Re-Envisioning Mathematics Pathways to Expand Opportunities
i
RE-ENVISIONING
MATHEMATICS
PATHWAYS
TO EXPAND
OPPORTUNITIES
The Landscape of High School to Postsecondary Course Sequences
2022
Re-Envisioning Mathematics Pathways to Expand Opportunities
ii
Executive Summary _____________________________________________________________________________________________________ 1
Introduction ______________________________________________________________________________________________________________ 3
Reimagining Mathematics in Postsecondary Education __________________________________________________________ 3
Implications of Reimagining K–12 Mathematics ___________________________________________________________________ 4
Mathematics Standards and Instructional Materials __________________________________________________________________ 6
Mathematics Coursework ______________________________________________________________________________________________ 8
Background on State Data Collection and Submission __________________________________________________________ 10
Middle School Mathematics ___________________________________________________________________________________________ 12
Middle School Course Sequences ________________________________________________________________________________ 12
High School Mathematics _____________________________________________________________________________________________ 15
High School Course Sequences ___________________________________________________________________________________ 17
Mathematics Course Taking in 11th and 12th Grades ____________________________________________________________ 19
Types of Mathematics Courses in 11th and 12th Grades _________________________________________________________ 21
Recent Revisions to Mathematics Coursework Requirements __________________________________________________26
Facilitating Students’ Seamless Transitions to Postsecondary ______________________________________________________ 28
Dual Credit Coursework in High School ___________________________________________________________________________29
High School Mathematics Alignment with Higher Education Requirements ___________________________________ 30
Assessing Students’ Understanding of Mathematics ________________________________________________________________ 32
Mathematics Assessment(s) Used for Federal Accountability __________________________________________________ 32
When Do States Assess Students’ Mathematics Achievement? ________________________________________________ 33
Recent Revisions to Mathematics Assessment Requirements __________________________________________________ 33
Lessons from the Field:
Recommendations and Considerations for Reimagining K–12 Mathematics ______________________________________ 35
Deciding on the Mathematics Courses and Content Students Need Today ___________________________________36
Barriers to Reimagining Mathematics Pathways _________________________________________________________________39
Actionable Steps and Opportunities to Move Forward __________________________________________________________ 42
Conclusion ______________________________________________________________________________________________________________ 47
Appendix ________________________________________________________________________________________________________________48
Endnotes ________________________________________________________________________________________________________________ 49
References ______________________________________________________________________________________________________________50
Acknowledgments _____________________________________________________________________________________________________ 53
CONTENTS
Re-Envisioning Mathematics Pathways to Expand Opportunities
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For too many students, the misalignment of high school and postsecondary
mathematics requirements is an unnecessary barrier to reaching their academic and
career goals. Although the nature of careers has evolved over time, mathematics
curriculum and instruction have largely remained unchanged in response to the modern
landscape. Therefore, some states’ postsecondary and K–12 systems have begun to
adjust mathematics course sequences to better align to the variety of dierent elds of
study available to students.
For K–12 education, states are grappling with the question of which high school mathematics content all
students should have as a foundation and when students should transition to specic courses that will help
them specialize their mathematical knowledge and skills for particular elds of study or areas of interest.
Understanding the work that states have been doing in this area, what lessons can be learned from that
work, and ultimately how to improve systems to better prepare students for success in postsecondary
education and careers is prudent.
This report — co-developed by the Charles A. Dana Center, Student Achievement Partners, and Education
Strategy Group — includes analyses of states’ available middle and high school student course-taking
data to examine whether recent high school mathematics pathways reforms have inuenced students’
mathematics course enrollment. It also examines how students’ mathematics course-taking patterns
vary within and across states and how state policy levers such as graduation course requirements might
be inuencing students’ mathematics course-taking decisions. The report includes a discussion of recent
changes to states’ standards and policies for adopting instructional materials as well as updates on the
student assessment landscape in mathematics. It also provides lessons learned and guidance from the eld
as well as recommendations and considerations for strategies to center equity and incorporate data into
states’ mathematics pathways eorts. Finally, the report includes noteworthy state-specic highlights,
mathematics focus group insights, and key questions for additional research.
EXECUTIVE
SUMMARY
Re-Envisioning Mathematics Pathways to Expand Opportunities
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Key ndings from the report include:
■ Course-taking data is not easily accessible and/or available in most states. Eighteen states were able
to provide data on students’ progression through mathematics course sequences in middle school and
high school, as well as mathematics course enrollment data for 11th and 12th grades. The remaining
states cited various reasons for their inability to provide data, including sta capacity, state data
request laws, timing, and data formatting issues. Some states also reported that course enrollment
data is not collected at the state level.
■ In middle school, the percentage of students following a traditional (enrollment in 6th-, 7th-, and
8th-grade mathematics courses), accelerated (completion of Algebra I or higher in 8th grade), or other
course sequence varied widely across states. The traditional sequence was the most common. The
median across the states in each sequence in middle school was 64 percent traditional, 28 percent
accelerated, and 8 percent other. In the 11 states that provided data disaggregated by race/ethnicity
and other student demographics, White students made up a higher percentage in the accelerated
sequence than Black students and students experiencing poverty.
■ In high school, a median of 27 percent of students progressed through the traditional sequence
(completion of Algebra I in 9th grade, Geometry in 10th grade, Algebra II or an equivalent course
in 11th grade, and another mathematics course in 12th grade), 13 percent of students followed the
accelerated sequence (completion of Algebra I prior to high school, Geometry in 9th grade, Algebra
II or an equivalent course in 10th grade, an additional course in 11th grade, and a fth mathematics
course in 12th grade), and 56 percent of students were in other courses outside of these sequences.
1
■ In 27 states, students are required to take three mathematics courses in high school.
2
Seventeen
states and the District of Columbia require students to take four mathematics courses prior to
graduation. The vast majority of states allow students substantial exibility with a range of course
options that satisfy mathematics requirements.
■ States reported the most common 11th-grade course to be Algebra II or Integrated III; however, the
percentage of students taking these courses varied widely. The median across states was 49 percent.
The states with policies requiring students to take four mathematics courses in high school reported
the highest percentages of students taking mathematics in 12th grade. States’ data revealed students’
12th-grade course selections varied much more than students’ 11th-grade selections.
■ An examination of changes to states’ graduation requirements in mathematics over the past ve years
revealed more exibility and less specicity for students. In many states, students have to “opt
in” to taking a set of courses that meets the requirements for entering the large, public four-year
postsecondary institutions that serve most graduates from their high schools.
■ The state policy scan also revealed that most states have one measure of student prociency from a
state assessment administered at a single point during a student’s high school experience, and most
of these measures are not tied to specic courses.
■ State mathematics leaders identied dierent potential barriers to planning and implementing
mathematics pathways for students. Some internal systemic barriers that state leaders identied
were related to existing ideologies, structures, and capacity. Some external barriers were related to
social contexts, equitable access, and building postsecondary connections.
Re-Envisioning Mathematics Pathways to Expand Opportunities
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The mathematics needed to engage in modern society increasingly relies on data
analysis and quantitative reasoning. Yet, high school preparation for postsecondary
mathematics remains largely centered on a pathway to Calculus at the expense
of a wider array of more relevant (or useful) mathematics (Herriott & Dunbar,
2009).
3
Similarly, “traditional entry-level college mathematics programs fail to
serve students well because they comprise disconnected courses whose content is
misaligned to students’ career and life needs,” and the attrition rates are alarming
(Liston & Getz, 2019, p. 1). Ample evidence shows that changes to mathematics
pathways at the K–12 and postsecondary levels, both in terms of opportunities and
content, would benefit students.
REIMAGINING MATHEMATICS IN POSTSECONDARY
EDUCATION
In response to these challenges, postsecondary education institutions across the country are re-
evaluating the content of their credit-bearing mathematics courses. College Algebra, a course originally
intended to prepare students for Calculus, has been the dominant gateway mathematics course in
higher education. But that need is no longer relevant. In fact, at most institutions, fewer than 20 percent
of students in College Algebra are in programs that require a yearlong calculus sequence (Herriott &
Dunbar, 2009). Each year, only 50 percent of postsecondary students pass College Algebra, and fewer
than 10 percent of students who pass this course enroll in Calculus (Gordon, 2008). Furthermore, many
incoming postsecondary students are placed into at least one developmental mathematics course each
year. Of those students placed into developmental mathematics sequences, only 33 percent complete the
sequence, and only 20 percent complete a credit-bearing college mathematics course (Bailey et al., 2010).
These barriers to degree completion are unfortunate realities at both community colleges and four-year
institutions (Liston & Getz, 2019).
INTRODUCTION
Re-Envisioning Mathematics Pathways to Expand Opportunities
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To address the outdated requirements and equity barriers to degree completion, postsecondary institutions
in many states have overhauled developmental education sequences and implemented mathematics
pathways that are better aligned to the dierent elds of study available to students (Burdman et al.,
2018). By 2015, more than half of two-year colleges had redesigned their developmental course sequences
to provide students earlier access to college-level or credit-bearing courses. In addition to opening up
access to credit-bearing mathematics courses, postsecondary education institutions are implementing
corequisite models to support student success in those credit-bearing courses. Mounting evidence shows
that a large majority of students, including those referred to developmental mathematics, can succeed in
accelerated college-level mathematics courses at higher rates and in less time compared to students in
traditional developmental sequences (Bailey et al., 2010; California Acceleration Project, 2015; Complete
College America, 2016; Logue et al., 2016; Rutschow & Diamond, 2015; Sowers & Yamada, 2015; Tennessee
Board of Regents, 2016).
Additionally, higher education institutions have shifted away from College Algebra for all to mathematics
gateway courses related to students’ majors and intended career elds (Blair et al., 2018). Growing evidence
shows that when students engage with mathematics that is relevant to their programs of study, they are
more motivated and more likely to succeed (Rutschow & Diamond, 2015). Students who take content-
specic mathematics courses (e.g., social science statistics, quantitative reasoning mathematics-based
courses for Humanities majors) are more motivated, earn higher grades, and are less likely to fail the
course (Rutschow & Diamond, 2015).
IMPLICATIONS OF REIMAGINING K–12 MATHEMATICS
As postsecondary institutions across the country expand their oerings in credit-bearing mathematics
coursework and strive to remove other barriers, what do these changes mean for K–12 mathematics? The
reality is that the most common high school mathematics sequences in the United States — referred to as
the “geometry sandwich” by Steven Levitt — still consist of Algebra I, Geometry, and Algebra II, with the
goal of putting students on a path to Calculus (2019). The focus on Algebra II, Precalculus, and Calculus
persists despite research suggesting that fewer than ve percent of workers and a smaller percentage of
community college students actually use the mathematics from these later courses (National Center on
Education and the Economy, 2013). Additionally, research for this report shows that students traverse
through the “geometry sandwich” and courses after Algebra II in more varied sequences than this linear
progression suggests, but K–12 systems are designed with this assumption. According to the group
Just Equations, the emphasis on Algebra II in high school can be primarily attributed to admissions
requirements at colleges and universities rather than a need for students to master the content taught in
the course (Burdman, 2019). One of the driving incentives for updating high school mathematics is simply
that “our schools do not teach what their students need, while demanding of them what they don’t need”
(National Center on Education and the Economy, 2013).
Re-Envisioning Mathematics Pathways to Expand Opportunities
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One eort to increase mathematical opportunities for students and bridge the gap between higher
education and K–12 in mathematics is being led by The University of Texas at Austin’s Charles A. Dana
Center (Charles A. Dana Center, 2020). This initiative, called the Launch Years, is directly aimed at
improving mathematical learning opportunities for all students in high school and better aligning high
school mathematics with students’ postsecondary and career aspirations. Ultimately, Launch Years seeks
to dismantle systemic barriers that have disproportionately limited equitable access, particularly for
students who are Black, Hispanic, or experiencing poverty, to the high-quality and relevant mathematics
courses needed to succeed in today’s workforce and postsecondary education and training.
