Degrees of Freedom:
Diversifying Math
Requirements forCollege Readinessand Graduation(Report 1 of a
3-part series)
Pamela Burdman
April 2015

linking knowledge, policy and practice

http://www.edpolicyinca.org

linking knowledge, policy and practice

http://www.learningworksca.org

linking knowledge, policy and practiceDegrees ofFreedom:
Diversifying Math
Requirements forCollege Readinessand Graduation(Report 1 of a
3-part series)

Pamela Burdman

http://www.edpolicyinca.org

linking knowledge, policy and practice

http://www.learningworksca.org

Executive Summary
Since the mid-20th century, thestandard U.S. high school andcollege math curriculum has been
based on two years of algebra anda year of geometry, preparing students to take classes in pre-calculus followed by calculus. That
pathway became solidified after
the 1957 launch of the Soviet satellite Sputnik motivated reforms
in U.S. science and engineering
education to boost the nation’s
technological prowess. Students’
math pursuits have been differentiated primarily by how far or
how rapidly they proceed along aclearly defined trajectory that haschanged little since then.
But evolutions in various disciplines and in learning sciences are
calling into question the relevanceand utility of this trajectory as arequirement for all students. The
emerging movement is towarddifferentiated “math pathways”
with distinct trajectories tied to
students’ goals. Alternatives emphasizing statistics, modeling,
computer science, and quantitative reasoning that are cropping
up in high schools and colleges
are beginning to challenge thedominance of the familiar mathsequence.
The drive toward acknowledging

the importance of multiple domains within math is prompted
largely by two developments:• Technological tectonics. The
advent of new technologies iscreating novel applications ofmath in various academic disciplines, elevating the importanceof statistics, data analysis, modeling, and computer science inthe undergraduate curriculum.
It is also leading faculty members outside of math departments to pay more attention
to their students’ quantitative
preparation.
• Demand for deeper learning.
Learning scientists and matheducators are emphasizing theimportance of students’ developing the capacity to use mathskills and knowledge to solve
problems in various contexts
rather than simply learning
isolated procedures and facts.
American students’ poor performance in traditional mathsequences as well as the highproportion of college students
taking remedial math havesome reformers asking whether
more applied courses would
better lend themselves to theeffective instruction needed tosupport college success.
Continued on page 2.

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

1

EXECUTIVE SUMMARY (Cont.)
The dialogues converge in broader
conversations about how colleges’
expectations shape what students
need to learn in K–12 schools—
as well as in community colleges,
which send hundreds of thousands of transfer students annually to four-year institutions. Decisions about math requirements
and expectations will have a major
impact on the academic opportunities of millions of students nationally.
Technological Tectonics
Math and science associations areurging changes in undergraduate
quantitative preparation to reflectchanges already occurring in various disciplines. For example, atsome research universities, biology
professors are offering new mathcourses that emphasize statistics
more than the traditional calculusbased course. Medical schools arerevising entrance requirements toemphasize grounding in statistics.
And social science departments
such as sociology, psychology, andpolitical science typically require
statistics either for entering the
major or completing a degree.
Students’ college math enrollments are dictated by the demands of their intended major
or by institution-wide graduation2

requirements that became common
over the last 50 years. As the requirements brought more students into
math departments, remedial enrollments skyrocketed. Statistics enrollments also mushroomed, while enrollments in more standard courses,
such as College Algebra, declined.
Concern about math’s relevance tostudents and responsiveness to other disciplines is sparking a numberof reforms:• promoting mathematicians’ understanding of how math is usedoutside of math departments,• redesigning College Algebra
courses to emphasize modelingand other practical applications,and• allowing alternative courses suchas statistics and quantitative
reasoning to fulfill quantitative
graduation requirements.
Demand for Deeper Learning
Discussions about improvements
to pre-college math instruction are
driven by alarm over weak achievement in math and the need to improve instruction in K–12 as wellas college remedial courses. The vision of millions of college students
spending time and money on highschool material is unsettling to policymakers, parents, and students—
even more so as research reveals that

