, Burris and Garrity (2008) state that when multiple levels of
mathematics courses are on offer, when starting to de-track, it is best to eliminate the lowest
track first.
Many school systems, districts, schools, and math departments have worked toward the
goal of creating systems that are structured in such a way to create rich learning experiences and
fair student outcomes for all (Gutiérrez, 1996). In this study, eight high school math departments
were looked at. Gutiérrez, (1996) used a new conceptual framework that she coined “organized
for advancement” or OFA. Rather than focusing on tracking itself, they looked at the common
practices or traits between these math departments that allowed students to effectively advance
through the mathematics curriculum at the school. The author described these schools as either
OFA or non-OFA, an OFA school was one that was better structured to support students taking
more advanced mathematics as opposed to non-OFA schools. All the schools studied offered
some form of tracking, but OFA schools had a higher proportion of what would be deemed
“advanced” classes. High OFA schools also offered fewer choices of courses, the curriculum was
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more compressed, allowing students to advance through the curriculum more efficiently. Non-
OFA schools offered more lower-level classes. This allowed students more lateral movement
options, thus avoiding some of the more challenging curriculum offered. Gutiérrez identified
four common traits that supported advancement in OFA schools. The first of which is a
commitment to a rigorous and common curriculum. Second, is an active commitment and
collective responsibility to see students achieve. This should not be the job of a just individual
teacher; the entire school community needs to work towards this goal. Third is a commitment to
the collective enterprise. In OFA schools, this could be seen by teachers’ willingness to teach a
variety of courses. Teachers in these schools seemed to have a greater autonomy in their own
classrooms, but fit this within a collective vision for the department they were a part of. Last,
OFA school were committed to innovative instructional practices, and there was a belief that
teachers could learn and use them, an example of teacher self-efficacy.
Research Based Structures That Support De-tracking
While innovative instructional practices are valuable tools, alone, they are not enough to
ensure equitable instruction in the classroom. Changes need to be made at an institutional level,
not just by individual teachers (NCTM, 2020; Rubin, 2006). One institutional change is how
teachers are assigned to the courses they teach. In many schools, there is a system of unofficial
teacher tracking that runs parallel to the tracking of students (Gutiérrez, 1996; Mooney et al.,
2021; NCTM, 2020). Another factor in successfully de-tracking is the level of academic support
that is offered to students that are now being asked to engage in more challenging tasks which
they may not be fully prepared for (Rubin, 2006). This support can and should be offered within
and outside of the math classroom. Within classes this can be achieved by utilizing learning
support teachers as a resource to better support the needs of all learners, or at the very least every
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effort should be made to minimize the number of students per class in order to make it easier for
teachers to differentiate for a wider range of students. Outside of class, this may take the form of
a structured support class where students can be given additional time to deepen conceptual
understanding of pre-requisite knowledge, they may otherwise lack, or given more time and
instruction on the topics they are currently learning. One cannot simply thrust students into a
more advanced course and expect them to succeed without additional support. This is one reason
that if the goal of de-tracking is simply integration, without also striving to improve the learning
experiences of all students, it may lead de-tracking to be ineffective and possibly harmful to the
students involved (Rubin & Noguera, 2004).
As with many new initiatives, time and training needs to set aside for teachers to
implement changes to the curriculum and instructional practices necessary to successfully
manage a de-tracked mathematics classroom. It is not enough to simply set aside this time prior
to implementation. There should be ongoing, regularly scheduled time for teachers to meet and
plan collaboratively (Rubin, 2006). Doing this work is hard. Bannister (2016) points to a norm
that is as true for teachers as it is for students: “None of us alone is as smart as all of us together”
(p. 344).
A practice that some schools have found to be effective is offering optional after-school
classes for parents/students that want to accelerate (Boaler, 2016). Accelerating students, which
may involve skipping a grade or class has been shown an effective way to meet the needs of the
highest achieving students, but only if they can show they have a deep conceptual understanding
of the course material they are trying to move past already (NCTM, 2020).
In a qualitative research study, a group of students were asked about their experiences with
curricular differentiation, and what processes and structures should be in place to support them in
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their learning. Their own perceptions often aligned with the current educational research (Stanley
& Venzant Chambers, 2018). Many of these students didn’t see tracking so much as the cause, or
source of inequity, rather a symptom of a problem that already existed. With that being said, all
students felt that tracking should not occur before students reach high school. The participants
felt strongly that the best way to ensure all students were prepared for more challenging
mathematics by the time they reached high school was to focus on early preparation. Even
students that supported tracking in theory highlighted that students rarely had an equitable
background in mathematics, and that this needed to be addressed. Another common response was
the importance of allowing students to fulfil their individual strengths, and that a one-size-fits-all
model was often not effective. The need to build strong communities among students was
emphasized, especially among minority groups, so they do not feel as isolated. Ways to
strengthen these communities could be through celebrating backgrounds and establishing cohorts
of minority students to take advanced placement courses together. Although the participants in
this study were African American, many of these same principles could likely apply to female
students that often find themselves underrepresented in these same courses. Students expressed a
lack of knowledge about the tracking policies at their schools, and the need to better inform
students about policy and the consequences of entering a specific track that few students ever
move out of. Last, several of the participants pointed to a lack of minority teachers, that by
increasing the number of faculty that looked like them would likely lead to improved
performance.
Changes to Instructional Practices and Philosophy
Within this section, the focus will be what happens within the classroom and the
philosophy behind it. There are several theories about how to create curricular structures that will
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challenge all students while allowing more equitable access to engaging content. To begin with,
there will be a brief discussion of the multidimensional classroom (Boaler, 2016; Boaler &
Staples, 2008), culturally relevant pedagogy (Ladson-Billings, 2009), and complex instruction
(Bannister, 2016; Boaler, 2016; Cohen et al., 2002; LaMar et al., 2020; Louie, 2017).
Multidimensional Classroom
In many mathematics classrooms, the primary focus is on procedural fluency. A teacher
shows students an algorithm, then asks them to apply this algorithm by solving increasingly
complex problems, while gradually removing the level of teacher support required. Students that
are able to quickly learn and apply these algorithms, with minimal calculation errors, are
rewarded and often promoted into higher track math classes. These traditional classrooms can be
best be described as one-dimensional. The one skill that is valued above all else is being able to
see something modelled and then repeat that procedure on your own. In order to effectively de-
track, classrooms need to become more multi-dimensional in their approach to teaching and
evaluating mathematics (Boaler, 2016). Things that have traditionally been de-emphasised:
asking good questions, proposing ideas, showing connections between different methods and
representations, reasoning, rephrasing, justifying methods, using manipulatives, and helping
others need to play a more prominent role in the curriculum. “Put simply, when there are many
ways to be successful, many more students are successful” (Boaler & Staples, 2008, p. 630).
Culturally Relevant Pedagogy
Ladson-Billings (2009) coined the phrase culturally relevant pedagogy and laid out some
underlying principles that have direct application when looking at ways to teach a diverse set of
learners in a heterogeneously grouped classroom. These include: building on students’ own
interests and knowledge (Rubin, 2006), incorporating real-life experiences, and teaching through
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activities that showcase student’s strength Ladson-Billings (2009) describes culturally relevant
pedagogy as playing a part in a broader context of good teaching practice, of which strong
knowledge in pedagogy, subject matter, and methods of assessment also play vital roles.
Escondido Union High School District (EUHSD) in California worked to incorporate many of
these changes in their own district through systematic changes to their math curriculum (NCTM,
n.d.). Some of these changes included promoting a collective responsibility among teachers,
using a more problem-based curriculum and de-tracking their math classes. This district also
committed to a four-year “teacher curriculum” aimed at improving its quality of instruction,
collaboration, and building a shared vision. The result of this ongoing work has been an increase
in teacher satisfaction, students working at grade level, and increased enrolment in advanced
courses.
Complex Instruction
Complex Instruction is a practice that was based on the theoretical work of Elizabeth
Cohen and then further developed by Rachel Lotan. The underlying principles of complex
instruction have been directly applied to many efforts in de-tracking math classrooms (Bannister,
2016; Boaler, 2016; Cohen & Lotan, 2014; Cohen et al., 2002; LaMar et al., 2020; Louie, 2017).
Much of the basis for complex instruction comes from students collaboratively working together
on what Cohen and Lotan describe as group worthy-tasks. To start, it is the teachers’
responsibility to intentionally assign students to different groups. This is an important step in a
heterogenous class. This was highlighted in a study focused on a de-tracked social studies class,
examining the experiences and perceptions of six students and their two motivated teachers
(Rubin, 2006). The teachers in this study used intentional grouping to mix high and low-ability
students. When students were left to choose their own groups, students would choose to sit in
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segregated clumps that have similar academic identities, or in other words, higher achieving
students would only work with other high-achieving students, and the same for lower achieving
students. When students naturally segregate themselves in this way, it eliminates many of the
benefits that a positive peer effect can have in a heterogenous classroom. Even with these
intentional groupings, group work was still a challenge for several of the student participants
because of the social dynamics they brought into the classroom with them, highlighting the
importance of teachers to help build relationships between students.
The second part of the teacher’s job in complex instruction is assigning clearly defined
roles to each member of the group, highlighting group members’ competence. This may be
achieved through highlighting when students show an excellent strategy, getting them to explain
it to one another, especially with low-status students (Bannister, 2016). As part of complex
instruction, developing a norm of community responsibility is important. Students are taught that
learning is a collective responsibility, that they should be activity involved in, and that it should
not be left up solely to the teacher to be the dispenser of knowledge in the classroom. When
student realize their knowledge and skills are important to the learning of others, it is not
uncommon to see previously quiet and disengaged students become active classroom participants
(Rubin, 2006; Rubin & Noguera, 2004).
The last major component of complex instruction is the creation of group-worthy tasks.
Group-worthy tasks focus on big ideas, are open-ended, and benefit from students that bring
different strengths to the group (Bannister, 2016; Cohen et al., 2002); Rubin, 2006). In her study
of a de-tracked curriculum that uses complex instruction Boaler (2016) captures the essence of a
group-worth task when students realize “No one is good at all of these ways of working, but
everyone is good at some of them” (p.121). Complex Instruction is certainly not without its
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challenges. As mentioned previously, when creating groups, the academic levels of students are
only one factor. Each group of students has their own social dynamic that can be influenced by
many variables that teachers have no way of knowing. Developing truly group-worthy tasks can
be a challenge as well. Boaler (2016) found that even when teachers had received training in
complex instruction and were effectively taught through group work, students still struggled to
work together collaboratively when the assigned tasks were not open-ended enough. When given
closed-ended tasks, the higher achieving students that were more proficient with quickly
completing calculations, finished first and were frustrated by classmates that took a longer
amount of time. Those students that did not finish as quickly mindset were negatively affected
because of the common misconception that speed is an attribute of higher ability. Students also
found the use of extrinsic awards demotivating (LaMar et al., 2020).
One thing all these different approaches to de-tracking have in common is the centering
of students in the learning process. This is in direct contrast to literally centuries of how math has
traditionally been taught, so it comes as no surprise that change may be slow and resistance high.
A shift in mindset around how mathematics is taught is what is needed, but there are some
specific, concrete strategies that teachers can employ in their classrooms to help achieve these
goals, regardless of whether students are grouped together heterogeneously or not, which is what
will be discussed in the final part of this section.
Expanding on the idea of group-worthy tasks, problems presented in class ideally should
be “low-floor” and “high-ceiling”, meaning that students with limited background knowledge
can still access some parts of the task, while the highest achieving students are not limited to a
closed, fixed correct response (Boaler, 2016). These types of tasks must be open-ended, and
allow a diverse group of students to all engage in meaningful mathematics, regardless of the
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background knowledge they enter a class with. Just a few recent examples of these types of
resources can be found by exploring Dan Meyer’s Three-Act Problems (Meyer, 2020), Open
Middle Math (Kaplinsky, 2020), Mindset Mathematics (Boaler et al., 2019), and Making Number
Talks Matter (Humphreys & Parker, 2015). Focusing solely on short, closed-ended question
reaffirms to students that are fast and strong procedurally that they are good at math, while for
students that may be strong problem solvers but not have the same strengths think they are bad
(Boaler, 2013). Another helpful practice is making sure the curriculum taught is spiraled,
meaning that topics are revisited, just covered in more depth each time students encounter them.
This allows students that have previously mastered certain content to dive deeper into more
abstract concepts, while allowing more time for other students to revisit previously taught
content that they may have not fully understood (Rubin, 2006). Organizing content delivery in
such a way minimizes the curricular differentiation so prevalent in most tracked systems. Last is
how students are provided feedback. Feedback should always be criterion reference, evaluating
students based upon developmentally appropriate criteria, rather than norm referenced, which
emphasizes competition between students and a greater likelihood that student mindset may be
negatively affected through peer comparison (Marsh et al., 2008).
Examples of De-tracking
In this last section, there will be a brief discussion of some national educational systems
that have had success through the elimination or minimization of ability grouping. Then three
different schools will be offered as examples of school systems that have de-tracked their
mathematics programs, leading to more equitable outcomes for their students and overall
improved academic achievement.
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When examining data from The Trends in International Mathematics and Science Study
(TIMMS), which compares the mathematics and science achievement of various countries,
Beaton and O’Dwyer (2002) found some of the most successful countries where ones in that had
moved away from ability grouping. Examples within Asia include both Korea and China, which
routinely perform well compared to the rest of the world and heterogeneously group students.
One of the best performing countries is Finland, even once factors such as socio-economic status
have been accounted for. In the Finish school system, there has been no tracking in grades one to
nine since 1985. The success of Finish schools is not only found on the TIMMS assessment, but
they are also one of the highest scoring countries in math and language arts according to Program
for International Student Assessment (PISA) (Burris & Garrity, 2008). In this same study
(Beaton & O’Dwyer, 2002), the U.S., which is one of the participating countries that is most
heavily tracked (Loveless, 2013), showed one of the widest ranges of variability when
comparing student performance between different races and socioeconomic groups.
Three specific school districts will be provided as examples of successful de-tracking;
San Francisco Unified School District [SFUSD] Mathematics (n.d.), Rockville School District in
New York, and Railside School (Burris & Welner, 2005; NCTM, 2019; SFUSD, n.d.).
Several recent studies have looked at the success the SFUSD has had since de-tracking
their math programs (NCTM, 2019; SFUSD, n.d.). Starting in 2014-15, the SFUSD undertook an
ambitious plan to de-track their math curriculum, offering only heterogenous classes through the
tenth grade. Since that time, the repeat rate for Algebra I has plunged for all racial and ethnic
groups, from an average of 40% to 10%. These figures include students from a low
socioeconomic status, English language learners, and other minority groups. The repeat rate for
black students dropped from 52% to 19%, and for Latino students, from 57% down to 14%.
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Concurrently, with these reductions in repeat rates, SFUSD has seen an increase in enrolment in
advanced high school math courses by this same group of students. The results at SFUSD have
been so successful that a similar framework is being proposed for all public schools in
California, most notably not tracking students until Grade 11 (California Math Archives.
YouCubed, n.d.).
Second, is a study by Burris and Welner (2005) of the Rockville School District, a
diverse suburban school in Long Island. Late in the 1990s, the school decided to de-track its
math classes after they discovered the second Regent Exam in math. A graduation requirement
proved to be a stumbling block for many students, keeping them from earning diplomas. The
response at the time was to eliminate the lowest track math classes and enroll all students into
heterogenous math classes that taught the same curriculum previously only students that were in
the advanced classes were exposed to. Struggling students were offered support classes, called
math workshops, and offered after-school help multiple times during the week. The results
provided more evidence that de-tracking can raise academic achievement. The rate of students
earning their first Regent diplomas went from 58%, to 90% of all incoming freshmen having
already passed it. The largest gains came from minority populations, reducing the substantial
achievement gap that existed during this time. This was in contrast to the rest of New York, that
did not see the same narrowing of the achievement gap. Rockville approached this process with
caution, at first keeping a lower-level math class, that was well-supported, for the lowest
achieving students, often with special learning needs, hoping these smaller, well-supported
classes, staffed with additional learning support and teaching assistants, would better serve this
group of students. What they found was the opposite, that this group fell further behind because
of a “low-track culture” that had developed in which teachers spent the majority of their time
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trying to manage the behavior of apathetic students. School officials realized the failure of this
program and moved these students into heterogenous classrooms with the rest of their peers.
Rockfield found that “closing the ‘curriculum gap’ was an effective way to close the
‘achievement gap’” (p. 7).
Last is the case of Railside school, a diverse high school in California, that took part in a
five-year longitudinal study of over 700 students that compared Railside to two other schools
(Boaler & Staples, 2008). Of notable difference between Railside and the other two schools in
this study was that Railside heterogeneously grouped its students in math, in contrast to the other
schools that did not. One impact on the way students were grouped was on the teaching
approaches employed at each school. Railside used more conceptual, open-ended, collaborative
tasks compared with the other two schools. Students at Railside spent 72% of their time engaged
in collaborative activities and only about 4% of their time listening to teacher lectures. In
contrast, in the tracked schools, 21% of the students’ time was spent listening to teacher lectures,
and the nature of the problems posed were much more procedural. Students at Railside reported
enjoying math more, and the achievement gap was less than in the tracked schools. Upon
entering high school, all students were given an algebra test, in which Railside students
performed worse. By the end of the first year, they had drawn even with the other two schools,
and by the end of the second year were significantly outperforming students that had taken part
in the more traditional, tracked sequence of courses.
Conclusion
As a researcher reviewing the literature on tracking, one can easily be overwhelmed by
the seemingly contradictory findings. The power of a meta-analysis is the generalizability of
their findings because of a larger and more diverse sample of students and schools. However,
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when looking at a system such as tracking, generalizing findings may be inherently difficult
because of the complexity of interacting variables that are involved. The context of a study may
be a significant cause of the varied findings in the literature. The make-up of a student
population, the community in which it is a part of, the age at which tracking occurs, how students
are placed, how achievement is measured, and the type of tracking (within-school, between-
school, or ability groupings within a heterogenous class) are all variables that may influence or
amplify the positive or negatives effects of tracking. For this reason, generalizing based on meta-
analysis should be done with great care, similar to trying to generalize based on a single study
within a specific context. Perhaps the question practitioners should ask is not is tracking effective
and equitable, but is tracking effective and equitable within a specific context.
In moving toward a de-tracked mathematics curriculum, there are several things schools
should avoid to help transition more smoothly, and to have the largest impact on student
achievement. First, is avoiding what is described as “teaching to the middle”, or the practice of
delivering content that will challenge most of the students, but simultaneously bores high
achievers while making curriculum inaccessible to low achievers (Rubin & Noguera, 2004).
Throughout this paper, several strategies to avoid this have been discussed, such as assigning
group-worth tasks that are open-ended and have a low-floor, high-ceiling. Second, there is a
danger of “re-tracking” students once they are placed in a heterogenous classroom (Rubin &
Noguera, 2004). This might be done by separating students into smaller groups based on ability
or offering different assessments within the same class. Care needs to be made that teachers are
offering equivalent feedback for all students and maintaining similar, high expectations for all
students.
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Along with the challenges, there are also benefits that schools and students have
discovered beyond improving academic outcomes for their students (Burris & Garrity, 2008).
Often teachers get to know students better as learners when they are forced to differentiate.
When delivering the same content in the same way, to a homogenous group of similarly
achieving students, there is less need to focus on individual student needs. With a wider range of
learners in their classrooms, teachers need to focus more on the needs of each individual student.
Teaching non-traditional learners can also be a motivator for changing outdated instructional
strategies, encouraging teacher flexibility, and limiting rigid thinking.