Access to clearly dened mathematics pathways is critical to ensure that all students and families have the
option and information to select the mathematics courses that best align with students’ college and career
plans. States are in varying stages of redesigning high school mathematics pathways and course oerings,
but there has yet to be an analysis of how state policy conditions and higher education requirements
inuence mathematics pathways implementation and students’ course selections. As eorts to revise
mathematics education in K–12 and higher education continue across the country, Education Strategy
Group (ESG), the Charles A. Dana Center, and Student Achievement Partners (SAP) sought to benchmark
how states are attending to related mathematics policies such as aligned standards and assessments,
graduation expectations, and postsecondary transitions.
Defining Mathematics Pathways
The term “pathways” has dierent meanings across states and parts of the education system.
For the purposes of this report, a mathematics pathway is a mathematics course or sequence
of courses that students take to meet the requirements of their program of study. Mathematics
pathways enable students to take dierent paths through the mathematics curriculum, making
the mathematics students learn relevant to their programs of study and careers.
This report includes analyses of states’ middle and high school student course-taking data that oer an
early look at whether high school mathematics pathways reforms have inuenced students’ mathematics
course enrollment. Lessons learned from focus groups with state mathematics leaders in various stages of
implementing mathematics pathways provide guidance from the eld. Finally, a set of recommendations
and considerations are included to provide states with strategies to center equity and incorporate data into
their mathematics pathways eorts.
Re-Envisioning Mathematics Pathways to Expand Opportunities
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To fully realize the goal of revising mathematics pathways for learners, education
leaders must think about the practical implementation of pathways and existing state
policy structures. State policies such as those guiding standards revisions and those
designed to support district adoptions of high-quality instructional materials are two
key related policies examined in this report.
A review of states’ mathematics standards found that nearly a third of states (16 states) last reviewed or
adopted their mathematics standards in 2012 or earlier. In 26 states, mathematics standards were last
reviewed/adopted between 2013 and 2020. Nine states reviewed their mathematics standards in 2021 or
are currently in the process of reviewing their standards. These states include Georgia, Idaho, Minnesota,
Oklahoma, Oregon, Rhode Island, Tennessee, Virginia, and Wisconsin.
As states review, revise, and adopt new mathematics standards, and as they think about how the standards
are organized into courses and pathways in middle and high school, districts and schools must review and
update the instructional materials that teachers use to guide students’ learning of the standards. High-
quality curricula are a key component for supporting student learning (Ed Reports, n.d.). In most states,
local education agencies (LEAs) decide which instructional materials will be used in their schools, but this
review found a range of state approaches to supporting LEAs’ decisions. On one end of the continuum, each
LEA independently researches and decides which curricular materials to purchase with no guidance from
the state education agency. On the other end of the continuum, states require LEAs to select textbooks
from a state-approved list. Between these two extremes are diering degrees of state and LEA choice and
responsibility. For example, Tennessee’s State Textbook and Instructional Materials Quality Commission
recommends an ocial list of textbooks and instructional materials, but LEAs may submit a waiver request
if they wish to use textbooks or instructional materials that are not on the approved list. Massachusetts
convenes panels of educators to review and rate evidence on the quality and alignment of specic curricular
materials and then publishes their ndings to support local decision-making processes (Massachusetts
MATHEMATICS
STANDARDS AND
INSTRUCTIONAL
MATERIALS
Re-Envisioning Mathematics Pathways to Expand Opportunities
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k
Department of Elementary and Secondary Education, n.d.). The state also provides incentives, including
statewide master service agreements for approved materials, that make these materials easier for districts
to procure. Finally, the Massachusetts Department of Elementary and Secondary Education collects
and publishes information about the curricular materials that are in use in districts and displays this
information on a map.
In October 2021, the Oregon State Board of Education adopted the
Oregon Mathematics Standards. Key revisions from the previously
adopted mathematics standards included adding a K–12 data reasoning
domain; merging measurement content with geometry content;
revising the K–12 domains to reect the learning pathways of Algebraic
Reasoning, Numeric Reasoning, Geometric Reasoning & Measurement,
and Data Reasoning; and identifying a core two-course requirement
in high school that aligns to the Oregon 2+1 course design. Additional
resources such as standards-level guidance documents, learning
progressions, and crosswalks to the previous standards have been
developed (Oregon Department of Education, n.d.).
Re-Envisioning Mathematics Pathways to Expand Opportunities
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MATHEMATICS
COURSEWORK
Every mathematics education system decides what content is foundational for all
learners and what content is better suited to prepare students with particular interests
(e.g., a path toward science, technology, engineering, and mathematics [STEM]
professions may diverge from others). The mathematics courses and course sequences
students take may look relatively similar in middle school, but they often diverge in
high school. They then may diverge even further in postsecondary education depending
on a student’s program of study and interest.
States are grappling with the pivotal question of if and when high school student course sequences should
branch in mathematics. That is, should students stop taking the core mathematics that all students need
as a foundation for higher-level mathematics and start taking courses that help them specialize their
mathematical knowledge and skills for a particular eld? And if so, when should they make that switch?
Some states have decided to branch after the second year and some after the third year. Some states have
decided to branch after Algebra II specically, citing concerns about district capacity to oer multiple
mathematics courses in the third and fourth year and concerns about equitable access to those courses.
States including Georgia and Washington have instead chosen to add more mathematics content to
Algebra II, such as modeling, statistics, and foundational concepts for data science. This approach to
modernize Algebra II is meant to better prepare students for any course they choose to take following the
third year of high school mathematics.
Upon learning content typically taught in Algebra II or an equivalent course, students have built the
mathematical foundation to be successful in Statistics, Quantitative Literacy, and Precalculus or similar
courses.
4
Rather than aiming for everyone to complete mathematics course sequences in preparation for
Calculus, students should instead have opportunities to engage in high-quality mathematics experiences
that better align to their desired program of study — or at least a broad category of programs. For example,
a student may not know their exact major but be most interested in the social sciences, criminal justice,
and psychology — all of which benet from courses in statistics. On the other hand, the natural sciences,
Re-Envisioning Mathematics Pathways to Expand Opportunities
9
mathematics, and engineering benet from courses on the path to Calculus. Given this changing landscape,
students deserve the opportunity to access the courses that best prepare them for success in postsecondary
requirements and to start on the mathematics pathway that best prepares them for their career aspirations
while still in high school.
KEY TAKEAWAY
Although consensus on the best approach to designing mathematics pathways or the
best way to branch course sequences is not universal, states should rst gain a deeper
understanding of the mathematical expectations of employers and postsecondary
institutions. With this understanding, states can assess the mathematics content of current
courses and sequences to ensure greater alignment with postsecondary education and
workforce requirements. Creating mathematics opportunities aligned with college and
career expectations can help counselors and families assist students with enrolling in
mathematics courses that are aligned with their college and career aspirations.
Some states have been engaged in implementing mathematics pathways in higher education for more than
10 years. Mathematics pathways in higher education aim to ensure that students take the mathematics
course(s) required for their program of study starting in their rst year of postsecondary education and
are able to more quickly access credit-bearing mathematics courses rather than trudging through long
sequences of developmental mathematics courses. The shift in higher education from most students,
regardless of their major, being required to take a gateway course in the path to Calculus — typically College
Algebra — to taking the course that best prepares them for the content of their program and ultimately
their career has implications for high school mathematics education as well.
Approximately 20 states have worked to better align K–12 mathematics, especially at the secondary level,
with changes in higher education. In Arkansas, for example, higher education state leaders worked with
faculty across the state to dene postsecondary mathematics pathways (Charles A. Dana Center, n.d.).
The Arkansas Division of Higher Education made recommendations for which programs should require
College Algebra and which should require Quantitative Literacy based on a survey of faculty across
disciplines. The changes in higher education prompted leaders from the Department of Elementary and
Secondary Education to create a Math Alignment Task Force. Through the work of state leaders and this
task force, the number of students taking Quantitative Literacy in high school has steadily increased over
the past three years. Currently, 18 percent of 12th graders and three percent of 11th graders in Arkansas
are enrolled in Quantitative Literacy. Arkansas leaders have also convened K–12 and higher education
faculty to improve the alignment of the high school Quantitative Literacy course to the college-level
course. Regional task forces of higher education and K–12 educators across the state worked to draft goals
and recommendations to expand the implementation of mathematics pathways in K–12 districts and will
implement the recommendations in the 2022–23 school year.
Re-Envisioning Mathematics Pathways to Expand Opportunities
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Regardless of postsecondary pursuits, students need access to high-quality mathematics courses and
instruction throughout their elementary, middle, and high school education. This research sought to reveal
how students are progressing through mathematics course sequences in middle and high school. It also
examined whether students are taking 11th- and 12th-grade courses aligned to postsecondary mathematics
pathways, whether enrollment in particular courses varies across dierent student demographic groups,
and what policies and practices might inuence students’ course-taking patterns. The goal of this research
eort was to understand how students’ course-taking patterns vary within and across states, how state
policy levers such as graduation course requirements might be inuencing students’ course taking, and
ultimately how to improve systems to best prepare students for success in postsecondary education and
their career aspirations.
BACKGROUND ON STATE DATA COLLECTION AND
SUBMISSION
For this report, states were asked to submit data on students’ progression through mathematics course
sequences in middle school and high school, as well as mathematics course enrollment data for 11th and
12th grades. The data specically focused on enrollment in courses and not students’ success. See the
Appendix for the data template sent to states and the methodology used in the data request process. The
template and research assumed that states have data systems with varying capacity to pull the requested
data and varying sta capacity to provide the data in the given timeframe. The years of the submitted data
vary from state to state and are based on the most recent high school cohort with the most comprehensive
data available.
Two categories of data were requested from states:
■ The rst category was course sequence data in middle school and high school. The goal was to gather
data on what percentage of students progressed through particular sequences of mathematics courses
and understand dierences within and across states.
■ The second category of data requested focused specically on the distribution of students across
11th- and 12th-grade mathematics courses. The goal was to learn whether students were taking
mathematics in these grades, which courses students were taking, and whether there were dierences
across states and across student groups.
Communication between the research team and various state agency sta began in December 2021 and
continued through April 2022 to gather the data, revise the data request to align with data system and sta
capacity, and learn why states were not able to provide the mathematics pathways data during this time
period. States fell into one of ve categories based on whether or not data is included in this report.
Re-Envisioning Mathematics Pathways to Expand Opportunities
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ALL DATA SUBMITTED: NINE STATES (Arkansas, California, District of Columbia, Georgia, Illinois, Indiana,
Oregon, Utah, and Virginia) submitted all requested data, including data disaggregated by student groups.
5
Few states had systems in place that allowed them to examine data related to cohort course-taking
sequences.
SOME DATA SUBMITTED: NINE STATES (Alabama, Colorado, Connecticut, Idaho, Nebraska, New Mexico,
Ohio, Texas, and West Virginia) submitted some of the requested data; their data included mathematics
course enrollment data for 11th and 12th grades and in some cases 8th grade.
6
Four of these states (Alabama,
Nebraska, Texas, and West Virginia) were able to provide data disaggregated by student group — race/
ethnicity, students with disabilities, English learners, and eligibility for the National School Lunch Program.
SUBMITTED RELATED DATA: SIX STATES (Iowa, North Carolina, South Carolina, Vermont, Washington, and
Wisconsin) provided related course-taking data (e.g., as part of a state public report) but not in the format
of the provided template that could be used for this analysis.