remedial courses have no positiveeffects in terms of student success.
Nationally, about $2 billion isspent on remedial math education. Among entering community
college students, roughly 60 percent test into remedial math, thevast majority of whom never earna college degree.
Tension over Algebra 2. Efforts
to address this problem have diverged over the question of how
much to emphasize Algebra 2, achallenging course that prepares
students for Calculus. Algebra 2
became a fixture of high school
curricula due to Ivy League entrance requirements, and a drive
to offer it to all students grew fromstudies correlating the course withcollege success. While many agree
about the value of Algebra 1 for abroad population of students, thenecessity of Algebra 2 as a requirement for all, regardless of careerplans or college major, has long
been debated.
Concerns about college readinesshave led some states to require
Algebra 2 for high school graduation, while others are eliminating
it to ensure that more students
who are not pursuing science andtechnology-oriented majors canContinued on page 3.

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

linking knowledge, policy and practice

EXECUTIVE SUMMARY (Cont.)
be considered college-ready (atleast for community colleges).
The emerging trends in K–12 andcommunity college math reform
are moving in opposite directionsin terms of addressing Algebra 2:
Common Core implementation. The Common Core mathstandards being implemented
in California and most otherstates retain the traditional emphasis on two years of algebra
and a year of geometry. Yet,they represent a fresh approachto math content within traditional topic lists because theyemphasize “practice standards”
that measure students’ mathematical maturity.
Community college alternative remedial pathways. The
idea of alternative pathways
is gaining particular traction
as a strategy among leaders ofefforts to improve remedial
education and thereby increase
graduation rates at community
colleges. Fueled by foundation
support and policymaker interest, experimentation is taking place in a number of stateswith alternatives that prepare
students for statistics and quantitative reasoning. In 2014, theAmerican Mathematics Associ-ation of Two-Year Colleges (AMATYC) passed a resolution sayingthat intermediate algebra should
not be a “universal prerequisite
for all college-level mathematics
courses.” The experiments havereported surprisingly strong results, tripling and quadrupling
students’ passage of college-level
math courses.
Diversification and its Dilemmas
Given mathematics’ traditional
prominence in the undergraduate
curriculum, the implementationof alternatives faces a host of challenges:• Rigor. Selective institutions thatcannot admit all students often
use rigorous courses as a screen toeliminate students. Community
colleges and other broad-accesspublic institutions, under pressure to increase graduation rates,
are focused on rooting out unnecessary barriers and reject theidea of rigor for the sake of rigor.
There is an inherent tension between the two perspectives.
• Articulation. Universities’ policies for admitting transfer students are also creating dilemmas
for expanding alternative remedial pathways in some states, including California. Likewise, thesuccess of the remedial courses, if
it continues, could call into ques-tion high schools’ continued reliance on more traditional mathcourses. But the idea of eliminating Algebra 2 gives manyeducators pause, since they fear
this could prematurely close off
opportunities for students.
• Barriers. While proponentsof alternative pathways view
algebra-intensive curricula asa potential barrier to students’success, critics fear that an education without advanced algebra itself constitutes a barrieror dead end. One solution being explored is to make bridge
courses available for students inalternative courses who discover an interest in STEM.
Looking Ahead
California’s higher education institutions need to consider practical approaches that will best serve
the students they have now, while
simultaneously investing in andevaluating longer-term strategies
to better prepare students for college-level courses.
The second and third reports inthis series will include specific recommendations about policies andpractices that do this. This reportconcludes with a set of principles
Continued on page 4.

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

3

EXECUTIVE SUMMARY(Cont.)
that will underscore those specific recommendations:• Keep the focus on maximizing student success statewide;• At the college level, base remedial policies and otherrequirements on equipping
students for subsequent
coursework, career, and citizenship (rather than abstract
notions about desired knowledge);• Increase communication sosegments can better understand the implications of
their policies on students inother segments and minimize
misalignment;• Apply greater scrutiny to academic requirements noted forhigh failure rates; and• Make room for experimentation and innovation whilegathering evidence.