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Chapter Three: Methodology
In this study, the researcher was trying to determine if a relationship existed between
tracking students into different levels of mathematics classes at the beginning of Grade 6 and
their:
1. academic growth in middle school, and
2. access to higher-level math classes in high school. This chapter will provide
justification for the sample of students chosen, the choice of instrumentation, and statistical
techniques that were used to analyze the data. The purpose of the study was to determine if
student’s academic growth during middle school, and ultimate academic outcomes, as measured
by the number of higher-level math courses taken in high school, was influenced by the level of
math class they are initially placed in when tracking first begins.
There were two sources of data used in this study. The first were student transcripts in
which all identifying information was removed except for a numeric code to match each
student’s high school records with the courses they took in middle school and corresponding
standardized test scores. These were used to identify which math courses the selected sample of
students were enrolled in from Grade 6 through Grade 12. The second source of data was
standardized test scores on the Measures of Academic Progress (MAP) assessment. The MAP
assessment is an adaptive test that all SIS students take bi-annually, from grade three to grade
eight, in the fall and spring of each academic year. The researcher was given access to, and
permission to, use this data by SIS administration in order to help evaluate the effectiveness and
equity of their current tracking program in the middle school. This was part of a larger k–12
curriculum review process the school was working through. The administration was made aware
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that the researcher would use this data to complete this study as part of the researcher’s
dissertation.
This quantitative study may be limited in scope due to the population of students it
serves, but may be useful for SIS as it considers structural changes to their current math
programs as well as other private, international schools that may be questioning their own math
programs.
Context
The current structure of SIS’s mathematics tracking program officially begins when
students are placed at the end of Grade 5 into one of two math courses for the following
academic school year, Math 6 or Math 6+. An illustration of the pathways in the middle school is
shown in Figure 2, the source of which is the organization’s website.