DATA NOT AVAILABLE: NINE STATES (Alaska, Kansas, Maine, Massachusetts, Michigan, Montana, Nevada,
North Dakota, and South Dakota) either did not collect the course enrollment data requested or were
unable to run the necessary reports to compile it in the way requested without excessive sta time devoted
to the data assembly.
STAFF CAPACITY, TIMEFRAME, OTHER: Data from the remaining 18 STATES (Arizona, Delaware, Florida,
Hawaii, Kentucky, Louisiana, Maryland, Minnesota, Mississippi, Missouri, New Hampshire, New Jersey,
New York, Oklahoma, Pennsylvania, Rhode Island, Tennessee, and Wyoming) is not included in this report,
but the data may be available through future requests. The most common reason given was sta capacity
devoted to other pressing priorities during the given timeframe — for example, a state legislative session
or federal data reporting requirements. Other reasons for not being included are state data request laws
that do not allow the state to fulll the data request, diculty in getting the request for data in the right
format to the right sta member to consider the request in the given timeframe, or no stated reason (which
could include that the requested data is not collected at the state level).
Figure 1: Number of States That Submitted Course-Taking Data
MT
IL
VT
HI
MD
DE
NJ
NH
MA
RI
CT
NC
DC
PR
CA
WA
OR
AK
NV
NM
AZ
UT
OK
MO
IA
NE
WY
IN
WI
MN
ND
SD
OH
PA
NY
LA
MS
GA
SC
TN
WV
VA
ME
MI
ID
AL
CO
TX
KS
FL
AR
KY
Re-Envisioning Mathematics Pathways to Expand Opportunities
12
The mathematics courses states expect students to complete as part of their middle
school experience are relatively standardized and stable across states. Federal law
requires students to be tested in mathematics at the end of 6th, 7th, and 8th grades.
Students’ middle school mathematics courses are important for positioning students
to complete secondary, and eventually postsecondary, mathematics pathways.
MIDDLE SCHOOL COURSE SEQUENCES
States were asked to provide data on students’ course sequences in 6th, 7th, and 8th grades. Researchers
wanted to learn more about which students were taking grade-level 6th-, 7th-, and 8th-grade mathematics
sequences and which students were taking something more or less accelerated. Researchers also wanted
to know whether certain student groups were overrepresented or underrepresented in particular course
sequences. Middle school course sequence data was broken into three categories — traditional, accelerated,
and other.
■ The traditional course sequence is defined as students taking 6th-, 7th-, and 8th-grade
mathematics courses in those grades.
■ Students who progressed through an accelerated sequence took Algebra I or above, including Geometry
or Algebra II, in 8th grade.
■ Other course sequences include sequences that students took that dier from the ones described in
traditional or accelerated (e.g., students repeated a course or courses, students took a mathematics
course under special education services).
MIDDLE SCHOOL
MATHEMATICS
Re-Envisioning Mathematics Pathways to Expand Opportunities
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Few states had systems in place that allowed them to provide data related to cohort course-sequencing
data, which requires following a cohort of students across several grades based on the mathematics
courses they took. Twelve states (Alabama, Arkansas, California, District of Columbia, Georgia, Illinois,
Indiana, Oregon, South Carolina, Texas, Utah, and Virginia) provided this data. In the case of the District
of Columbia and South Carolina, their data is included in this set based on 8th-grade course enrollment
rather than following a cohort from 6th through 8th grade.
States reported a wide range in the percentage of students following each sequence. The traditional
sequence was the most common, with anywhere from 49 percent to 84 percent of students following this
sequence across dierent states. States reported a range from three percent to 50 percent of students
following an accelerated middle school course sequence. The range in the percentage of students who
did not follow either of these sequences (“other”) varied from zero percent to 38 percent across states.
The median across the states in each sequence was 64 percent traditional, 28 percent accelerated, and
8 percent other.
Table 1: Percentage of Students in Each State That Progressed Through Middle School Mathematics
Course Sequences
COURSE SEQUENCE RANGE MEDIAN
Traditional Middle School 49%–84% 64%
Accelerated Middle School 3%–50% 28%
Other Middle School 0%–38% 8%
Notably, the states with the lowest percentages of students completing other course sequences were
Arkansas (less than one percent), Texas (less than one percent), and Virginia (less than three percent).
Virginia also reported the highest percentage of students in the accelerated sequence at 50 percent of
students. The percentage of 8th-grade students taking Algebra I or higher in middle school is one of the
performance measures for the current Virginia Board of Education Comprehensive Plan (Virginia Board
of Education, 2017). In Virginia, all middle schools are required to oer Algebra I. In addition, the state’s
Algebra Readiness Initiative provides funding to every school district to address readiness for Algebra I in
middle school (Virginia Department of Education, n.d.).
7
Re-Envisioning Mathematics Pathways to Expand Opportunities
14
Student Subgroup Results
In the 11 states (Alabama, Arkansas, California, District of Columbia, Georgia, Illinois, Indiana, Oregon,
Texas, Utah, and Virginia) that provided data disaggregated by race/ethnicity and other student
demographics in middle school, White students made up a relatively higher percentage in the accelerated
sequence. Black students and students experiencing poverty made up a lower percentage of the students
in the accelerated sequence as compared to the traditional or other sequences. Hispanic students in each
state most often made up a lower percentage of the students in the accelerated sequence. In Georgia,
Hispanic students made up a slightly higher percentage of the students in the accelerated sequence
(18 percent) than in the traditional sequence (17 percent). This data point is worth noting given that Georgia
is an outlier from other states and also should not diminish the urgency needed to address equitable access
to an accelerated pathway.
For Additional Research
Why are middle school students taking courses other than the 6th-, 7th-, and 8th-grade
mathematics sequence or something more challenging? How do middle school courses aect
high school and postsecondary courses? State-level data most certainly masks district and
school dierences in course-taking patterns. States able to generate this data should take a
closer look to determine how course-taking patterns vary by district, school, and student groups
within their states.
Re-Envisioning Mathematics Pathways to Expand Opportunities
15
Most states require three or four years of mathematics in high school. Students who
take Algebra I in 9th grade and who have learned the content that is commonly found
in Algebra I, Geometry, and Algebra II or an equivalent course are well positioned to
access college-level courses (if available) in high school. Furthermore, taking a third
year of mathematics that includes most of the content found in Algebra II in many
cases means that students are prepared to take a college-level Quantitative Literacy,
Statistics, and Precalculus courses. States should encourage and support districts to
provide students access to college-level courses (i.e., through dual credit, Advanced
Placement, or International Baccalaureate) in high school.
State policies for high school mathematics requirements dier in the number of graduation options
available to students, the number of courses required, and the content of those courses. In all states, the
state sets the graduation minimum; districts, schools, and students may supplement the state minimum
with additional coursework or experiences, though this exibility assumes that students know which
courses and experiences they need to be successful in preparing for their postsecondary goals (e.g., two- or
four-year college, technical training, apprenticeship, or the military) and that students are able to access
these courses and experiences. Students may face barriers in accessing rigorous coursework, especially if
the courses are not required by the state (U.S. Department of Education Oce of Civil Rights, 2018).
HIGH SCHOOL
MATHEMATICS
Re-Envisioning Mathematics Pathways to Expand Opportunities
16
C
j
To ensure that students have access to courses, Arkansas has a list of
38 required courses that must be oered in high school. For mathematics,
four of the courses schools must oer include Algebra I, Geometry,
Algebra II, and Precalculus. Schools must also oer two of the following
courses: Advanced Topics and Modeling in Mathematics, Algebra III
(Transitional), Calculus, Statistics, Quantitative Literacy (Transitional),
Transitional Math Ready (Transitional), and Technical Math for College
and Career. At least one Advanced Placement course and one
transitional course from this content area must also be oered (Arkansas
Division of Elementary and Secondary Education, 2021).
Number of Graduation Options
Some states oer a single option for graduation while others oer several diploma options. Nationally,
states oer more than 115 dierent high school graduation options for students, an increase from the 95
high school graduation options available for the class of 2015. These graduation options may take many
forms, including endorsements, seals, or distinct diplomas. High school graduates in 18 states had three
or more paths to graduation in 2022. Eleven states oered two paths to graduation, and 21 states and
the District of Columbia had one state-dened path to graduation in 2022. As states continue to oer
students more options, it is crucial that they take the necessary steps to ensure that these options lead
to quality postsecondary opportunities — whether at a two-year college, four-year college, technical
school, etc. The analysis that follows focuses on the graduation requirements that students — absent
any action on their part — are expected to complete. In other words, these are the “default” graduation
expectations for students.
Oklahoma defaults students into its College Ready/Work Ready
curriculum but allows students to “opt out” with parental consent into
the Core curriculum. The consent letter is transparent and direct: “The
Core curriculum does not meet college entrance requirements, nor
requirements for the Oklahoma’s Promise scholarship available to
students whose family income is $55,000 or less annually and who earn
a 2.5 GPA in the college preparatory/work ready curriculum.”
Re-Envisioning Mathematics Pathways to Expand Opportunities
17
KEY TAKEAWAY
States should communicate the benets and potential drawbacks of specic graduation
options. These choices, made in early high school and even in middle school, may aect
students’ college admissions and scholarship eligibility. Students and families need clear
communications about the ripple eects of decisions to choose one pathway over another.
States can make better policy and practice decisions — and ultimately improve student
outcomes to achieve equitable results across student groups — when they have information
about how students are doing. Understanding whether a student enrolls and succeeds in
specic high school course sequences is one key measure for informing adjustments along
the way to ensure that the student is ready for the next steps after graduation. Moreover, if
states are investing in reimagining mathematics pathways, they must understand the courses
students are actually enrolling and succeeding in.
HIGH SCHOOL COURSE SEQUENCES
For this analysis, states were asked to submit the percentages of students completing traditional,
accelerated, and other course sequences in high school.
■ The traditional high school sequence includes Algebra I in 9th grade, Geometry in 10th grade,
Algebra II or an equivalent course in 11th grade, and another mathematics course in 12th grade.
■ The accelerated high school sequence includes Algebra I prior to high school, Geometry in 9th
grade, Algebra II or an equivalent course in 10th grade, an additional course in 11th grade, and a fth
mathematics course in 12th grade. For both the traditional and accelerated sequences, Integrated I, II,
and III are treated as Algebra I, Geometry, and Algebra II or an equivalent course for categorization
purposes.
8
■ The other high school sequences include any sequence of courses that students took that dier from
the traditional and accelerated sequences. This category includes “super accelerated” students who
took Algebra II in 9th grade. Students who did not take mathematics in 12th grade are also part of
the other category. This report goes into greater depth in subsequent sections about whether or not
students took mathematics in 11th and 12th grade and provides analysis of which mathematics courses
students typically took.
Re-Envisioning Mathematics Pathways to Expand Opportunities
18
Based on data from nine states (Arkansas, California, District of Columbia, Georgia, Illinois, Indiana,
Oregon, Utah, and Virginia), a median of 27 percent of students progressed through the traditional
sequence, 13 percent followed the accelerated sequence, and 56 percent were in other courses outside of
these sequences. Of the states that were able to compile and submit this data for the report, the percentage
of students in each of the sequences varied widely. Note especially that states reported a range of zero
percent to 74 percent of students who took courses in a sequence outside of the traditional or accelerated
sequences as dened by this report.
In the data request, researchers attempted to further break down the percentage of students who are
captured in the table below in the other category. For example, researchers requested the percentage of
students who took the traditional high school and accelerated high school sequences through 11th grade
but then did not take mathematics in 12th grade. Unfortunately, this and other sequence descriptions that
would have provided more information as to the variation between states in the other sequence category
were either not possible to parse out in data systems or otherwise represented in ways that were not
comparable across states.
Table 2: Percentage of Students in Each State That Progressed Through High School Mathematics
Course Sequences
COURSE SEQUENCE RANGE MEDIAN
Traditional High School 13%–88% 27%
Accelerated High School 3%–29% 13%
Other High School 0%–74% 56%
Of the states
that submitted high school course sequence data, Georgia and Arkansas had the highest
percentage of students taking the combined traditional and accelerated sequences at 100 percent and 92
percent, respectively. In other words, these states had few students taking other course sequences. From a
policy perspective, one reason may be that both states require students to take four mathematics courses
in high school, and while students have some exibility in which courses they take, the state’s expectation
is that students will complete at least an Algebra I, Geometry, and Algebra II (or Mathematics I, II, and
III) sequence. From a practical perspective, Arkansas and Georgia have both built robust statewide course
code management systems that assist them in understanding which students are taking which courses.
Generally speaking, states reported that Black and Hispanic students and students who are eligible for
the National School Lunch Program made up a higher percentage of the traditional sequence than the
accelerated sequence. For White students, it was the reverse. Furthermore, the students eligible for the
National School Lunch Program in most cases had the biggest dierence in representation from traditional
to accelerated sequence.
Re-Envisioning Mathematics Pathways to Expand Opportunities
19
For Additional Research
An original assumption of this research eort was that the ndings would help make the case that
in states where students are required to take courses X, Y, and Z, students in fact take courses X, Y,
and Z. Or it would make the case that states with policies that allow for students to choose a third-
or fourth-year mathematics course that aligns with their postsecondary and career goals would
be able to provide data on which students were choosing which courses. Unfortunately, many
states’ data systems were either not able to follow students’ course-taking sequences or the lift
to generate this information was too onerous. Especially as many states have created policies
that permit students to substitute a litany of courses for mathematics courses, this information is
critical to ensuring that the policy is serving students’ best interests. The ndings show that states
are, for the most part, in a space of “trust but cannot verify.”
MATHEMATICS COURSE TAKING IN 11TH AND 12TH GRADES
Most states (27) require students to take three mathematics courses in high school.
9
Seventeen states and
the District of Columbia require students to take four mathematics courses prior to graduation. Three
states require students to take two mathematics courses, and three states do not specify how many courses
students are required to take — these decisions are set by individual districts. This research examined
whether these state policy dierences in the number of required courses would yield substantive dierences
in the percentages of students taking mathematics courses in their nal years of high school.
In the 13 states that submitted this data (Arkansas, California, District of Columbia, Georgia, Illinois,
Indiana, Nebraska, New Mexico, Oregon, Texas, Utah, Virginia, and West Virginia), a median of 90 percent
of students took a mathematics course in 11th grade, and 74 percent did so in 12th grade. The percentages
varied considerably among the states, as shown in Table 3 on p. 20. Georgia, Arkansas, Texas, and the
District of Columbia reported the highest percentages of students taking mathematics in 12th grade.
Georgia, Arkansas, and the District of Columbia require students to take four mathematics courses in high
school.
Re-Envisioning Mathematics Pathways to Expand Opportunities
20
Table 3: Percentage of 11th and 12th Graders Enrolled in a Mathematics Course
STATE 11TH GRADERS 12TH GRADERS
NUMBER OF
MATHEMATICS
COURSES REQUIRED
Arkansas 96%
87% 4
California 84% 63% 2
District of Columbia 90% 82% 4
Georgia 99% 98% 4
Illinois 96% 76% 3
Indiana 87% 64% 3
Nebraska 95% 72% 3
New Mexico 90% 74% 4
Oregon 90% 56% 3
Tex as 98% 85% 3
Utah 92% 59% 3
Virginia 90% 65% 3
West Virginia 82% 80% 4
Median 90% 74% 3
Note: States may oer students more than one graduation option with dierent numbers of required mathematics
courses. For purposes of this analysis, the “default” number of courses is included.
In some states, dozens of course options are available to students to satisfy graduation requirements. This
exibility could be benecial when well aligned with a student’s career interests and high-quality career
pathways, but it should not result in placing students into a lower (or less rigorous) track. States are not
serving students’ best interests by allowing them to graduate not having taken an appropriately rigorous
course of study or not having demonstrated that they are ready for their next steps. Further, navigating
the myriad options without strong student advising and counselor supports, particularly for students from
underserved populations, can be a challenge.
Looking exclusively at top-level mathematics course requirements (e.g., “three mathematics courses”)
is insucient because these three courses look very dierent across states and within states that allow
districts to decide. While states may “require” a set of mathematics courses for graduation, the vast
majority allow students exibility with a range of course options that satisfy mathematics requirements.
What qualies as a “mathematics” course depends on the state or the district in which a student resides.
For example, a state that requires Algebra II might also allow the requirement for the third course covering
Algebra II to be met by a mathematics course with content comparable to Algebra II or by a computer
science, career and technical education/vocational education, economics, science, or arts course as
Re-Envisioning Mathematics Pathways to Expand Opportunities
21
determined by the local school district governing board or charter school. In another state, the third-
year mathematics course may be replaced by Accounting, Mathematics of Personal Finance, Medical
Mathematics, Modern Mathematics, Introductory Statistics, or Computer Programming. In other words,
students have a tremendous amount of exibility. This review found that 21 states allow computer science
to be substituted for a mathematics course; 15 states permit career and technical education courses to
substitute for mathematics. At least seven states count nancial literacy/consumer mathematics courses
toward a student’s required mathematics coursework.
TYPES OF MATHEMATICS COURSES IN 11TH AND 12TH GRADES
States reported that postsecondary mathematics faculty and leaders are often surprised by the large
percentage of students that do not take a mathematics course in grade 12. Higher education leaders and
mathematics faculty in state and regional mathematics alignment task forces indicated that they see
continuous mathematics course taking through 12th grade as important given the high percentages of
students that enter postsecondary programs needing supplemental support to succeed in college-level
mathematics. More than two-thirds of community college students and 40 percent of students enrolled in
four-year universities take at least one developmental mathematics course (Ganga & Mazzariello, 2019).
The use of multiple measures for placement, including grade point average, and the use of corequisite
models for remedial support are gaining traction and improving students’ access to and success in
college mathematics for students who enter college underprepared (Complete College America, 2021;
Ganga & Mazzariello, 2019). Still, the goal of K–12 systems should be that students graduate from high
school prepared for success in credit-bearing mathematics courses immediately upon enrollment in
postsecondary education institutions.
States reported the most common 11th-grade course to be Algebra II or Integrated III; however, the
percentage of students taking these courses varied widely. In the 17 states that reported this data (Arkansas,
California, Colorado, Connecticut, District of Columbia, Georgia, Idaho, Illinois, Indiana, Nebraska, New
Mexico, Ohio, Oregon, Texas, Virginia, Washington, and West Virginia), between 21 percent and 82 percent
of students in each state took these courses, the vast majority being Algebra II or Integrated III, in 11th
grade of the year reported. Even in the state where just 21 percent of students took Algebra II in 11th grade,
this percentage was higher than any other course. The median across states was 49 percent.
Students’ 12th-grade course selections varied much more than their 11th-grade selections. Twelfth-grade
course-taking data in a comparable format includes data from 16 states and the District of Columbia.
Table 4 on p. 22 includes the median percentage of students taking these courses across the states.
Re-Envisioning Mathematics Pathways to Expand Opportunities
22
Table 4: Percentage of 12th Graders Enrolled in Selected Mathematics Courses
STATE
ALGEBRA II OR
INTEGRATED III PRECALCULUS STATISTICS
QUANTITATIVE
LITERACY/
REASONING
Arkansas 8% 6% 6% 18%
California 9% 8% 15% 1%
Colorado 16% 12% 14% n/a
Connecticut 6% 14% 25% n/a
District of Columbia 7% 12% 52% n/a
Georgia <1% 38% 21% 18%
Idaho 13% 6% 15% n/a
Illinois 7% 13% 27% 7%
Indiana 11% 26% 26% 9%
Nebraska 15% 13% 15% n/a
New Mexico 8% 9% 8% n/a
Ohio 30% 15% 21% 3%
Oregon 9% 6% 10% n/a
Tex as 17% 29% 13% 9%
Virginia 7% 6% 17% 21%
Washington 6% 8% 8% n/a
West Virginia 4% 8% 3% n/a
Median 8% 12% 15% 9%
Note: States oer a wide variety of mathematics courses in 12th grade. The courses included in this table were those
that had the highest enrollment and were comparable across states. Thus, state-specic courses such as 12th-grade
transition courses are not included. Students also may be taking more than one mathematics course and thus may be
captured in the data in more than one course. “Statistics” includes a high school-level statistics course, Statistics for dual
credit, and Advanced Placement Statistics. Additionally, Georgia’s reported enrollment data for Precalculus and Statis-
tics includes percentages of 11th and 12th graders.
Re-Envisioning Mathematics Pathways to Expand Opportunities
23
Notably, all states reporting course enrollment data for 12th graders had a relatively signicant number of
students enrolled in Statistics. The review of state statutes and approved course lists found references in at
least 17 states to Probability and Statistics courses.
Quantitative Literacy or Quantitative Reasoning is one of the commonly required gateway mathematics
courses for programs of study in higher education institutions in states that have dened their mathematics
pathways. States reported that a smaller percentage of students take Quantitative Literacy compared to
Statistics in high school. Eight of the 17 states providing 12th-grade enrollment data reported students
enrolled in Quantitative Literacy or Reasoning. In Arkansas, for example, the percentage of students across
the state taking Quantitative Literacy increased by 58 percent from the 2017–18 school year to the 2020–21
school year. The review of state statutes and approved course lists found references in at least eight states
to Quantitative Literacy or Reasoning courses that were approved as mathematics courses.
States working to reimagine mathematics pathways and encourage specic courses such as Statistics and
Quantitative Literacy or Reasoning need to have course-taking data to understand the extent that these
courses are being oered by districts and schools and taken by students. Given the tremendous variation in
mathematics course requirements, graduation pathways, and student exibility within and across states,
it is important for state leaders to know which students are completing which graduation options and
taking which sequences of courses and how their outcomes dier within and beyond high school. Absent
attention to students’ course-taking experiences in high school, states will be unable to pinpoint where
students and schools are successful — and replicate and scale those eorts as needed — and where policies
may need to be adjusted to better serve students.
NOTE: In future presentations of this data, it would be useful to present the data in a way
that mirrors the most common postsecondary pathways — the path to Calculus, Statistics,
and Quantitative Reasoning. Combining Precalculus, College Algebra for dual credit, and
Calculus, for example, would give a clearer picture of the percentage of students on that
mathematics pathway rather than just breaking the data down by course. The researchers
did not request data on Calculus but rather on courses that follow Algebra II or an
equivalent course for comparison.
Re-Envisioning Mathematics Pathways to Expand Opportunities
24
J
For Additional Research
Which districts and schools are oering new(er) courses such as Statistics and Quantitative
Literacy or Reasoning? Why are states oering dierent types of these courses? Why would
students take high school-level Statistics rather than a college-level dual credit Statistics course?
What purpose does each course serve, and how are students counseled into them? What
systemic policies, practices, and supports need to be in place to avoid mathematics pathways
becoming a new form of biased tracking? For states with clear mathematics pathways in higher
education institutions, are students being advised into Statistics or Quantitative Literacy or
Reasoning based on their well-informed program of study and career aspirations?