Figure 2
Middle School Math Pathways at SIS

Note. Within this study all students scoring less than a 232 on their grade five MAP test were
ultimately placed in Math 6, while all students scoring more than 256 were ultimately placed in
Math 6+.
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Although Grade 6 is the first-time students are separated into different courses, the SIS
elementary school offers a “Math Lab” for its top performing math students, as identified by
their teacher. This is a pull-out program students attend sporadically throughout the year.
Placements at the end of Grade 5 are made by the student’s classroom teacher and are based on
academic achievement as measured by performance in the classroom, as well as MAP scores
from grades three, four, and five. Student’s learning behaviors, such as work completion and
classroom behavior, are also factored into teacher’s decisions. Parents had input on their child’s
placement, however, ultimately, the final decision rested with the teacher. The distribution of
MAP scores and placement is summarized in Table 1. Singapore International School does not
have cut-off values for MAP scores to be placed into Math 6+ as it is only one of many criteria
used to determine placement. For the population of students in this study, all students scoring
less than 232 were placed in Math 6, and all students scoring more that 256 were placed in Math
6+. For the scores in the middle, ranging between 232 and 256, some students were placed in
Math 6 and some in Math 6+.
Table 1
Distribution of MAP scores and Math Placement
Grade 5 MAP
scores
Math 6 Math 6 Math 6+ Math 6+
199–232 175
233–256 174 309
257–283 95

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Once students were placed on a track, most remained within that same track throughout
middle school. For the six-year period studied, 122 out of 753 students, or 16.2% of this study’s
sample, changed tracks while in middle school. The vast majority of these students, 118 or
96.7% of the students that changed into a different track, moved up a level. The movement
between tracks can be represented by the diagonal arrows in Figure 2. Since this time, the
mathematics department has attempted to create more opportunities for students to move
between levels. To move into a higher track, students are given additional work to supplement
what they are doing in their regular classes, and diagnostic exams are given to all students at the
beginning of each academic year, as well as at the end of the year for students wishing to change
levels.
Both levels of math courses offered follow the Common Core State Standards (CCSS),
however the more advance plus classes accelerate students, teaching some above grade-level
standards in order to ensure students are prepared to take Algebra 1 (Math 8+) as Grade 8
students. Besides accelerating students through the curriculum, the advanced courses require a
greater degree of critical thinking and depth of knowledge than the regular math courses at each
grade level. Math six, seven, and eight are often taught with a co-teacher who specializes in
learning support. Some students may also receive support out of class as part of the learning
support program. This is a daily 40-minute class in which they work on areas of need.
When students enter high school, they are placed into a course based on the class they
had taken in grade eight, as well as their performance in that class. A student that has completed
Math 8 will typically be enrolled into Algebra 1 as a ninth-grader, and may also may be
recommended to take a Math Lab course in conjunction if it is deemed they need extra support.
Most students that complete Math 8+ will be enrolled into geometry as a ninth-grader. If a
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student struggles in Math 8+ they may be assigned a Geometry Math Lab as well to better
support them, or sometimes, be required to retake Algebra 1. The strongest Math 8+ students will
take Accelerated Math I or skip ahead to Algebra II/Trigonometry, placement into these classes
are based on high academic achievement in Math 8+, MAP data, teacher recommendation, and a
placement test that is administered by the high school teachers. Figure 3 illustrates the SIS high
school pathways that are available to students, the source of which is the organization’s website.
Some higher-level courses do also require students to achieve a minimum grade in a pre-requisite
course.
Figure 3
High School Math Pathways at SIS

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Sample and Population
The data for this study was taken from a sample of students that had graduated over a six-
year period. The first cohort of students included in this study is those that graduated in 2019.
The reason for this time frame is that SIS first began administering the MAP test in 2011, which
means the class of 2019 was the first class of graduating students that will have MAP data dating
back to their grade 5 year.
A total population of 753 students was used for this study. The criteria used to select
these students were based on the time each had attended SIS. The 753 students comprised all the
students from the six cohorts that had attended SIS since Grade 5. The researcher’s goal was to
look at the potential influence tracking in middle school had on student’s academic achievement
and growth, as well as what higher-level math classes they eventually took in high school.
Therefore, students that had moved into SIS after Grade 5 were not considered in this study
because the researcher could not be sure of their math experience they had prior to arriving. In
order to achieve the largest sample size possible, this six-year period was used. Subsets, or sub-
samples of this population, were used when analyzing each research question. Further
information about how and why these sub-samples were chosen will be provided in the data
analysis section.
Research Design

This study had a non-experimental design. The choice for design is both a practical and
ethical matter. Practically, the researcher did not have control of the independent or explanatory
variables for two reasons. First, is because the data used for this study came from the past.
Second, even if the researcher had used current students, he did not have the authority to
randomly assign students to a particular track when the school in which this study was conducted
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in already has clear guidelines about the criteria used to place students. With this study, the two
comparison groups for this study were students that were assigned to Math 6 versus students that
were assigned to Math 6+, the regular and advanced math courses on offer, respectively. Another
reason for the choice to use a non-experimental design is that it does not require the researcher to
take significant steps to control the influence of extraneous variables. However, because of this
design choice, this study could not make claims about causality, a significant limitation of non-
experimental designs (Lochmiller & Lester, 2017).
This non-experimental study is best described as explanatory, and having a retrospective
casual-comparative design. This choice applies in this study because the researcher had
identified two explanatory variables, the first of which was categorical, the track students were
placed into in grade six. The second explanatory variable that was used was the student’s prior
academic achievement as measured by their Grade 5 Spring MAP score. The researcher analyzed
if there was an association between these two groups and the two dependent or outcome
variables. The first outcome variable, academic growth, was measured using the difference in
MAP scores between the fall of grade six and spring of grade eight. An analysis of this data was
used to answer the first research question. The second outcome variable, number of higher-level
math courses completed, was used to answer the second research question. As we started with
the cause, in this case the track students are a part of, as opposed to the effect, academic growth
and high school courses taken, this is retrospective (Lochmiller & Lester, 2017).
Instrumentation
Scores on the MAP assessment from grades five through eight were the source of
quantitative data used to answer the first research question for this study; what is the relationship
between the level of math courses a student takes in middle school and their academic growth in
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middle school? The two scores that were used as the main points of comparison were each
student’s MAP scores at the start of grade six and where they finished at the end of grade eight.
Grade five scores were gathered and used to determine a student’s prior knowledge, as well as
part of the analysis of how students were initially placed into these two different tracks.
The MAP is an adaptive test that adjusts the difficultly level of questions to provide a
meaningful measure for whatever level the student taking it is at. This means each student’s test
is individualized based on the current performance level. The MAP assessment uses a Rasch
Unit (RIT) to measure growth and the levels of academic difficulty. Because the RIT scale
extends across all grades, it makes it possible to compare a student’s score at various points
throughout his or her education (NWEA, n.d.). Because the goal of the first research question
was to measure student’s growth over time, this makes MAP data a suitable choice. RIT
measures are superior to percentile and grade equivalent scores because they are curriculum
referenced, instead of being based on a comparison to a specified group of students (Institute for
Objective Measurement, n.d.). This point particularly applies to the context of this study, because
of many of the advantages students at SIS enjoy compared to the vast majority of students that
take the MAP test. Some of these advantages include socioeconomic status, well-resourced
school, small class sizes, high levels of parental involvement, and an abundance of private
tutoring many of the students receive. Percentiles would show most SIS students are already
well-advanced compared to national averages, but this would not help to answer the first
research question about academic growth throughout middle school.
Student MAP scores were chosen as a source of data for this study because the MAP
assessment is the standardized instrument SIS has used to measure its student’s academic growth
and achievement from grade three to grade eight. It has been used for nearly a decade in the
55

school and scores are widely understood by the faculty and administration of SIS, and to albeit a
lesser extent, the parent community. Therefore, when looking at program effectiveness as part of
the curriculum review process that is currently taking place, MAP data can be used as a
subjective measure over time. Another option available for this study was student grades. The
challenge with using teacher assigned grades is the difficulty with standardization because of
different teachers with different grading practices. For this reason, teacher assigned grades were
not factored into this study.
Validity & Reliability of Instrument
The MAP database used has thousands of questions which a computer algorithm chooses
and assigns to students based on previous responses. These questions have been written by
trained teacher teams, and field tested with tens of thousands of students to ensure optimal
quality and fairness. The entire calibration process took ten years to research, develop, and field
test prior to eventually implementation (Institute for Objective Measurement, n.d.). The
extensive process in which these tests were developed by trained educators strongly suggests that
MAP scores can be a valid tool to measure student knowledge.
MAP data for math achievement has been shown to have high reliability. Teachers who
proctor this assessment are given training in how to administer each test and provided with clear
and prescribed teacher and student directions. Each year, MAP tests are given at roughly the
same time of the academic year, and the same time of day. As part of this computer-based
program, teachers are alerted if students are answering questions too quickly, showing they may
just be guessing, the test is paused so that the teacher can discuss with the student the importance
of taking the test seriously so that it can be used as a tool to help inform instruction. Last, each
56

participant would have taken the MAP test six times in this study, meaning one bad day or test
score would have minimal impact. These factors ensure maximum reliability for this data.

Data Collection
The data measuring academic growth was gathered from the selected student sample for
the years 2010 through 2017 for grades five through eight. The data was archived in the SIS data
repository. Subsequently, these student’s records were then retrieved to determine which high
school math classes they ultimately ended up taking. The data for this study was retrieved by the
SIS data information specialist. Student names and any other personal information were not
included, instead a numeric code was used to organize student data so that the researcher would
not know the identity of the participants. This data was kept on password protected files on the
researcher’s computer in order to maximize the security of the data.
During this study, the researcher was employed as a grade seven math teacher within the
school that was the focus of this study. The researcher had been employed at the school since
2016, therefore some students involved in this study would have been taught by the researcher
during his time at SIS, but by ensuring data did not include identifying information, any potential
bias should be limited.
Two sources of data were used to answer the research questions for this study. The first
research question; what is the relationship between the level of math courses a student takes in
middle school and their academic growth in middle school, uses MAP test scores of the
participants while in middle school. The purpose of this study was to compare academic growth
between groups of students in the two different tracks offered at SIS, therefore, MAP growth will
be compared between these 2 groups. Comparing to national averages would not have helped to
answer the first research question. The researcher determined because SIS average scores were
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so much higher than national averages, little useful information about the influence of tracking at
this institution could have been found out by this comparison. The second research question;
what is the relationship between the level of math courses a student takes in middle school and
the number of higher-level math courses taken in high school, required the use of high school
transcripts that showed course completion by the participants. Similar to the MAP scores,
transcript information was provided by the school data specialist with no identifying information,
other than the same numeric coding used to organize the MAP data.
The researcher had permissions from the school’s administration to use this data to help
inform the math curriculum review process that was currently underway at the school and with
the knowledge it would be used in this study. The data provided through MAP scores was
quantifiable and could be used in statistical analysis. No human subjects took part in this
research.
Data Analysis
The data analysis for this study was broken into two parts based on the research question
that is being analyzed. To answer each question, the researcher will draw on the same target
population of students, the six cohorts of students, including students graduating in the years
2019, 2020, 2021, 2022, 2023, and 2025. Data from the cohort of students graduating in 2024
has not been included because of a lack of data. This cohort did not take the MAP test their grade
eight year because all students were taking part in home-based learning because of Covid
restrictions, thus the MAP test was not administered during this cohort’s grade eight year. Each
question will use a different sample, with each a subsequent sample being a sub-sample of the
one that precedes it.

58

Analysis of Research Question 1
Research Question 1 asked the following: What is the relationship between the level of math
courses a student takes in middle school and their academic growth in middle school?
To answer this first research question, an ANOVA and regression analysis were used.
The choice for this model was based on that there was one explanatory variable used (Hoffmann,
2010). The explanatory variable was the number of advanced level math courses were completed
in middle school. The outcome variable will be student’s academic growth while in middle
school. The choice for explanatory and outcome variables is based on the current literature,
researcher’s conceptual framework, and the context of this study. Much of the current literature
has focused on the track students are placed in and their academic achievement. Because
achievement at SIS is relatively high to begin with, the researcher focused on growth during the
first three years students were tracked into different courses.
The Null and Alternative Hypotheses will be as follows:
Ho: There is no relationship between course placement in middle school and academic
growth
H1: There is a relationship between course placement in middle school and academic
growth
The sample for the primary analysis will include the entire population of student from the
previously stated six cohorts. Students at SIS tend to significantly outperform the general
population of students that take the MAP test, therefore I will begin by calculating each students’
Conditional Growth Index (CGI) (or Z-score) as follows in order to more effectively compare
growth within this high achieving sample:
(Student's Actual RIT Gain)-(SIS average RIT Gain)
Within SIS Standard Deviation of Growth

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Next, index variables were based on the number of years each participant was enrolled in an
advanced course.
Table 2
Index Variables for Regression Analysis
Value Definition
0 No years of advanced math courses in middle school (comparison group)
1 one year of advanced math courses in middle school
2 two years of advanced math courses in middle school
3 three years of advanced math courses in middle school

With the explanatory and outcome variables defined, a regression analysis was run using
STATA, and significance will be determined by interpret p-values and goodness-of-fit statistics.
As part of a secondary analysis for the first research question, the process outlined
previously was repeated using a sub-sample of the student population. The expected growth of
students on the MAP test is based on several factors, one of which is their previous year’s score.
Students that score at very high levels have a smaller expected growth. Even though the MAP is
an adaptive test, there is the possibility the highest achieving students may “max out” the
assessment, thus limiting growth. To account for this, as part of the supplementary analysis, the
lowest and highest achieving students on their Grade 5 MAP scores based on course placement
were removed to determine if it affected the results.
The sub-sample was comprised of students that scored between 237 and 246 on their
MAP. More detail is provided in the results section on how these values were chosen. Within this
band of scores, some students were placed in Math 6 while others were placed on Math 6+.
Presumably, these decisions were based on prior academic achievement in their elementary
school classes, learning behaviors, and each teacher’s professional judgement. The goal was to
compare students that have relatively similar prior academic achievement, comparing their
60

growth during middle school, to see if the track in which they took part in may have affected
their growth. The analysis for the first research question will conclude with a discussion of how
the results compared to what the researcher expected to find based on his conceptual framework
and the current research on tracking.
Analysis of Research Question 2

Research Question 2 asked the following: What is the relationship between the level of math
courses a student takes in middle school and the number of higher-level math courses taken in
high school?
To answer the second research question, only the first three cohorts of students were
considered. Because of the time frame of when this study was carried out, these are the only
students that had completed high school when the data was analyzed. Any students from these
three cohorts that left prior to graduation were excluded. Each student’s high school academic
records were used to determine how many higher-level math courses they completed by the time
they graduated. Higher-level math courses were defined as any Advanced Placement (AP) or
Advanced Topic (AT) course offered at SIS. Advanced Topic courses are higher-level courses
that have been developed within SIS in order to further challenge students. A total list of
advanced courses on offer are: AP Calculus AB, AP Calculus BC, AP Statistics, AT
Multivariable Calculus, AT Linear Algebra, AT Post-Euclidean Geometry, AT Finite Math
Modeling, and AT Data Analytics. Advance Topic courses are semester long, while AP courses
last the entire academic year. The last analysis considered only students who completed AP
Calculus, as access to this course seems to be of key concern for many students and parents
because of its perceived importance in gaining admittance to many competitive universities that
students from SIS are vying to attend.
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To answer the second research question, an ANOVA and binary logistic regression
model were used. Here, a regular regression model similar to what was used for research
question one would not be appropriate because the outcome variable is non-continuous
(Hoffmann, 2010). The sample that was used to analyze this research question were students
from the study’s population that attended SIS from grades five through grade 12. Data from
participants’ high school transcripts were used, specifically the number of higher-level math
courses taken in high school. The explanatory variable for this analysis will be the number of
higher-level math classes taken in middle school, the same coding as research question one will
be used. The outcome variable was the number of higher-level math courses taken in grades nine
through 12.

Table 3
Outcome Variables for Academic Outcomes
Value Definition
0 Did not take any higher-level math classes in Grades 9 through 12 (comparison
group)
1 Took one higher level math classes in Grades 9 through 12
2 Took two higher level math classes in Grades 9 through 12
3 Took three higher level math classes in Grades 9 through 12

The choice for explanatory and outcome variables is based largely on the context of this
study. The pathways that students have available to them at SIS are structured in such a way that
course placement in Grade 9 likely plays a role in how many higher-level courses a student has
access to. The purpose of this study is therefore to determine to what degree is this true.
The Null and Alternative Hypothesis will be as follows:
Ho: There is no relationship between course placement in middle school and the number
of higher-level math courses they take in high school.
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H1: There is a relationship between course placement in middle school and the number of
higher-level math courses they take in high school.
Similar to the analysis for research question one, a secondary analysis was carried out
with a sub-sample, removing the highest and lowest achieving students based on prior
achievement in grade five. Similarly, the purpose will be to look at a more homogenous group of
students and track their eventual outcomes in high school, seeing to what extent the track they
were placed in Middle School may be associated with these outcomes. The analysis for research
question two will conclude with a discussion of how the results compared to what the researcher
expected to find based on his conceptual framework and the current research on tracking.
A final analysis was carried out that only looks at whether students complete AP
Calculus, either the AB or BC course. The reasoning for this separate analysis is that, within the
context of this study, taking AP Calculus seems to be of increased importance for parents and
students in the SIS community. This may be because of a perceived need to score well in AP
Calculus to gain admittance to more prestigious universities. The analysis was the same as for
the second researcher question with an adjusted outcome variable of completion of calculus.
Table 4
Outcome Variable for Calculus Completion
Value Definition
0 Did not take AP Calculus (AB or BC) in Grades 9 through 12 (comparison group)
1 Took AP Calculus (AB or BC) in Grades 9 through 12

The Null and Alternative Hypothesis will be as follows:
Ho: There is no relationship between course placement in middle school and if a student
takes calculus in high school
H1: There is a relationship between course placement in middle school and if a student
takes calculus in high school
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Summary
The researcher’s focus for this study centered on a comparison between students in two
different math tracks, looking at their academic growth as measured by annual standardized
testing, and outcomes as measured by the classes these students ultimately completed while in
high school. Recent research has predominantly favored de-tracking math curriculums, but there
remains debate about which students benefit from tracking, and which may be harmed. The
researcher used a quantitative approach in attempting to compare the current homogenous
grouping that is used at SIS, a private international school, in Southeast Asia. The results will be
presented in the following chapter.

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Chapter Four: Results
The purpose of this study was to examine the potential relationships between tracking
students in middle school mathematics, academic growth, and student outcomes within the
context of a private, international school in SE Asia. This chapter will be divided into two main
sections associated with the two research questions presented in this study. What is the
relationship between the level of math course a student takes in middle school and their
academic growth from Grade 6 through Grade 8? What is the relationship between the level of
math course assigned to a student in middle school and the higher-level math classes completed
by graduation?
To answer the first research question, standardized test data from grade six to grade eight
was collected from six cohorts of students that attended SIS from the end of grade five through
the end of grade eight. This data was analyzed by comparing academic growth between groups
of students based on the number of years they took part in SIS’s advanced math courses,
anywhere from zero to three years. To minimize the impact that student’s prior knowledge may
have on the results, the process used to analyze the data was repeated, but after removing the
highest and lowest achieving grade five students. The goal of this was to look at a more
homogenous group of students.
Results Research Question 1, Academic Growth
The data gathered for the first research question of this study appeared to support current
research that a relationship exists between academic achievement and tracking. This analysis will
be broken up into two sections. The first section will consider all 753 students from the six
different cohorts. Second, in order to minimize the potential impact of students’ prior knowledge
65

on their academic growth, the highest and lowest achieving students will be excluded, and the
analysis of this more homogenous sample will be examined.
Academic Growth for All Students
In this initial analysis, data from 753 students was examined, a summary of their growth
in middle school is shown in Table 5. The tool used to measure academic achievement was the
MAP standardized assessment. Academic growth was measured by finding the difference
between each student’s score at the beginning of grade six and their score at the end of grade
eight. Students were grouped based on the number of years they took part in the advanced math
courses offered at SIS. The average growth for students in each of these groups was calculated,
as well as the average Conditional Growth Index (CGI). The CGI was included to norm
reference students within SIS, and calculated as follows:

(Student's Actual RIT Gain) - (SIS average RIT Gain)
Within SIS Standard Deviation of Growth

Table 5
Academic Growth of Middle School Students
Years in
advanced math
course
n Percent Average
growth
Conditional
growth index
(CGI)
0 237 31.5% 19.0 -0.5
1 63 8.4% 24.4 0.2
2 59 7.8% 26.6 0.4
3 394 52.3% 24.5 0.2

The most significant difference was seen in students that completed at least one year of
advanced math in middle school versus students that remained on the lower track for all three
years. It should be noted that the vast majority of students who moved between tracks during the
time of this study were from the lower track to the higher track. 122 students changed tracks
during middle school. Only four out of these 122 moved from the more advanced course offering
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to the lower. Therefore, of the 516 students with at least one year of advanced math, 512 finished
middle school in the advanced grade eight course. The growth for these two groups is
summarized in Table 6.
Table 6
Growth for Students Taking at least One Year of Advanced Math
Years in advanced math n Average growth
0 237 18.8
1-3 516 24.7

The MAP test was administered to students near the beginning of the school year in the
fall, then again near the end of the school year in the spring. The differences in academic growth
increased each year while in middle school, summarized in Table 7.