Georgia has created a number of policies that have resulted in strong
consistency in course-taking patterns through the 11th-grade year. For
example, Algebra II, now Advanced Algebra, is a required course for all
students. Eighty-two percent of 11th graders in Georgia took Advanced
Algebra. The course is also oered with corequisite supports for students
who need it. All other states reporting this data had between 21 percent
and 56 percent of students taking Algebra II in 11th grade. Additionally,
Georgia requires four years of mathematics and reported that 98 percent
of 12th graders took a mathematics course; the other states’ reported
data ranged from 56 percent to 87 percent. Because most students in
Georgia take Advanced Algebra by the end of 11th grade and because
four years of mathematics are required as part of the state’s graduation
requirements, students have the opportunity to take a mathematics
course in 12th grade that is college and career aligned. Georgia’s 12th-
grade course-taking patterns revealed that students were distributed
across the path to Calculus, Statistics, and Quantitative Literacy or
Reasoning courses. Some of the course-taking data includes both 11th
and 12th graders and some only 12th graders, so the data cannot be
directly compared. Still, there was clear distribution across the three
mathematics pathways with 18 percent of 12th graders in Quantitative
Literacy or Reasoning and 21 percent of 11th and 12th graders in
Statistics.
Re-Envisioning Mathematics Pathways to Expand Opportunities
25
r
Utah has an innovative approach to the Integrated pathway with two
secondary mathematics pathway options. Given the uniqueness of Utah’s
approach, it is not comparable to other states and thus not represented in
Table 4, but it is a model worthy of consideration. Students can choose to
take Integrated I, II, and III foundation courses or Integrated I, II, and II courses
plus extended topics. If students choose the extended option, they will
complete the content typically found in Algebra I, Geometry, Algebra II, and
Precalculus in a three-year period through the Integrated I, II, II “extended”
mathematics pathway. Of 12th graders in Utah, 49 percent took Integrated III,
which includes those in the foundation and extended paths. For an imperfect
comparison, in the other states that submitted 12th-grade course-taking
data, a median of 8 percent of students took Algebra II or Integrated III, and a
median of 12 percent of students across states took Precalculus.
NOTE: Iowa, Vermont, and Wisconsin could not be included in the report data tables because
the data provided was outside of the template. Key data shared by these states includes:
■ In Iowa, 76 percent of students took four years of mathematics, six percent of students
took less than three years of mathematics or contained an interruption, 18 percent of
students were accelerated to Algebra I or above in 8th grade, and 12 percent of students
took Calculus.
■ Vermont follows the course-taking patterns of other states that submitted data, with
Algebra I, Geometry, Algebra II, and Precalculus being the highest enrolled courses.
After those courses, the most common courses Vermont students took were Advanced
Placement Calculus, Advanced Placement Statistics, Statistics, and Consumer Math at
similar rates.
■ In Wisconsin, districts have the authority to choose which course codes to assign
to courses, making aggregating information across the state extremely dicult. For
example, districts have a wide variety of course codes even for the courses that most
students in Wisconsin take, such as Algebra I and Geometry. Trying to gather data for this
report prompted state leaders in Wisconsin to create a list of recommended codes for
mathematics courses so that they could compare data across districts. Other states with
local control governance in which districts make most decisions about courses to oer,
course codes, and instructional resources expressed similar challenges.
Re-Envisioning Mathematics Pathways to Expand Opportunities
26
RECENT REVISIONS TO MATHEMATICS COURSEWORK
REQUIREMENTS
For mathematics pathways to be eectively implemented in states, students will need to be able to access
and succeed in specic coursework. Students must understand which courses are appropriate for specic
pathways. One approach states can use to increase access is to ensure that the courses are approved
as mathematics courses for students. However, a review of trends in changes to states’ graduation
requirements in mathematics over the past ve years revealed more exibility and less specicity. The
most substantive changes are in states where students now complete two core years of mathematics
during the rst half of their high school experience (e.g., Algebra I and Geometry), followed by one or two
“personalized” courses. Kentucky, Oregon, South Dakota, Washington, and West Virginia have adopted
this model in recent years with Texas doing so in 2013. This approach is a departure from the more typical
model in which students are generally expected to take a commonly dened sequence of three or four
mathematics courses. Another common trend in states such as Kentucky, South Dakota, Washington, and
West Virginia is the removal of the requirement — or default expectation — that students complete an
Algebra II course or its equivalent to graduate. These states also are shifting to policies that allow students
more exibility around their options for their third and/or fourth mathematics course and, in some places,
creating policies for students to add “endorsements.” See Table 5 on p. 27 for examples of these changes.
Re-Envisioning Mathematics Pathways to Expand Opportunities
27
Table 5: Notable Changes to State Mathematics Graduation Course Requirements
STATE MATHEMATICS THEN … … AND MATHEMATICS NOW
Kentucky Students entering grade 9 in 2018–19 and
earlier (2022 graduates) are required to
take three credits: Algebra I, Geometry,
and Algebra II; a mathematics course or
equivalent must be taken each year.
Students entering grade 9 in 2019–20
and thereafter must take four credits:
Algebra I, Geometry, and two other
personalized credits covering the
remaining required Kentucky Academic
Standards for Mathematics.
South Dakota Students must take three units, which
must include one unit of Algebra I,
one unit of Algebra II, and one unit of
Geometry. With school and parent/
guardian approval, a student may be
excused from Algebra II or Geometry, but
not both, in favor of a more appropriate
course.
In 2018 the mathematics requirements
were revised to include three units of
mathematics, which must include one
unit of Algebra I.
Washington Students must take three credits, which
must include Algebra I or Integrated
Mathematics I, Geometry or Integrated
Mathematics II, and Algebra II or
Integrated Mathematics III. A student
may elect to pursue a third credit of
mathematics other than Algebra II or
Integrated Math III if the elective choice
is based on a career-oriented program
of study identied in the student’s
High School and Beyond Plan and the
student, parent or guardian, and a school
representative meet, discuss the plan,
and sign a form.
Beginning with the class of 2019,
students must take three credits, which
must include Algebra I or Integrated
Math I, Geometry or Integrated Math II,
and a third credit of mathematics that
aligns with the student’s interests and
High School and Beyond Plan, with the
agreement of the student’s parent or
guardian.
West Virginia Students must take four credits,
including Math I; Math II; Math III
STEM, Math III Liberal Arts, or Math
III Technical Readiness; and Math IV,
Math IV Technical Readiness, Transition
Mathematics for Seniors, or any other
fourth course option.
In July 2020 the mathematics
requirements were revised to include
four credits, including Math I or
Algebra I, Math II or Geometry, and
two additional personalized credits
from course options.
Re-Envisioning Mathematics Pathways to Expand Opportunities
28
FACILITATING
STUDENTS’
SEAMLESS
TRANSITIONS TO
POSTSECONDARY
A high school graduate’s path to and through postsecondary education is not always
linear. One student might graduate from high school, enter the workforce, and later
decide to return to higher education while working full time. Another student might
enlist in the military, serve for a number of years, and decide to pursue an associate
or bachelor’s degree. Another student might enroll full time in a two- or four-year
institution, scale back to part time, and then switch majors. Still other students might
accrue postsecondary credit in high school and look to build upon those credits at a
technical college. Because students’ experiences are so dierent, there must be on-
ramps and o-ramps to support students’ choices and provide exibility and support
where needed. And students must have access to the information they need to make the
best choices for their own career goals.
One of the goals of this research was to better understand how states are creating policies to support and
streamline students’ educational experiences, including through allowing dual credit course taking in high
school, administering high school assessments that signal to students their readiness for credit-bearing
postsecondary coursework, and expecting students to complete coursework in high school that aligns with
the coursework required for entrance into postsecondary education opportunities.
States oer a wide variety of mathematics courses that students can take to earn credits toward graduation
from high school. The list of courses, however, is not necessarily vetted to ensure that students will be
best prepared for a postsecondary credential, two-year degree, or four-year degree. Students also can
often take course sequences that include courses that are a lower level than a previous course they took, so
they are not advancing and expanding their mathematical knowledge and skills. Students may not even be
aware of dual credit course options that would better support them in their education goals.
Re-Envisioning Mathematics Pathways to Expand Opportunities
29
DUAL CREDIT COURSEWORK IN HIGH SCHOOL
States reported minimal course-taking data regarding the mathematics courses students take for dual
credit. Some states shared that the K–12 agency or department does not collect and/or have access to this
data. Other states shared that pulling data on specic courses for dual credit or just mathematics courses
was dicult. One key nding is that for states reporting student enrollment in dual credit, students most
frequently enrolled in College Algebra. In Georgia, of the 11th and 12th graders who took any mathematics
course, 10 percent took College Algebra for dual credit. The state also reported disaggregated data on
students enrolled in the following mathematics courses for dual credit: Precalculus (three percent),
Statistics (two percent), Calculus (one percent), and Quantitative Literacy or similar course (one percent).
While these percentages of students taking courses for dual credit are small, the data indicates that the
infrastructure is in place to oer a range of gateway college mathematics courses for dual credit in high
school, as well as to monitor and report data. Eight states provided data on the percentage of students
taking “all other mathematics courses that require Algebra II as a prerequisite course or have a prerequisite
requirement like a score on a standardized test.” States reporting this data had between three percent and
29 percent of 12th graders who took a mathematics course categorized this way.
KEY TAKEAWAY
The use of data and the ability of state systems to pull relevant data to guide dual enrollment
oerings and advisement on what dual enrollment courses align with postsecondary gateway
courses and requirements vary widely across states. This fact hinders the progress that can
be made toward increasing equitable access to and success in postsecondary-aligned
mathematics courses and pathways.
For Additional Research
Which dual credit mathematics courses are available to students? Are dual credit options
available to students in a range of courses, and do they align with students’ career interests? To
what extent are students exercising choice when electing to take a dual credit course in Calculus?
Re-Envisioning Mathematics Pathways to Expand Opportunities
30
HIGH SCHOOL MATHEMATICS ALIGNMENT WITH HIGHER
EDUCATION REQUIREMENTS
In many states, students have to “opt in” to taking a set of courses that meets the requirements for entering
the large, public four-year postsecondary institutions that serve most graduates from their high schools.
This approach means that the graduation expectations for many students are set lower than what is needed
for admission into four-year schools. Students may be left scrambling to make up coursework later in their
high school experience as this misalignment between state requirements and college entrance expectations
becomes more evident. Given other equity gaps, this gap between high school graduation requirements
and college admissions requirements undoubtedly disadvantages Black and Hispanic students and those
experiencing poverty. States should do more to ensure that all students are required to take courses in high
school that do not limit the opportunities available to them after they graduate.
This review found that states take a number of dierent approaches when it comes to the transition
between high school exit and postsecondary entrance requirements:
1. The state requires students to complete X, Y, and Z mathematics courses to graduate, and the higher
education system also requires X, Y, and Z mathematics courses to be considered for admissions.
2. The state requires students to complete X and Y mathematics courses to graduate but oers at least
one graduation endorsement/pathway that requires students to complete X, Y, and Z mathematics
courses. Students who elect to complete this more rigorous option will meet the higher education
system admissions requirements.
3. The state requires students to complete X and Y mathematics courses to graduate, but the higher
education system requires X, Y, and Z mathematics courses to be considered for admissions. Students
interested in pursuing postsecondary education at the higher education system are responsible for
understanding the gap between K–12 exit and higher education admissions requirements and address
it accordingly in high school.
4. In many states the exibility for the courses a student may take in K–12 and/or the lack of specicity
about students’ required coursework for admission to the higher education system make knowing
whether students are experiencing gaps extremely challenging. The K–12 system species courses
that students may take that count toward their mathematics or science courses for graduation, but
whether the higher education institution would count those same courses as meeting its admissions
requirements is unclear. A student who completes the specied K–12 coursework may fall short of
the coursework needed for entry into many or all institutions of higher education and also may nd
themselves unprepared because of the courses they did or did not take.