Table 7
Annual Growth in MAP Scores by Grade in Regular and Advanced Math Courses
Grade
Regular
fall MAP
scores
Regular
spring
MAP
scores
Growth
Advanced
fall MAP
scores
Advanced
spring
MAP
scores
Growth
Difference
in growth
6 225.3 231.7 6.3 243.1 249.6 6.5 0.2
7 230.6 237.6 7.1 249.9 257.4 7.5 0.4
8 235.9 242.6 6.6 257.8 264.9 7.2 0.6

The results in Table 7 show that each year students in the advanced course, they slightly
out gain their classmates that remain in the regular course. While these differences may seem
minor, it illustrates what other studies have found to be the cumulative effect of tracking, that
over time, students placed in a more rigorous mathematical track will out gain their peers that are
taking a lower-level course, and that this causes a difference in academic achievement that
widens over time (Mooney et al., 2021; Oakes, 1980; Oakes, 1987). Also worth noting is that the
growth shown in the regular class may be slightly skewed by the 118 students in this study that
moved up a level at the end of grade six or grade seven. In order to make this transition, students
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are given additional work and learning resources outside of the regular curriculum to complete. It
seems reasonable to assume that these students may, on average, out gain their classmates that
are not doing this extra work. Last, student scores in the advance courses improved over the
summer by 0.35 point on average, while student scores in the regular courses decreased by 1.05
points over this same time span.
Last, to gauge the impact each year of participation in the advanced math track had on
academic growth, an ANOV A was performed. The independent variable used was the number of
years they were enrolled in an advanced math class in middle school. The dependent variable is
the academic growth in middle school. The appropriateness of using an ANOV A to evaluate
these variables depended on the student’s academic growth being normally distributed, which is
illustrated in Figure 4.
Figure 4
Academic Growth in Middle School

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Once normality was established, an ANOVA was run two times. The first, broke the
explanatory variables into two groups, those that took at least one year of advanced math versus
those students that did not. The results are shown in Table 8. The effect of one year of advanced
math enrollment was significant, F(1,752) = 89.88, p < .001. This suggests that students taking at
least one year of advanced math was a significant factor in their academic growth.

Table 8
Total Growth of Students with At least One Year of Advanced Math
Sum of squares df Mean squares F p
Between groups 5649.154 1 5649.154 89.875 < 0.001
Within groups 47204.830 751 62.856
Totals 52853.984 752

Next, a second ANOVA was run to determine to what degree the total number of years a
student participated in advanced math had on academic growth. The results are shown in Table
8. The effect of number of advanced math courses was again significant, F(3,752) = 31.32, p <
.001. This suggests the total number of years was significant when determining academic
growth. The study did provide a surprising result, that more advanced math did not always lead
to greater gains. This was seen in Table 5 and once again in Table 10. In both cases, student with
two years of advanced math showed the greatest gains. This will be explored in greater detail in
chapter 5.

Table 9
Total Growth Based on Number of Years of Advanced Math
Sum of squares df Mean squares F p
Between groups 5890.551 3 1963.517 31.315 < 0.001
Within groups 46963.433 749 62.702
Totals 52853.984 752

Finally, a regression analysis was run using the total number of years in advanced math to
determine to what extent the years in advanced math had on academic growth. Students that did
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not take any advanced math courses in middle school, Years in Advanced Math = 0, was used as
the reference category. The model was statistically significant F(3, 752) = 31.32, p < .001
suggesting that the number of advanced math classes predicts academic growth. This suggests
the total number of years in advanced math is significant. The adjusted R Square value of 0.108
tells us that 10.8% of the variability in academic growth can be attributed to the number of years
in advanced math, suggesting there are other variables likely affecting student growth that are
not included in the current model.

Table 10
Regression Analysis for Number of Years of Advanced Math
Independent Variable Slope
Std.
error t-ratio Prob.
Years in Adv. Math = 1 5.586 0.514 4.976 < 0.001
Years in Adv. Math = 2 7.800 1.122 6.770 < 0.001
Years in Adv. Math = 3 5.663 1.152 8.700 < 0.001
Constant 18.827 0.514 36.603 < 0.001

R
2
= 0.111
Adjusted R
2
= 0.108

F Change = 31.315 p <
0.001

n = 752

Academic Growth Excluding the Highest and Lowest Achieving Students
Many factors contribute to a student’s academic achievement. A limitation stated in many
studies around tracking is having difficulty measuring the impact tracking has on students versus
a host of other variables. The biggest difference in the population of students being analyzed in
this study was their prior knowledge as they entered middle school, which may affect their
potential growth from grades six through eight. To minimize this impact when answering the
first research question, the highest and lowest achieving students were excluded prior to
analyzing the data a second time. The method for identifying the subset of students to be used
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began by looking at the spread of grade five MAP scores of students that were placed into both
the regular and advanced course offerings in grade six. As seen on box-and-whisker plots in
Figure 5, there was significant overlap when comparing the MAP scores of students that were
placed into each course. This is consistent with the school’s placement policy, which states that a
multitude of factors are considered when placing students, including classroom grades and in-
class behaviors.
Figure 5
Range of Grade Five MAP Scores Versus Grade Six Math Placement

The sub-sample of students chosen for analysis was based on grade five MAP scores,
specifically between 237 to 246, the median score for students placed into the regular math class,
Math 6, to the median score for students placed in the advance math class, Math 6+, respectively.
Again, the purpose for analyzing this subset of students was to remove students that either may
be the most likely to have benefited from outside tutoring, and also the lowest achieving students
that may have specific learning needs that could hinder their growth in middle school. This
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sample consisted of 232 students, 80 of which were placed into Math 6 and 152 of which were
placed into the Math 6+. A summary of their growth is shown in Table 11.

Table 11
Academic Growth for Students Academically in the Middle
Years in
advanced math
course
n Percent Average
growth
Conditional
growth index
(CGI)
0 33 14.2% 19.1 -0.5
1 22 9.5% 22.6 0.0
2 31 13.4% 27.8 0.6
3 146 62.9% 23.6 0.1

Comparing these values to those in Table 5, the most obvious similarity is between the
students that did not take any advance math courses throughout middle school, with only a 0.1
difference in their average growth from grade six to grade eight (19.0 v. 19.1). The other groups
of students that took between one and three years in advance math showed similar academic
growth as well, with our sub-sample slightly underperforming the overall population for one year
of advance math, (22.6 v. 24.4), outperforming when taking two years of advanced math, (27.8 v.
26.6), and underperforming when taking three years of advanced math, (23.6 v. 24.5).
Table 11 shows what happens when we group students based on whether or not they took
any advanced math. Similar to the larger data set, students that took at least one year of advanced
math outgained students that did not. Unlike the growth when considering all students (see Table
4), this sample did not show a consistent trend when annual growth was compared (Table 13).

Table 12
Growth for Middle Achieving Students Taking At least One Year of Advanced Math
Years in Advanced Math n Average growth
0 33 19.1
1-3 199 24.1

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When looking at the sub-sample in Table 13, with the highest and lowest achieving
students removed, there was an even greater disparity in academic growth between the students
in the advanced course versus students in the regular course in grade six and grade seven.
However, in grade eight, this trend reversed itself with the regular students out gaining advanced
students by 0.5 points on average.

Table 13
Annual Growth by Middle Achieving Students
Grade
Regular
fall
Regular
spring
Growth
Advanced
fall
Advanced
spring
Growth
Difference
in growth
6 232.1 238.0 5.9 237.9 244.8 6.9 1.0
7 236.2 242.6 6.5 244.9 252.3 7.5 1.0
8 243.3 251.3 8.0 253.5 260.9 7.5 -0.5

Another ANOVA was carried out to determine if exposure to advanced math in middle
school still had a significant impact on academic growth using the sub-sample of students. First,
normality was established for our sub-sample to ensure the appropriateness of using an ANOVA,
as illustrated in Figure 6. Once it was shown that academic growth was normally distributed, an
ANOVA was run for the groups of students that had at least one year of advanced math (Table
13), then using the students grouped by total years of advanced math (Table 14).

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Figure 6
Academic Growth for Middle Achieving Students

The effect of one year of advanced math was significant, F(1,231) = 11.92, p < .001,
Table 14 and the effect of number of year of advanced math was also significant, F(3,231) =
6.91, p < .001, Table 15. Both ANOVA’s showed that exposure to advanced math was a
significant and positive variable in academic growth with p-values less than 0.001.

Table 14
Total Growth of Middle Achieving Students with At least One Year of Advanced Math
Sum of squares df Mean squares F p
Between groups 729.060 1 729.060 11.918 < 0.001
Within groups 14069.215 230 61.171
Totals 14798.276 231

Table 15
Total Growth for Middle Achieving Students Based on Number of Years of Advanced Math
Sum of squares df Mean squares F p
Between groups 1232.544 3 410.848 6.905 < 0.001
Within groups 13565.732 228 59.499
Totals 14798.276 231

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Lastly, a regression was run to determine the impact the total number of years in
advanced math had on academic growth. Once again, students that did not take any advanced
math courses in middle school, Years in Adv. Math = 0, was used as the reference category. The
results are summarized in Table 16.