Re-Envisioning Mathematics Pathways to Expand Opportunities
31
X
a
KEY TAKEAWAY
More states now oer varying high school diploma options that have implications for
postsecondary opportunities for students. Therefore, states should be explicit in their
communications and advising that opting out of particular mathematics course sequences
or failing to complete particular courses may aect a student’s eligibility for certain
postsecondary institutions.
The Missouri Department of Elementary and Secondary Education’s
guidance for high school graduation requirements crosswalks the
requirements to graduate from high school with the standard diploma
and the additional courses/content that are necessary for students
to meet the Missouri Coordinating Board for Higher Education’s
Recommended High School Course Work (Missouri Department of
Elementary and Secondary Education, n.d.).
North Carolina has created charts detailing what courses are required for
admission into a University of North Carolina System institution, admission
into community college, or direct entry into a career after high school
(North Carolina Department of Public Instruction, n.d.).
Re-Envisioning Mathematics Pathways to Expand Opportunities
32
ASSESSING
STUDENTS’
UNDERSTANDING
OF MATHEMATICS
In addition to reviewing course-taking data, how can states understand the ways
students are moving through their K–12 mathematics courses, whether these students
are successfully meeting expectations, and what interventions and supports are
necessary to ensure their success? Given the variation in the mathematics courses
students take in high school, how dierent are the assessments states use to monitor
how students are progressing in their mathematics journeys? This report aims to
understand which assessments states administer for federal accountability, when
students are being assessed, which mathematics content is assessed, and whether
the assessments have consequences for students. The policy scan revealed that most
states have one data point on students from one assessment administered at a single
point during a student’s high school experience, and most assessments are not tied to
specic student coursework.
MATHEMATICS ASSESSMENT(S) USED FOR FEDERAL
ACCOUNTABILITY
The Every Student Succeeds Act requires states to assess students’ academic achievement in mathematics
at least once in high school for federal accountability, but it gives states the exibility to decide how often
students are assessed, at what grade(s) students are assessed, the content and type of assessment(s) used,
and where to set prociency benchmarks. One approach to categorizing states’ assessments is by end of
year (e.g., at the end of 11th grade) or end of course (i.e., tied specically to a course upon completion of
that course, regardless of the grade the student is in).
Re-Envisioning Mathematics Pathways to Expand Opportunities
33
This review of states’ high school assessments used for federal accountability revealed the following:
10
■ Twenty-one states administer the ACT or SAT assessments.
■ Three states administer the ACT Aspire or PSAT assessments.
■ Seventeen states administer state-developed end-of-year assessments. This number includes seven
states that administer the Smarter Balanced Assessment Consortium assessment.
■ Fifteen states administer end-of-course exams to students. Each of these 15 states administers an
Algebra I assessment. Eight of these states also administer assessments in Geometry and/or Algebra II
for accountability, though participation may not be required for all students. One state requires all
students to be assessed using an Algebra II end-of-course assessment. One state requires all students
be assessed with a Math III end-of-course assessment.
WHEN DO STATES ASSESS
STUDENTS’ MATHEMATICS
ACHIEVEMENT?
States most commonly assess students in high school
mathematics in grade 11 (28 states) and grade 10 (seven states).
Two states assess students in grade 9; two states assess
students in grades 9 and 10; and one state assesses students
in grades 9, 10, and 11.
11
The remaining states assess students
using an end-of-course assessment (or assessments).
RECENT REVISIONS TO MATHEMATICS ASSESSMENT
REQUIREMENTS
States’ assessment systems are not xed. The past three years have brought substantive changes to
some states’ assessments that meant changing types of systems, most commonly moving from end-
of-course exams to the use of college admissions tests: the ACT and/or SAT. Arizona, Indiana, and New
Mexico replaced their state-developed end-of-course or end-of-year exams with the SAT as the state’s
high school mathematics assessment. New Jersey has moved to replace the state’s longer end-of-
course exams with a single end-of-grade assessment in grade 11. Maine no longer administers the SAT
to students, opting to use the NWEA assessment for high school mathematics. Other states have made
Grade 9 Grade 10 Grade 11
5
10
29
Figure 2: When Do States Assess High
School Students?
Note: Categories not mutually exclusive.
Re-Envisioning Mathematics Pathways to Expand Opportunities
34
more minor but substantive changes, reducing the number of high school mathematics assessments
students take. For example, Georgia no longer administers statewide end-of-course assessments in
Analytic Geometry or Geometry.
Outside of what is required for federal accountability, at least four states have added national assessments
in the early part of students’ high school experience: Arizona (ACT Aspire), Hawaii (PreACT), North Carolina
(PreACT), and South Carolina (PSAT, PreACT, or ACT Aspire). Each of these four states also administers the
ACT/SAT to students in grade 11.
More states than ever before are administering the ACT and SAT assessments. In 2021–22, 31 states
administered the ACT or SAT to students (including a few that fund but do not require the assessment), an
increase from 26 states in 2018–19. This review further found that 14 states administered the ACT Aspire/
ACT or PSAT/SAT assessment suites in 2021–22, an increase from nine states in 2018–19.
Which States Have Stakes for Students Associated with
Assessments?
This review found that less than a third of states (15 states) place student stakes on high school mathematics
assessments — i.e., students must pass state assessment(s) to graduate and/or the assessment(s) are
factored into the course grade.
■ In 2021–22, nine states (Florida, Louisiana, Maryland, Massachusetts, Mississippi, New Jersey, New
York, Texas, and Virginia) required students to meet certain assessment thresholds, typically in
Algebra I, to graduate from high school. However, even among these nine states, some oer students
exibility in how they reach the minimum thresholds and provide alternatives to meeting specic
assessment thresholds. For example, a student might need to accumulate a certain number of points
across multiple subject-area tests, allowing them to compensate for a lower score in one content area
with a higher score in another.
■ Six states (Florida, Georgia, Louisiana, North Carolina, South Carolina, and Tennessee) factor
assessment results into a student’s nal grade in a course, most commonly in Algebra I. In these states,
the assessments comprise 15 percent to 30 percent of the student’s nal course grade. Two of these
states administer assessments that students must pass to graduate and that are also factored into
students’ course grades.
There has been a noteworthy shift in states removing assessment stakes for students. For example, New
Mexico allows students to use assessment performance as one way among many to satisfy graduation
requirements. Indiana, Pennsylvania, and Washington joined Ohio in creating policies whereby
assessments are only one of several ways for students to demonstrate readiness to graduate.
Re-Envisioning Mathematics Pathways to Expand Opportunities
35
In March 2022, SAP conducted a series of stakeholder engagement eorts to
understand the successes and challenges states have encountered throughout
their work to implement or attempt to implement high school to postsecondary
mathematics pathways. Furthermore, SAP also completed several interviews with
students and mathematics educators to gain insight into how they experience
mathematics education policies and practices within the classroom and while preparing
for postsecondary transitions. This next section consists of ndings from focus groups
with representatives from states at dierent stages of mathematics pathways work.
The ndings are accompanied by quotes that also raise awareness of the impacts of
particular decisions or barriers.
In total, individuals from 13 states representing various policy and political contexts participated in the
focus groups. Additionally, states were purposely sampled to represent one of the following two scenarios
to try to better understand the specic challenges that might arise at various stages of the planning and
implementation process:
■ Scenario 1: States are early in the work of implementing high school to postsecondary mathematics
pathways.
■ Scenario 2: States have made substantial progress in implementing high school to postsecondary
mathematics pathways.
LESSONS FROM
THE FIELD
RECOMMENDATIONS
AND CONSIDERATIONS
FOR REIMAGINING K–12
MATHEMATICS
Re-Envisioning Mathematics Pathways to Expand Opportunities
36
Figure 3: States Included in Participant Interviews
Throughout the focus groups, questions centered on:
■ Understanding the decisions that states made about which high school mathematics content was
non-negotiable in redening high school to postsecondary mathematics pathways;
■ The unexpected challenges that surfaced during the planning and implementation phases; and
■ The lessons learned that might benet other states that are considering launching work related to
reimagining mathematics pathways.
Additional attention was paid to issues that arose related to reimagining mathematics pathways through
an equity-oriented lens.
DECIDING ON THE MATHEMATICS COURSES AND CONTENT
STUDENTS NEED TODAY
Prior to deciding on a path toward implementation, states generally took a step back to investigate how
they might revise their high school mathematics standards and content in a manner that was dierent
from “business as usual.” State leaders that participated in the focus groups considered multiple career
pathways students might take after high school and the variations in specic content and mathematical
skills that might be needed for dierent groups of students. More specically, they asked themselves some
of the following questions:
■ Which mathematics is good for a particular pathway [after high school] versus which mathematics is
needed for all high school students regardless of their specic future goals?
HI
MD
PR
CA
WA
OR
UT
OK
IA
IN
OH
GA
VA
AL
Re-Envisioning Mathematics Pathways to Expand Opportunities
37
■ Should there be multiple versions of Algebra II, or should we reimagine and update Algebra II for all
students?
■ How much mathematics beyond grade 8 do students need?
■ How do we leverage the guidance and essential concepts in the National Council of Teachers of
Mathematics book Catalyzing Change in High School Mathematics: Initiating Critical Conversations to
support content selections and to help students discover the wonder, joy, and beauty of mathematics?
■ What would you be willing to withhold a high school diploma for? (This question was posed by one
state to get a content committee to be extremely selective in the choices they were making.)
These questions, and others, allowed states to shift their
mindset going into decisionmaking about mathematics
pathways and provide a model for how to start conversations
aimed at reimagining mathematics pathways at the high
school level. States described these questions as key to
reframing a conversation that would have otherwise been
limited to the more traditional ways of thinking about
mathematics courses and pathways. Additionally, such
questions led to a dierent set of non-negotiables in terms
of content and academic standards than what they currently
had in place because states were now thinking critically about
what mathematics content students might need across the
spectrum of postsecondary opportunities available to them.
Within a minimum requirement of three courses of mathematics, Oregon
chose to establish a two-course core for all students consisting of algebra,
geometry, and data science to be followed by a third course aligned with
students’ postsecondary goals.
Alabama decided to reduce the number of standards by about 10 per
course, emphasizing statistical reasoning and modeling in alignment with
the mathematical skills students need in today’s postsecondary landscape.
“ Start with encouraging
the mathematics students
need for their careers.”
— High School Mathematics Teacher
k
B
Re-Envisioning Mathematics Pathways to Expand Opportunities
38
With a focus on multiple postsecondary paths, states decided content such as quadratic functions and
polynomial long division might be less critical for all pathways and could be opted into, especially when
thinking about the “core” of mathematics that all students should have access to. However, other content
areas and courses were deemed non-negotiables for high school students. More specically, these non-
negotiables fell into two categories:
1. Content or courses that needed to be
added or expanded to better round out
mathematics pathways for students in today’s
technological world.
2. Content or courses that needed to be
reimagined to provide students with a
better foundation for a STEM postsecondary
pathway.
For the rst category, states overwhelmingly pointed to the incorporation of data literacy, data science,
and mathematical modeling content for high school students. Furthermore, states emphasized the need
to provide more opportunities for students to develop mathematical reasoning skills related to specic
content pieces. Georgia even connected this line of thinking to the broader K–12 mathematics pipeline,
emphasizing that some of these general concepts could be introduced as early as kindergarten to provide
the building blocks for students in later years. These shifts in which content or courses states prioritized
for students reect a substantial deviation from the historical trends in the content of high school
mathematics courses, taking a strong position on the need for a more updated approach to teaching and
learning in high school mathematics.
For the second category, states most noticeably pointed to reimagining Algebra II for high school students,
noting that “everyone deserves a modernized version of Algebra II” regardless of their postsecondary
pathway. Washington also noted that it found that the current version of Algebra II was not adequately
preparing its students for subsequent college courses. Key to the work in Washington is seeking to answer
the question, “What mathematics do all students need to see before they get to mathematics not all
students need?” Additionally, related to the previous category of non-negotiables, Algebra II was seen as
a course in which data science and statistical reasoning content might be incorporated.
Finally, for some states, the reimagining of mathematics pathways was undergirded by a desire to adequately
prepare students for the various STEM elds that exist in the current economy as well as for occupations
created in the future. Some states noted they specically wanted students to have access to precalculus-level
material in their high school mathematics courses as a way of better preparing students for dierent STEM
careers, while also acknowledging that not all pathways require the traditional calculus route in college. The
state representative from Ohio elaborated on how this was done in their state, saying, “We partnered with
higher ed[ucation] to dene what was needed to be prepared for a college credit-bearing mathematics course.
If I want calculus-based STEM, I need a traditional path. If I want to go into cyber security, I may be able to
take a non-calculus-based pathway.” This broader goal was connected to how states decided on the required
number of courses for students, the depth versus breadth of content standards per course, and the shift in
the emphasis of course taking toward relevance for students’ future career aspirations. More often than not,
these decisions resulted in the paring down of standards within courses. Alabama, for example, decided to
reduce the number of standards by about 10 per course but noted that the state now emphasizes statistical
reasoning and modeling — in line with the previous category of adjustments made to content and courses.
Re-Envisioning Mathematics Pathways to Expand Opportunities
39
BARRIERS TO REIMAGINING MATHEMATICS PATHWAYS
While states had clear goals for reimagining mathematics in hopes of creating better pathways for high
school students, they had to contend with substantial barriers before and during implementation. In
most cases, these barriers slowed planning and implementation. The barriers that states grappled with,
and continue to grapple with, during the planning and implementation stages of this work can best be
summarized as internal and external systemic barriers. There were also important nuances in state-level
context that have implications for how to engage in this work moving forward. The following section
further discusses the internal and external barriers that were true across states as well as a handful of
state-specic circumstances that led to additional barriers in reimagining mathematics pathways.
Types of Barriers
■ Internal barriers refer to the series of challenges that states encountered related to the
ideologies, policy and practice structures, and capacity of states’ educational systems and
school districts.
■ External barriers refer to challenges states encountered related to social contexts,
equitable access, and building postsecondary connections.
Internal Barriers Related to Existing Ideologies, Structures,
and Capacity
As most states represented in these conversations set out to reimagine the current mathematics pathways
in their schools, they rst had to examine how existing structures and resources for mathematics education
in their states did not necessarily align with or support their aspirational goals for new mathematics
pathways. Multiple states spoke about the long history of mathematics education in their states that
shaped traditional understandings of mathematics pathways for educators and other stakeholders, which
included a strong emphasis on promoting a path to Calculus for all students and as the standard of rigor.
Such histories meant that getting mathematics education stakeholders to envision a modernized Algebra II
or understand that additional mathematics course options would be viable in the long run was dicult.
Relatedly, states at varying stages of mathematics pathways implementation also discussed general
pushback they received about detracking and expanding access to higher level mathematics classes given
the aforementioned histories of mathematics ideologies regarding rigorous education in their states.
Re-Envisioning Mathematics Pathways to Expand Opportunities
40
One state went on to describe people in its education systems trying to hold onto mathematics as “an elite
social club” of sorts, in which multiple stakeholders were invested in limited access to more accelerated
mathematics classes. In the same vein, states were forced to defend the rigor of their new proposed
pathways or consider if opening up access to advanced mathematics classes would reduce the rigor
of these classes. To be clear, these oppositions came from educators and non-educators alike. Some
states have addressed this barrier by listening to and responding to the opposition, generating clearer
communication about the comparable rigor across dierent pathways, and more clearly articulating the
alignment of different pathways to the particular mathematical skills needed for a wide variety of
career options.
In addition to the persistent and counterproductive ideologies noted previously, states were beholden to
the resource and stang constraints that were in place at the onset of their eorts. States that were further
along in the process and states that were early in the process both described challenges related to stang
and professional learning for their teachers across the K–12 spectrum. The bulk of the conversation
centered on the fact that many school districts did not have the sta they would need to support new
coursework — a problem that was further exacerbated by heightened turnover during the COVID-19
pandemic as well as the challenges of stang mathematics classrooms generally, especially in rural, more
isolated locations. Additionally, the sta that did remain in schools were not necessarily trained in the
approach to mathematics education that was required under these reimagined pathways. States such as
Utah attributed this lack of the necessary training to their existing teacher education and certication
processes, which were largely competency based and, thus, were in tension with the reimagined vision of
mathematics education. There was evidence across states that changes needed to be made throughout the
teacher education pipeline in tandem with their eorts related to mathematics pathways if there is any
hope that these changes will become systematically implemented and sustainable in the future.
The nal point that came up in several states’ comments with respect to stang and training was related
to the building blocks of mathematics instruction that students received prior to high school. In most cases,
a major theme that states drew out of their planning conversation was that there was a need to address the
mathematics education that was happening in K–8 classrooms at the same time that they were trying to
change the mathematics content and standards in high school classrooms. For example, one state noted
that elementary school teachers were not well trained in number sense and were not teaching students
number sense. This gap in content, of course, then had ripple eects for whether students were prepared
for certain types of mathematics education by the time they reached high school. Another state discussed
implementing mathematics pathways as early as kindergarten to allow caregivers to be brought on board
with supporting their child’s mathematics education earlier in the process. Regardless of the context,
there was overwhelming consensus that what was decided for high school to postsecondary mathematics
pathways would have an impact on the mathematics content in earlier grades.
Re-Envisioning Mathematics Pathways to Expand Opportunities
41
External Barriers Related to Social Contexts, Equitable
Access, and Building Postsecondary Connections
In addition to the internal barriers states contended with as they tried to reimagine mathematics pathways
for students, several external factors outside of the education system inevitably inuenced the planning
and implementation of new mathematics pathways at the state level. While states worked to prioritize
equity, incorporating diverse perspectives and the needs of all student communities in meaningful ways,
the most often cited external barrier included the states’ sociopolitical climate and opposition to equity-
driven strategies. The extent of pushback states received was the most surprising for those that were
already engaged in mathematics pathways work that was previously fully supported by state educational
leaders. As such, mathematics initiatives that are not designed to teach about race or racism can be
wrongfully categorized as being related to race and culture-related studies. This miscategorization was
the case for the mathematics pathways eorts in Georgia because the state emphasized how mathematics
could be used to explain things in the world with particular phrases such as “mathematics in everyday
life” and “contextualizing” mathematics. The use of such language meant the state’s eorts fell under
policymakers’ anti-critical race theory legislation. When situations such as this arose, states were forced
to make compromises they were not initially anticipating with respect to content, strategy, and framing.
These circumstances hindered progress in responding to the needs of communities that have historically
been underserved in K–12 schools and in incorporating diverse perspectives on mathematics education.
The challenge to this end became how to meaningfully incorporate the perspectives of dierent community
groups in ways that were not performative. States described this eort as making sure “the right people
were in the room” when decisions were made and initiatives were planned. For example, Oklahoma
described bringing Native communities into the conversation early on as this practice was common in
the state with respect to educational initiatives. The state
typically looked at Native student data at various levels
before making decisions and switched to virtual meetings
to be more inclusive about who could attend the meetings.
During the focus group, Oklahoma leaders discussed
needing to now consider how to incorporate feedback
from Native communities, as they were in the early
stages of taking on the work of reimagining mathematics
pathways.
In addition to incorporating the perspectives of
different racial and ethnic communities, other states
described wanting to incorporate the perspectives
of K–8 educators for the reasons highlighted in the
previous section on internal barriers. While states often
had large committees of multiple stakeholder groups
reviewing materials and student data, there was a
“ Leading with equity
without context makes
it very easy to target and
weaponize mathematics
redesign eorts. We lead
with words like ‘modern,’
‘rigorous,’ ‘exible,’ and
‘opportunity.’”
— State Mathematics Lead
Re-Envisioning Mathematics Pathways to Expand Opportunities
42
sense that K–8 educators were not well represented in the decision-making process for mathematics
pathways. As multiple states were interested in exploring the idea of K–12 mathematics pathways as
a precursor to high school to postsecondary pathways and in bridging the divides in teacher education
and content between elementary schools and later grades, this concern was pressing for states across
contexts.
The nal external barrier that rang true across multiple contexts was the misalignment — or perceived
misalignment — between the mathematics curriculum and testing requirements imposed by colleges and
universities and the aspirations states had with respect to reimagined mathematics pathways. In addition
to states encountering basic communication challenges, there was a concern that higher education
institutions across some states still required traditional Algebra II, even though not all Algebra II courses
suciently prepare students for success. In other instances, states noted that universities expressed
concern that a shift in mathematics content and requirements would result in “lower standards” for
mathematics education that would have ripple eects for postsecondary education outcomes. Some
postsecondary mathematics faculty and administrators strongly assumed that mathematics pathways are
designed to reduce the rigor of mathematical options to address inequity and prohibit acceleration.
ACTIONABLE STEPS AND OPPORTUNITIES TO MOVE
FORWARD
Despite the challenges that states faced in reimagining mathematics pathways, they remained strategic
and hopeful that they would be able to meet their goals with the support of local partners and other
states engaged in this work. States were steadfast because, as highlighted in the Introduction, there was
a consensus that this work was essential to best meet the needs of all students. As such, in that same
collaborative spirit, they shared various lessons learned during the process thus far that might be helpful
for other states looking to take on similar work. These takeaways from participating states ranged from
big-picture lessons about how systemic change happens to advice about the pace of work and where to
seek collaborative partners.
1. Build Collaborative Bridges with Higher Education and
Mathematics Networks
Because of the multiple barriers to doing this work, it is strongly encouraged that states collaborate
with higher education partners and networks that are ready to support their eorts. States that were
further along in the process urged their counterparts who are earlier in the mathematics pathways
implementation process to recognize this reality early on and strategically build partnerships. Working
to expand mathematics pathways in high school requires attention to a range of cross-sector policies that
have the potential to help or hinder student access to and enrollment in courses and particular programs.
These policies include dening the entrance criteria for high school students to access early postsecondary
opportunities such as Advanced Placement, International Baccalaureate, and dual credit; determining how
to dene success in a course; and guaranteeing credit transferability across public systems. Across states,
Re-Envisioning Mathematics Pathways to Expand Opportunities
43
the mismatch between the mathematics states required for high school graduation and postsecondary
admissions and the gateway expectations for college-level mathematics was a barrier. It is advantageous
to hold discussions with high school and higher education partners to collaboratively decide on the
mathematics content and skills students need for equitable access to higher education opportunities that
align most with their college and career aspirations. A suggestion for future work is for states to convene
K–12 and higher education faculty leaders to review the list of approved K–12 mathematics courses in the
state and limit the list to those courses that prepare students for success in postsecondary mathematics and
students’ ultimate career elds. States that do not have approved lists but leave it to districts to determine
what to oer should create an inventory of all of the courses oered and work to better understand which
students are taking which courses — and why. These intrastate decisions can have interstate implications
that can hinder student progression if students need to move to a dierent state.
In Washington, conversations with two- and four-year colleges
highlighted that the content students were receiving in traditional
Algebra II classes was not serving them well in those institutions — a
nding that supported the need for a modernized Algebra II.