Table 16
Regression Analysis, Number of Years of Advanced Math for Middle Achieving Students
Independent Variable Slope
Std.
error t-ratio Prob.
Years in Adv. Math = 1 3.576 2.123 1.684 .094
Years in Adv. Math = 2 8.714 1.929 4.516 < 0.001
Years in Adv. Math = 3 4.528 1.487 .003 .003
Constant 19.061 1.343 14.195 < 0.001

R
2
= 0.083
Adjusted R
2
= 0.071

F Change = 6.905 p < 0.001

n = 231

Similar results to the larger population were found. The model was statistically significant F(3,
231) = 6.905, p < 0.001 suggesting that the total number of years in advanced math is significant.
The adjusted R Square value of 0.071 tells us that 7.1% of the variability in academic growth can
be attributed to the number of years in advanced math, suggesting there are other variables likely
affecting student growth that are not included in the current model.
Results Research Question 2, Academic Outcomes
The second research question of this study focused on academic outcomes, specifically
the possible relationship between the level of math course assigned to a student in middle school
and the higher-level math classes completed by graduation? A subset of student data was used
from the analysis of research question one, which looked at six cohorts of SIS students that were
enrolled from Grade 5 through Grade 8. The first three cohorts were chosen to use for the second
75

research question. The reason for this choice of cohorts was that they were the only three classes
that had graduated from SIS at the time of this study, therefore allowing for an effective
comparison of the subgroups. Data from students that did not graduate from SIS were removed
from analysis. The total population that was used for this part of the analysis was 247 students.
The data was first organized by grouping students based on the number of years of
advanced math they completed during middle school, the total number of higher-level math
courses completed in high school, and specifically how many completed an AP Calculus course.
For this study, a higher-level math course will be defined as any AP or AT course offered at SIS.
As noted in Figure 2, SIS has a clearly defined flow chart that certainly affects which
courses students have access to based on where they enter high school at, either exiting middle
school from Math 8, the regular level math course, versus Math 8+, the higher level math course.
As different outcomes are to be expected, the goal is to measure the level of impact course
placement in middle school plays in eventual outcomes. With an emphasis placed on initial
placement in Grade 6.
The most common paths for students entering high school from Math 8 is taking Algebra
1, Geometry, Algebra II/Trigonometry, and finally Introduction to Pre-Calculus and Statistics.
For students entering high school from Math 8+, the most common path is Geometry, Algebra
II/Trigonometry, Precalculus, and finally one of the AP Calculus courses on offer. There are
several variations and other variables that affect what courses a student may choose or be
allowed to take. Sometimes students may “double-up” on math one year, complete a summer
course that allows them to skip a class, or show their understanding of the learning objectives in
a course by testing out, thus allowing them to skip to the next class entirely. Various classes,
particularly the higher-level courses, also have pre-requisites that students must meet to be
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allowed to enrol. These pre-requisites can be pre-assessments in which they show their
understanding of needed background knowledge, other classes they must have already
completed, and often minimum grade requirements they must have met.
To measure the impact of course placement in middle school, two statistical tests were
used. First, an ANOVA was carried out to determine if there was a significant relationship
between the number of higher-level courses a student completed in high school based on the
number of years in which they took part in an advanced math class in middle school. Second, a
binary logistic regression was used. The independent variable was which math class the student
was placed in Grade 6, and the dependent variable was whether those students ultimately
completed AP Calculus. Calculus has intentionally been chosen because of the importance
placed on it by community members and university admission offices alike. Similar to research
question one, this analysis was repeated once more for a narrower band of students. This was
based on their Grade 5 MAP results in order to measure the impact that initial placement in
either the regular or advanced tracks had on outcomes for students that enter middle school with
similar prior knowledge.
Academic Outcomes for All Students
First will be an analysis of all the student data from our first three cohorts of students.
Table 17 summarizes the average number of high-level courses each group of students
completed in high school, and who eventually took calculus. The ANOVA to evaluate the impact
of total years of advance math courses in middle school on the number of high-level course in
high school is shown in Table 18.

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Table 17
High School Academic Outcomes
Years of advance
middle school
math
Totals
Average number of
higher level courses
completed
AP calculus Percent who completed
AP calculus
Yes No
0 80 0.3 14 66 17.5%
1 13 1.0 6 7 46.2%
2 19 1.4 14 5 73.7%
3 135 1.7 112 23 83.0%

Table 18
Higher Level Courses
Sum of squares df Mean squares F p
Between groups 103.846 3 34.615 59.091 < 0.001
Within groups 142.348 243 .586
Totals 246.194 246

The dependent variable was the number of years of higher-level math courses students
completed in high school, while the independent variable was the years of advanced math
courses in middle school. The independent variable was shown to have a significant positive
impact on the number of higher-level courses complete in high school, F(3,246) = 59.09, p <
.001.
Next, a binary logistic regression was carried out to look specifically at the likelihood of
students completing either AP Calculus AB or BC. The reason for using this method is that our
dependent variable, completion of calculus, is a categorical variable with only two possible
outcomes. For this reason, an ANOVA would not be a suitable choice for analysis. A binary
logistic regression model estimates the probability that some binary variable, in this case,
completion of calculus, takes on a value of one rather than zero (Hoffmann, 2010). The reference
or outcome variable in this case is students that were initially assigned to Math 6. The results are
summarized in Table 19.
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Table 19
Logistic Regression Completion of Calculus
Independent Variable Slope
Std.
error Prob. Exp (B)
Math 6 (yes or no) 2.343 0.343 < 0.001 10.410
Constant -0.0834 0.208 < 0.001 0.434

n = 231

The Expected (B) value indicates that, holding all other predictor variables constant, in
this population a student enrolled in Math 6+ is 10.410 times more likely to complete a calculus
course while in high school B = 2.34, SE = .343, 95% CI [5.40, 18.88], β = 10.41, p = .001.
Academic Outcomes Excluding the Highest and Lowest Achieving Students
Similar to the analysis of the first research question, the highest and lowest achieving
students were excluded prior to analyzing the data a second time. The purpose of looking at this
sub-sample was to analyze a more homogenous group of students. This was to help determine
the impact tracking had on academic outcomes of students with similar prior knowledge. The
method for identifying the sub-sample of students used was the same as for research question
one. In Figure 7, overlap can be seen when comparing MAP scores of students that were placed
into each course. In the final part of our analysis, we will repeat the ANOVA and binary logistic
regression with this sample, n = 107. A different band of scores was used from our analysis of
research question one because the median scores for this sum-sample changed.

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Figure 7
Range of Grade Five MAP Scores Versus Grade Six Math Placement

Using this sub-sample, a second ANOVA was run, with years enrolled in an advanced
math class as our independent variable and number of higher-level math classes completed in
high school our dependent variable. Similar to our larger population, the years a student took
advanced math in middle school proved to have a significant impact on the number of higher-
level math classes they took, F(1,106) = 85.98, p < .001.

Table 20
Higher Level Courses Excluding Highest and Lowest Achievers
Sum of squares df Mean squares F p
Regression 10.098 1 10.098 85.978 < 0.001
Residual 12.332 105 .117
Total 22.430 106

Last, a second logistic regression was carried out with our sub-sample. The results are
shown in Table 21. The p-value rose to 0.021 but remained significant. In addition, with this sub-
sample of students, whom are more homogenous in terms of their prior knowledge, the
Expected(B) value dropped from 10.410 to 2.794. This showed that Math 6+ students within this
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sub-sample are still almost three times more likely to eventually complete a calculus course
while in high school, B = 1.03, SE = .446, β = 2.79, p = .21.

Table 21
Logistic Regression Completion of Calculus Excluding Highest and Lowest Achievers
Independent Variable Slope
Std.
error Prob. Exp (B)
Math 6 (yes or no) 1.028 0.446 0.21 2.794
Constant 0.182 0.350 0.602 1.2

n = 231

Summary
Overall, the results supported prior research that suggests students placed into advance
math tracks benefit academically, compared to their peers whom were placed into lower-level
courses. Those benefits include greater academic growth in middle school and their long-term
academic outcomes. In chapter five, these results will be explored in greater depth, and an
analysis of possible reasons for the differences seen here will be discussed.

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Chapter Five: Discussion
This chapter starts by summarizing the results of the study, followed by a deeper
discussion of each research question. It concludes by examining the implications of this research,
limitations of the study, recommendations for future researchers, and ends with a brief
conclusion.
Findings
The efficacy of tracking students in mathematics has been debated extensively, without a
clear consensus emerging as to its value. The purpose of this study was twofold, aligning with
the two research questions. First was to examine the relationship between the level of math
course a student took in middle school and their academic growth from grade six through grade
eight, focusing on the immediate impact tracking had on growth. This was measured through
looking at the change in standardized test scores in a school that tracks its students into two
levels of mathematics, beginning in Grade 6. The student’s growth on their MAP scores between
grade six and grade eight was used as the outcome variable for this part of the study. Second,
was to see if the level of math course assigned to a student in middle school affected the higher-
level math classes they completed by graduation. This question focused on longer-term
outcomes, to see if there was a measurable difference between students that were initially placed
on an advanced track, versus those that were not. The number of higher-level courses taken,
particularly completion of AP Calculus, were used as outcome variables for the second research
question.
One reason for the debate around tracking is the number of variables that contribute to a
student’s academic achievement. The context of this study helped to minimize many of these
variables. A student’s socio-economic background has been shown to have a significant impact
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on academic achievement. This has often been tied to their race (Guiton & Oakes, 1995; Moses
& Cobb, 2002; Oakes, 1980; Oakes, 2005; Wells, 2018). A limitation of many studies around
tracking is a lack of clarity if differences in achievement should be attributed to. Some of these
variables include a student’s racial background, their family’s socio-economic status, the quality
of the school(s) they attend, or what track they were placed in. These variables do not occur in
isolation, with many researchers finding that race and socio-economic status often influence
course placement and school quality. Because of the high fees charged by the private
international school in which this study was carried out, it can be assumed that all students come
from a similarly high socio-economic background. There are certainly different racial groups
represented at this international school, but it would be difficult to identify a single group of
students that are historically marginalized in the context of this community.
Another variable that is commonly stated as a potential cause of unequal growth is the
practice of teacher tracking (Fitzpatrick & Mustillo, 2020; Guiton & Oakes, 1995; Mooney et al.,
2021; NCTM, 2020; Oakes, 1980). This is when seniority is used to assign more experienced
teachers to teach advanced math classes, leaving less experienced teachers to work with students
that need the most help, usually in lower-level classes. When teaching assignments are allocated
in this way, it can contribute to a difference in the quality of instruction. This is often stated as a
limiting factor in studies. When unequal growth occurs between groups of homogeneously
grouped students, it is difficult to determine how much of an influence that the practice of
tracking had, versus a difference in the quality of instruction. Once again, this potentially
confounding variable has been limited to the context of this study. At SIS, each grade level in
middle school has three teachers, all of which teach students in both the regular and advanced
classes. Singapore International School has an excellent reputation and attracts teaching
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candidates that are highly sought after and experienced. Because of these reasons, one could
reasonably assume that students in both tracks in the middle school are receiving a similar level
of quality instruction. However, this doesn’t mean that all teachers have equally high
expectations for both classes, another factor that has been shown to impact academic
achievement.
Through this lens, the relatively narrow context of this study can be looked at as a
strength as well as limiting factor. While it may limit the generalizability of the results, it could
also provide a clearer picture that the impact tracking has on growth, as opposed to another
confounding variable.
Discussion of Research Question One, Academic Growth
The results from this study were consistent with other research that found students
benefitted academically when placed in a more advanced math course (Burris et al., 2006; Ehlers
& Schwager, 2020; Fitzpatrick & Mustillo, 2020; Mooney et al., 2021). Average growth was
greater for students that took at least one advanced math course in middle school in both our total
population, (24.7 v. 18.8) from table 2, and in the sub-sample of students with the highest and
lowest achievers removed, (24.1 v. 19.1) from Table 8. Somewhat surprisingly, was that it
seemed to be less important how many years a student was enrolled in an advanced math course.
There wasn’t any significant difference between academic growth in middle school for students
that took part in the more advanced track for one, two, or all three years. All made similar gains.
One explanation for this may include the additional work students are expected to complete
preparing for optional, annual placement exams, that were given at the end of each year to
determine each student’s readiness to move into a higher-level course the following academic
year. It may also be because of the group of students willing to put in this extra, optional work is
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showing that they are inherently more self-motivated. If that is the case, one would expect to see
greater growth from this group of students. In Table 1 and Table 7 we see the largest growth
occurring in students with two years of advance math experience, or those that moved from Math
6 to Math 7+ between their grade 6 and 7 years. Because of a difference in pacing, the
curriculum diverges more and more each year, thus moving to a higher track becomes more
difficult each subsequent year. The biggest difference in growth occurs when students are
grouped whether they had completed any advanced courses, illustrated in Table 2 and Table 8.
Students that remain in the lower track for all three years of middle school are not exposed to the
same breadth or depth of content as their peers, consequently experiencing less academic growth.
Looking at the structures in place at SIS and the design of the MAP test, the greatest
growth should be the opposite of what was seen. Students at SIS in the lower-level class are
often supported by a learning specialist teacher. The role of this second teacher varies.
Sometimes they are used as a co-teacher in the regular classroom, other times they act as
intervention specialists, pulling out small groups of students either to reteach or pre-teach
concepts. They work with students to fill in gaps in their prior knowledge that might limit a
student’s ability to access the content of the course. Besides more teacher support from learning
specialists, the regular level classes are usually kept smaller to help students get more one-on-
one attention. Last is how the MAP actually measures annual growth. Based on the students’
previous years’ score, the MAP has an algorithm that projects the expected growth for the
following year. Lower-scoring students receive higher predictions in terms of overall growth.
This is because of the adaptive nature of this assessment. As a student gets more correct
responses, the questions become more difficult. These factors suggest that students in the lower
track at SIS should out gain their peers on the higher track. They receive more teacher support
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because of smaller class sizes and a higher teacher to student ratio. Despite all this, the data
indicates the opposite with these cohorts of middle school math students., leading the researcher
to believe that tracking may play a role in this discrepancy.
The ANOVA’s performed on both the population and sub-sample of students with the
highest and lowest achievers removed, showed that course placement was a significant factor in
academic growth. In table 6, we see that each year a student participated in the advanced math
course, the average MAP score grows by 5.6, 7.8, and 5.7 respectively. We see a similar pattern
when looking at our sub-sample in table 12, with MAP score increases for each year of advance
level math of 3.6, 8.7, and 4.5 respectively. Each of these variables was found to be statistically
significant, except for students from the sub-sample with one year of advanced math completed.
The only major differences between the population and sub-sample analysis occurred
when comparing annual growth. When looking at all six cohorts of students, we saw a gradual
widening of growth each year (Table 4), supporting research that has found that tracking students
has a cumulative effect time (Dweck, 2016). When looking at our sub-sample (Table 10), the
difference in growth is even greater in grades six and seven. The difference in annual growth
grew from 0.2 and 0.4 in grades six and seven with our population to 1.0 and 1.0 respectively in
the sub-sample. One explanation for this is that the sub-sample was comprised of students closer
to the average, or students that demonstrated similar levels of academic achievement in
elementary school as measured by their grade five MAP scores. The result is that this sub-sample
had students scoring in the highest range of students placed into Math 6 and the lowest range of
scores for students placed into Math 6+. This supports the notion that students placed into the
more advanced class may have benefited from a peer effect (Ehlers & Schwager, 2020) and
overmatching (Fitzpatrick & Mustillo, 2020) as seen in previous studies.
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The one exception to the trend of advanced students out gaining students in the lower
track occurred within the analysis of our sub-sample in grade eight. For this one group, students
in the lower-level course grew on average of 8.0 points, as opposed to those in the advanced
course, which grew on average 7.5 points. One explanation could be attributed to the curriculum
covered in the advanced grade eight math course. This is a traditional Algebra 1 course, therefore
covering little in the way of geometry, probability, and statistics, all of which are assessed on the
MAP test. Whereas students in the regular grade eight math course are exposed to more diverse
topics. Why might the growth in grade eight be different when looking at our sub-sample as
opposed to the entire population? The highest achieving students have an opportunity to either
skip geometry in high school, or move into an even more accelerated track. To do so, they
demonstrate an understanding of the key concepts in this geometry course, excel academically in
grade eight, as well as taking a placement, or “skip” test, as it is referred to at SIS. If they meet
these requirements, they are provided access to this accelerated track. If these highest achieving
students are doing additional work outside of class on their own, or with the assistance of private
tutoring, they very well might have exposure to topics not covered in Algebra 1, thus perform
better on their Spring MAP test. This is only an untested hypothesis based on the researcher’s
knowledge of the curriculum at SIS. Further data and research would be necessary to confirm if
this is true or not.
Discussion of Research Question 2, Academic Outcomes
Prior research has showed tracking students in math may lead to different outcomes
because of differing expectations placed on them by their teachers (Boaler, 1997). The belief that
placing students on a lower track can negatively affect long-term academic outcomes is often
used as a powerful argument to de-track math programs (Werblow et al., 2013). The results from
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this study support previous research that has found evidence that students placed in lower tracks
in middle school have lesser academic outcomes in mathematics, at least when measured by the
higher level courses they complete in high school.
This was clear when grouping students based on the number of years of advanced middle
school math, and then comparing the average number of higher-level courses completed, Table
14. Unlike with academic growth, the number of years that students were a part of the advanced
math track in middle school seemed to influence how many higher-level math classes they
completed in high school. The average number of higher-level courses completed grew by 0.7,
0.4, and 0.3 for each additional year of advance middle school math competed, respectively. The
ANOVA performed, Table 15, supported the claim that taking at least one year of advance math
was a significant factor in higher-level courses completed. The current pathways that exist at
SIS, see figure 2, certainly play a role in these findings. Students that finish middle school with
at least one year of advanced math are almost exclusively entering high school from Math 8+
(the equivalent of Algebra 1). This allows access to the full range of high school math offerings,
whereas a student entering from Math 8 must either take two math courses in one year, or
complete a summer course to access courses like AP Calculus.
One of the most notable findings of this study was the discrepancy in students who
completed an AP Calculus course when looking only at their math placement at the end of grade
five into Math 6 or Math 6+. Singapore International School provides students opportunities to
move up a level while in middle school, but it requires additional work. Even using the narrowest
sub-sample of students, with the most similar prior knowledge, a student placed to Math 6+ was
almost three times as likely to eventually complete calculus by the time they finish high school,
Table 17. When looking at the entire population of this study, with the highest and lowest
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achieving students included, this discrepancy grows. When looking at all students from the six
cohorts, a student initially place in Math 6+ was over ten times more likely to complete AP
Calculus, Table 15.
While tracking may play a part in this, it cannot be said with certainty tracking alone is
the cause. The structures and pathways currently set-up within the institution are also certainly
playing a role in these results. For this reason, this study was not designed to prove causality
between these variables.
Limitations
As previously discussed, a key limitation to this study is the generalizability of the
findings. The context in which the data was gathered was narrow. Singapore International School
serves almost exclusively students from families with a high socioeconomic status. High tuition
fees ensure the school is well-resourced; including class sizes that are capped at 24 students, as
well as learning support teachers that can push-in to support students in and outside of math
classes. Compared to public schools in the United States and throughout the world, teachers at
SIS are comparatively well-compensated, both in salary, and time to collaboratively plan and
prepare for classes. Care should be taken when trying to generalize findings to schools that serve
students from a more diverse socio-economic background, and may not be as well resourced.
A threat to the internal validity of this study is not being able to factor in the role that
private tutoring may play in the findings. Private tutoring is certainly part of the culture at SIS,
and Singapore. No exact figures were available, but various educators and administrators at the
school estimate that anywhere from one-third to one-half of students receive some sort of private
tutoring outside of the school. It is not clear if students on a particular track are more likely to
receive tutoring or not, which makes it difficult to gauge the potential impact on academic
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growth and outcomes. A result of elementary students receiving private tutoring may be that
these students demonstrate the highest achievement in Grade 5, and subsequently move into the
advanced math track in middle school. As shown, students that took part in the advanced track
improve academically at a greater rate than students placed in the regular track, but one cannot
be sure if this growth is because of their placement in advanced math, or if just a greater
percentage of these higher achieving students are receiving extra support, thus contributing to
their accelerated growth.
Academic achievement is only one area that tracking may affect. Another major factor is
the potential impact it has on student’s mindset, something this study did not consider. The
research on the impact that tracking has on student mindset is similarly two-sided. What most
educators and researchers agree upon is that mindset almost certainly plays a role in
achievement, that students with higher self-efficacy and self-belief are more likely to grow
academically. What is less clear is how tracking impacts mindset. Some studies have indicated
the act of tracking, thus labeling students, harms mindset and math identity, leading to lesser
gains academically (Boaler, 2013; Boaler, 2016; Dweck, 2016; Mooney et al., 2021). While
other studies demonstrated that tracking has a neutral impact on mindset, or sometimes may
improve it because students are not constantly comparing themselves to high-achieving peers
(Dai & Rinn, 2008; Marsh et al., 2008; Reed et al., 2015). This researcher at least partially
reconciles these differences by considering the various contexts in which the studies were
conducted. By not considering the relationship between tracking and mindset in this study, it can
only hope to paint a partial picture.
The final limitation is how academic growth and outcomes were measured. Using
standardized test data, while useful and convenient as a point of comparison, is only one data
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point. Using achievement data from classes, student interviews, or observations could also
unearth useful information. When looking at academic outcomes, only completion of higher-
level courses was considered, not the level of success students had while in those courses. The
current pathways that exist in the high school, as well as existing pre-requisite requirements,
certainly contribute to the higher-level courses students have access to. Because of this, it is
difficult to isolate the impact tracking has on these outcomes.
Implications for Practice
Because of the scope and design of this study, it is difficult to provide a definitive
recommendation as to the efficacy of tracking. Similar to countless other studies, the sheer
number of variables influencing student achievement cause causal interpretations difficult.
However, through this study, as well as a review of the existing literature, this researcher has
concluded that within the context of SIS, tracking students beginning in grade six occurs at too
early of an age. One argument against tracking is the subjective nature in which students are
placed (Ansalone, 2010; Oakes, 2005). At SIS, a student’s MAP score is only one measure that is
used to determine their readiness to enter Math 6+. Also factored in are student’s in-class
behaviors, achievement on internal, school-based assessments, and perceived academic potential.
This is done to reduce the stress caused by high stakes testing, as well as acknowledge the
variety of ways in which a student show their leaning and potential. The downside of a system
such as this is that teachers can make mistakes, and often have no way of knowing definitively
about a student’s academic potential years into the future. The structures currently in place allow
some movement between tracks, but to switch tracks, it requires a self-motivated and mature
student and informed family.
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Another key finding of this study arose when looking at the number of students that did
in fact change tracks, when they did it, and to where. If we accept teachers are fallible in their
ability to place students, it would be reasonable to assume that sometimes students would be
placed either in too low of a course or too high of a course, somewhat proportionally. That over
the years, some students would choose to move up a level and that some would choose to move
down. During the time this study, only 122 out of 753 students, or approximately 16%, switched
levels. Of these 122 students, 118, or approximately 97% moved up a level. This shows that
students are initially much more likely to be mistakenly placed on a less advanced track than the
other way around.
The causes for mistakenly being placed on a lower track are many, and worthy of their
own study. At SIS, student achievement data is much higher than national norms. Therefore,
some teachers may be norm-referencing their students to their peers, as opposed to the adopted
standards of the curriculum. Historical trends may also contribute to teacher perception. If
traditionally 50% to 60% of students are placed in Math 6+, as at SIS, teachers may be
uncomfortable recommending more than this amount, even if some students seem like they may
be ready for a more rigorous curriculum and faster pacing. It should be noted that under the
current structure at SIS, elementary school generalist teachers are being asked to be make
placement decisions. While undoubtably well-intentioned, these grade five teachers have a wide
degree of subject knowledge of mathematics, and they themselves may not understand the full
impact of their decision on long-term student outcomes.
Last, part of the challenge of placing students at such an early age can be evaluating
where they are at developmentally. Middle school is a time of rapid growth, and for most
students, a time to mature as individuals and learners. A student that seems to lack the needed
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motivation or executive functioning skills for Math 6+ might very well be ready for that
additional challenge a year or two in the future. The data, however, show that less than one in
five students attempt to move up a level. Each year, moving becomes more difficult as the gap in
content knowledge widens. By the time a student and/or their teacher realizes they might benefit
from the additional challenge a more rigorous course would offer, the work required to catch up
may seem to be too daunting to some.
If students continue to be tracked going into middle school, there may be some ways to
mitigate the long-term effects of their initial placement. First, teachers involved in placement
need to be educated about the impact it has on access to higher-level courses in the future. Great
care needs to be made to not norm-reference the students against one another, instead measuring
them against the school adopted standards. Likewise, parents and students need to be made
aware of the implications of how placement can affect their learning, and be given more voice
and power in where students are placed. Last, structures that allow students to move up a level
need to be clearly articulated, communicated to students and families, and barriers need to be
minimized.
Beyond structures that allow more freedom of movement between tracks, some
fundamental changes may need to occur in the curriculum itself. This may mean restructuring the
scope and sequences of content throughout middle school, as well as the pacing. To more closely
align regular and advanced courses, the differences between each should center more on the
depth of the content covered, not necessarily the breadth. Different schools have been able to
accomplish this in a variety of ways.
A final implication of this research is a need to re-examine the pathways available to
students once they enter high-school. Currently, at SIS, the pre-requisite requirements for
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students to access higher-level math courses are rigid, both in terms of the classes needed, and
grades earned in those classes. A heavy burden is placed on students that decide once they reach
high school they want to take more advanced math courses if they are not already on this track
from middle school. This includes either doubling up on a math class, usually geometry and
either Algebra I or Algebra II, or taking a summer course. To do this successfully, numerous
students at SIS turn to outside tutoring, something not all are willing or able to do. This study
found that students that were initially placed into Math 6+ were over ten times as likely to
eventually take AP Calculus. This figure includes the students that moved up to Math 7+ or Math
8+ in middle school, which would have made AP Calculus much more accessible. If the
researcher instead compared Math 8 and Math 8+ students, the difference in likelihoods of
completing AP Calculus would have undoubtably been even greater.
Future Research
As discussed throughout this study, the amount of research done around the benefits and
drawbacks of tracking students, particularly in math, is extensive. Because of the complex ways
in which we measure academic success, and the sheer number of variables that impact how
students learn most effectively, this question is one which will undoubtably continue to be
studied. This researcher believes the key to looking at the efficacy of tracking centers on the
context in which it occurs. There are different forms of tracking. Some countries have national
education systems that track students into entirely different schools. This can also be done on a
smaller scale within districts that have magnet schools that attempt to draw students with specific
strengths. Then there is the within-school tracking that was the focus of this study. But even
when looking at tracking within a school, variations arise when this occurs and what form it
takes. Within these different contexts, how tracking affects students may differ.
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More research is needed to determine the potential impact on higher-ability math students
when placed in a heterogenous classroom. A large body of recent research has shown that lower-
ability and average students benefit when placed in a more rigorous class. A finding that was
reinforced in this study. However, if there is a move to detrack middle school math curriculums,
more attention needs to be paid to higher achieving students to study if there is any sort of
negative impact on their achievement.
Within SIS, there is a current move to detrack the middle school mathematics program, at
least until grade eight. This is beginning by allowing students and families an informed choice of
which math class they would like to take. The same measures will assess their readiness and
reported to families, but the ultimate decision will be left in their hands. If these changes
eventually lead to a de-tracking of the math curriculum, future research comparing the
achievement and outcomes from this study to similar measures once the curriculum is de-tracked
could provide valuable information about whether tracking is helping or harming student
learning. Similar longitudinal studies at various institutions have been conducted, and more are
needed. In recent years, many schools have made the moved to detrack math to varying degrees,
therefore there should be ample opportunities for these types of studies to continue.
Finally, there seems to be a need for more mixed method studies that look at impacts
tracking has on both academics and mindset, as these two things are intertwined with one other.
Most current research available seems to focus on one or the other, but not necessarily both. By
looking at individual student’s academic achievement, as well as gaining insight into their
mindset through surveys, interviews, and possibly observations could help to shed a brighter
light on how tracking is affecting them as learners.