Additionally, having conversations earlier rather than later helped underscore the need to align work that
K–12, higher education, and intermediary stakeholders were already doing. These early conversations
provided states an opportunity to identify allies that were interested in and ready to do this work. States
at various stages of the process noted that they leverage mathematics networks or have formed teams to
attend learning opportunities related to mathematics pathways, such as participating in the Conference
Board of the Mathematical Sciences, the State Collaborative on Assessments and Student Standards, and the
Math Pathways Special Interest Group in the Association of State Supervisors of Mathematics, to make great
strides in implementing pathways. Each of the aforementioned spaces allowed states to gather information
that could inform their own work while forming collaborative networks with others engaged in the same
work. States were hopeful that continuing to engage in such networks and spaces would eventually lead to
deliverables states could use in their day-to-day work, such as a communications package.
u
Re-Envisioning Mathematics Pathways to Expand Opportunities
44
2. Center Equity and Leverage Data
Few states were able to provide cohort sequence data; just 12 states submitted that data for this report.
Some states’ data systems did not collect or were not able to access this data in this format. In other
states, the process for a sta member to aggregate data in this way would have been too time consuming.
However, course enrollment data by grade and student demographic groups was available in many states;
16 states submitted that data for this report. Requesting mathematics course enrollment data by student
demographics for 8th through 12th grade would likely yield higher participation from states relative to
course sequence data; this data request is clearer and easier to execute for state data systems sta. This data
also is an indicator of where most students start and end as they progress through secondary mathematics
courses. Course data for 11th and 12th grades may indicate whether the mathematics pathways that are
implemented are aligned to postsecondary mathematics pathways. Additionally, future research should
organize courses by those that align with the most common emergent secondary mathematics pathways
— the path to Calculus, Statistics, and Quantitative Literacy or Reasoning.
Both states that were early in mathematics pathways implementation work and states that were more
advanced in this process urged others to be intentional in their eorts to dismantle deep systemic
inequalities in mathematics education. Some states noted that the inequities related to access to technology
and grade-level content that were highlighted and exacerbated during the COVID-19 pandemic provided an
avenue through which to have these conversations. Though unexpected and under dire circumstances, the
ability to have conversations about inequitable access to COVID-19-related resources proved helpful and
provided opportunities for states to grapple with how they would engage in such discussions to maintain
momentum moving forward.
The policy and state data analysis ndings in the previous section signal a need for clearly dened
mathematics pathways and data to monitor and assess students’ progress in each pathway. If mathematics
pathways are implemented, a state should have the data to determine which students enroll, which
students succeed, and which students are hindered by courses along each pathway. Furthermore, clearly
dened mathematics pathways can help reduce uncertainty about mathematics course selections and
ensure that students have accessible guidance that claries which mathematics courses will best prepare
them for their college and career plans.
In states where students can opt out of or modify particular course sequences, knowing which courses and
course sequences students complete would be instructive. Are these students enrolled in courses of study
that align with postsecondary pathways and/or technical training programs leading to career opportunities?
Do some courses of study disproportionately leave students poorly prepared for postsecondary success
and ultimately lead to less successful postsecondary outcomes?
Re-Envisioning Mathematics Pathways to Expand Opportunities
45
States need to build longitudinal data systems that routinely collect the information necessary to enable
them to analyze both enrollment and success in course-taking patterns. This information enables state
education leaders to answer questions such as:
■ Are there gaps in successful participation in and completion of specic mathematics pathways based
on race, ethnicity, gender, family income, English language status, and special education status? Are
the gaps closing?
■ Are there signicant dierences within and across districts in the number of students who participate
in and complete specic mathematics pathways?
■ Are the students who have completed specic mathematics pathways better prepared to enter and
succeed in credit-bearing courses in postsecondary institutions? Are students less likely to need
remediation?
■ Are there mathematics pathways that disproportionately leave students poorly prepared for
postsecondary success — and with less successful postsecondary outcomes?
The Massachusetts Department of Elementary and Secondary
Education annually reports through the District Analysis Review Tool
for Success after High School the number and percentage of 12th
graders who have successfully completed a full year of mathematics
coursework (Massachusetts Department of Elementary and Secondary
Education, 2022). The data can also be disaggregated by student
groups. Four years of high school mathematics is an admissions
requirement for Massachusetts four-year public postsecondary
institutions and a meaningful measure. This dashboard empowers
districts, schools, parents, and advocates to conduct their own
analyses, within and across schools, as well as over time.
S
Re-Envisioning Mathematics Pathways to Expand Opportunities
46
3. Highlight Progress
Finally, states urged others interested in this work to understand that change happens incrementally
at times and that this reality does not discount the progress being made on reimagining mathematics
pathways. Put another way, given the pace of the work, states should acknowledge the small changes as
well as larger, systemic shifts in mathematics content and courses. Such a perspective requires states
to be exible and responsive to the ever-changing context of education in the United States. As the
representative from Oregon said at the end of one of the focus groups, “Reimagining a system means not
building within the system that exists.” This type of work requires persistence, clarity of purpose, and
intentional collaboration to achieve the intended, systemic changes that states aim to make for the benet
of students and their postsecondary goals.
Re-Envisioning Mathematics Pathways to Expand Opportunities
47
Access to clearly dened mathematics pathways is critical to ensure that all students and
families have the option and information to select the mathematics courses that best
align with students’ college and career plans. A growing number of states are working to
better align high school mathematics course content and oerings to higher education
mathematics pathways to increase relevance for students and increase equitable access to
and success in the courses needed for postsecondary and career-eld training. However,
there is little consistency or consensus on the best approach to creating relevant and
rigorous mathematics pathways. Ultimately, researchers and state mathematics leads
found that increased collaboration between K–12, higher education, and the workforce
is necessary to allow for better alignment between high school mathematics content and
postsecondary expectations.
Furthermore, the use of data and states’ ability to access and analyze relevant data to guide decisions
about improving student success in postsecondary-aligned mathematics in high school varies widely
across states. This fact hinders the progress that can be made to increase equitable access to and success in
postsecondary-aligned mathematics courses and to assess course enrollment and success with an equity-
centered lens. State policies and practices have a drastic inuence on the mathematics course-taking
patterns of students (e.g., when four years are required, higher percentages of students take four years
of mathematics compared to states that require fewer courses). This report raises a few implications for
future research from a policy perspective:
■ How do mathematics educators and leaders continue to improve collaboration structures so that states
benet from learning about work that is clearly providing students with better opportunities?
■ What can be done to improve access to relevant mathematics pathways data and encourage its use for
expanding equitable access to and success in postsecondary-aligned mathematics courses in high school?
As more states move to adopt modernized traditional mathematics sequences, these answers are critical to
ensure modern mathematics sequences are indeed equipping students for postsecondary success and to
meet today’s college and career expectations.
CONCLUSION
Re-Envisioning Mathematics Pathways to Expand Opportunities
48
METHODOLOGY
■ Policy Analysis: In January–March 2022, ESG collected publicly available information on statewide
high school mathematics standards and instructional materials, graduation requirements, summative
assessments, dual enrollment, and postsecondary admissions policies from state education agency
websites.
■ State Mathematics Course-Taking Data: Outreach to states began in mid-December 2021. Initial
communication was sent to state mathematics supervisors or sta in similar roles in state agencies. The
Dana Center, ESG, and SAP have relationships with many leaders in these roles. Given the complexity
of the data request, outreach began with leaders who would be willing to shepherd the data request
through state departments of education to the benet of their work at the department. Most states
required formal data requests or other formal channels to obtain the data. As needed, formal requests
were made to states in February and March 2022. Given that data collection varied in the past two years
due to the COVID-19 pandemic and that states have dierent lag times for compiling the most recent
data, researchers believed capturing data in the most recent year with the best data available would
best serve the purpose of the research, though this approach has limitations. For example, some of the
data is from school years that were aected most by the pandemic whereas other data is prepandemic.
View the data template/request to states.
■ State Mathematics Leadership Focus Groups: During March 2022, SAP conducted a series of focus
groups with representatives from state education agencies, institutions of higher education, and district
leaders as part of the Math Pathways project with the Dana Center and ESG. The intent of conducting
focus groups with representatives from dierent states was to gain a deeper understanding of the
successes and challenges states have encountered throughout their work implementing or attempting
to implement high school to postsecondary mathematics pathways. In total, 13 states representing
various policy and political contexts were included in the focus groups. Additionally, states were
purposely sampled to represent one of two scenarios — they were early in the work of implementing
high school to postsecondary mathematics pathways, or they had made substantial progress in this
process — to try to better understand the specic challenges that might arise at various stages of the
planning and implementation process.
APPENDIX
Re-Envisioning Mathematics Pathways to Expand Opportunities
49
1. Many states have written “Algebra II or an equivalent course” into their education code. Equivalent courses
vary and can include Integrated III or courses with content that is intended to prepare students equally well
for follow-on courses that are considered “above Algebra II” or that require “Algebra II” as a prerequisite.
The language of “or an equivalent course” is common enough that to get comparable data across states, it
was important to include these courses in addition to Algebra II.
2. States approach graduation requirements dierently, with some states dening student course
requirements by units, courses, or years. For the purposes of this report, the term “course” refers to a full
year, unit, and/or course for consistency.
3. The “path to Calculus” refers to courses that prepare students specically for success in Calculus, such as
Precalculus and College Algebra, through Calculus itself.
4. Many states have written “Algebra II or an equivalent course” into their education code. Equivalent courses
vary and can include Integrated III or courses with content that is intended to prepare students equally well
for follow-on courses that are considered “above Algebra II” or that require “Algebra II” as a prerequisite.
The language of “or an equivalent course” is common enough that to get comparable data across states, it
was important to include these courses in addition to Algebra II.
5. This report includes data from students enrolled in District of Columbia Public Schools (DCPS). DCPS serves
approximately 52 percent of K–12 public school students in Washington, D.C. Public charter schools serve
approximately 48 percent of students. See https://osse.dc.gov/sites/default/les/dc/sites/osse/page_
content/attachments/School%20Year%2021-22%20Annual%20OSSE%20Enrollment%20Audit.pdf.
6. Texas data was obtained from the Texas Education Research Center through a related research initiative.
7. Similar to school districts in other states.
8. Some states shared data that followed students from 8th-grade mathematics through the traditional
sequence or Algebra I in 8th grade through the accelerated sequence, while other states captured this
cohort of students starting in 9th grade.
9. States approach graduation requirements dierently, with some states dening student course
requirements by units, courses, or years. For the purposes of this report, the term “course” refers to a full
year, unit, and/or course for consistency.
10. States may appear in more than one category because they administer more than one mathematics
assessment for federal accountability; numbers will not sum to 51 states (including the District of
Columbia). For example, Rhode Island administers both the PSAT and the SAT to students as part of its
accountability system.
11. These numbers reect assessments used for federal accountability.
ENDNOTES
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50
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Re-Envisioning Mathematics Pathways to Expand Opportunities: The Landscape of High School to Postsecondary
Course Sequences was jointly developed by The Charles A. Dana Center at The University of Texas at Austin,
Education Strategy Group, and Student Achievement Partners. The three organizations wish to extend
gratitude for the contributions of Lindsay Fitzpatrick, David Kung, Shelly Ledoux, Susan May, and Elisha
Smith Arrillaga with the Charles A. Dana Center; Jhenai Chandler, Marie O’Hara, and Ryan Reyna with
Education Strategy Group; and Shelbi Cole, John Young, Sandra Alberti, Amy Briggs, Diana Cordova-Cobo,
Ruth McKenna, Hope Wilson, Ted Coe, and Vicky Gonzalez with Student Achievement Partners.
Special thanks to Kelly Van Beveren for her communications leadership and to Kathy Ames and her team at
Next Chapter Communications for their editorial and design work.
ACKNOWLEDGMENTS