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Conclusions
Whether to track students, in any subject, will remain a topic for debate and one in which
a simple answer may not exist. The reason that tracking students is so debatable is the logical
arguments that can be made for and against it. On the pro-tracking sides, educators and
researchers alike can point to the desire to differentiate for students, allowing students with a
particular strength opportunities to extend and challenge themselves. While providing students
that struggle more in that area, a space where they can receive extra support and work at a pace
that will enable to them to learn more deeply, hopefully understanding concepts that might
otherwise be rushed through. Proponents often argue that by tracking students, there are benefits
to student mindset, that by limiting comparisons to higher achieving students, it will allow lower
achieving student’s confidence to grow.
Detractors of tracking students have equally valid arguments. Academically, they argue
that by placing some students in a lower-level course; you are limiting their access to a more
rigorous curriculum. This may be because of a difference in the content covered, the instruction
they receive, who is assigned to teach these classes, or the expectations placed on them by their
teacher. Students may be negatively affected by their peers if there are different behavioral
expectations for lower-level classes. Rather than believing that mindset is improved for students
in lower-level classes because of a lack of comparisons to higher achieving students, many
researchers have found the act of labeling students by putting them in a particular course has
detrimental effects on their confidence and belief in their abilities to learn and do math.
It is important for researchers, educators, families, and students to keep in mind that the
purpose of this discussion is positive, to create a system that is equitable for all students. One
that challenges all students wherever they are at and supports them as needed. Just as in any
96

subject, some students will be naturally drawn to math, showing an aptitude for the subject and a
joy in discovering the mysteries that surround it. The same could be said for students that
develop a love for language arts, music, or science. However, how math education differs from
most of these other subjects is how schools often separate and label students at an early age. This
is unique to math, or at the very least, happens at a younger age and is far more common. If we
can develop great authors, artists, and scientists without tracking students, it begs the question of
if this type of structure is necessary to produce highly trained mathematicians as well.

97

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Abstract (if available)

Abstract The purpose of this study was to look at the relationship between tracking students in math into two different tracks beginning in grade six in a private, international school in Singapore and if there was an impact of course placement on (1) academic growth while in middle school, and (2) academic outcomes as measured by completion of higher-level courses by the end of high school. Students were grouped by the number of years they took advanced math while in grade six through eight, then the academic growth they made on a school administered standardized test was compared, as well as the number of high-level courses these students eventually completed by the time they graduated from high school. The analysis showed that course placement in middle school had a significant impact in both areas. With students placed on the advanced math track, showing greater growth and better academic outcomes. The conclusion of the researcher was that within the context of where this study was carried out, that delaying when students are tracked, and creating more flexible pathways for students, may very well lead to greater growth for students that currently have limited access to some of these more advanced courses.

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Creator Windust, Jason Scott (author)

Core Title Implications of a tracked mathematics curriculum in middle school

School Rossier School of Education

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Degree Conferral Date 2022-08

Publication Date 08/06/2022

Defense Date 05/03/2022

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