4

Degrees of Freedom: Diversifying
Math Requirements for College Readinessand Graduation
ForewordS

ince the mid-20th century, thestandard U.S. high school andcollege math curriculum has
been based on two years of algebra
and a year of geometry, preparing
students to take classes in pre-calculus and calculus. That pathway
originated with admissions requirements at Harvard and other elite
universities. It became solidified after the launch of the Soviet satellite
Sputnik motivated reforms in U.S.
science and engineering education
to boost the nation’s technologicalprowess. Since then, it has effectively been a matter of faith that the calculus-directed sequence of courses isessential content for college success.
Students’ math pursuits, then, havebeen differentiated primarily by
how far or how rapidly they proceed along a clearly defined trajectory that has changed little since the1950s. While the traditional course
pattern remains the norm, evolutions in various disciplines and inlearning sciences are calling into
question its relevance and utility asa requirement for all students. Alternatives emphasizing statistics, modeling, computer science, and quantitative reasoning are cropping up inhigh schools and colleges and beginning to challenge the dominance ofthe familiar sequence. The emerging
movement is toward differentiatedmath pathways with distinct trajectories tied to students’ goals. Consider these examples:• In Texas, instead of an Algebra
2 graduation requirement, highschool students can now opt totake courses in statistics or algebraic reasoning.
• In Los Angeles, a $12.5 million National Science Foundation grant is allowing some highschools to pilot a data scienceclass intended to replace Algebra
2 for some students.
• At about 50 institutions (mostly
community colleges) in 14 states,
the Carnegie Foundation for the
Advancement of Teaching is testing new approaches to remedial
math that emphasize statistics
and quantitative reasoning rather
than the traditional algebra-intensive curriculum.
• In Georgia, a state-level task force
concluded that college students
who aren’t pursuing majors inscience and technology may ful-Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

linking knowledge, policy and practicefill their math requirements withcourses called Quantitative Skills
or Introduction to Mathematical
Modeling instead of College Algebra.
• And at the University of California’s Los Angeles and Berkeley
campuses, life sciences professors
are offering new math coursesthat emphasize statistics morethan the calculus-based class traditionally required for biology
majors.
Proponents of such diversity say itbetter reflects the evolving role ofmath outside of math departments.
“Math has grown incredibly in thelast 25 years,” notes Uri Treisman, amathematician at the University ofTexas at Austin and leader of several
math reform initiatives. “It’s not justareas of engineering, although theyremain extremely important. It’s
Pixar, Netflix, Google, Facebook,tomography, epidemiology, mathematical pharmacology, machinelearning. If you look at the mathematics we have in the curriculum
now, it represents one powerfulstrand among a forest of powerfulstrands of mathematics.”
The drive toward acknowledging
the importance of multiple domains
within math is being played out inschools and colleges in California
and nationally, prompted by twodevelopments:• Technological tectonics. The
advent of new technologies is creating novel applications of mathin various academic disciplines,
elevating the importance of statistics, data analysis, modeling,
and computer science in the undergraduate curriculum. It is also
leading faculty members outsideof math departments to pay more
attention to their students’ quantitative preparation. Concerned
about ensuring the relevance of
their field, mathematics societies
are encouraging greater dialogue
between math departments andother disciplines that use math.
• Demand for deeper learning.
Learning scientists and math educators are increasingly emphasizing the importance of students’
understanding
mathematicalconcepts and developing the capacity to use math skills to solve
problems in various contexts
rather than simply learning isolated procedures and facts. Research suggests that the de-contextualized instruction common
in American math classrooms canactually impede learning (Stigler& Hiebert, 1999; Grubb, 2013).
Some suspect it can even deter
students from pursuing scientificfields (President’s Council, 2012).
Some educators, researchers, andpolicymakers are asking whether
more applied courses—such as
those emphasizing statistics andquantitative reasoning—would
better lend themselves to the effective instruction needed to support college success.
“When today’s parents were going
through the schools, the main focus
in mathematics was on mastery ofa collection of standard proceduresfor solving well-defined problems
that have unique right answers.
Learning mathematics had beenthat way for several thousand years.
Math textbooks were essentially recipe books,” Stanford mathematics
professor Keith Devlin told Forbes
Magazine.
“Now all those math recipes havebeen coded into devices, some ofwhich we carry around in our pockets. Suddenly, in a single generation,
mastery of the procedural math skills
that had ruled supreme for 3,000
years has become largely irrelevant.
Students don’t need to train themselves to do long computations, as
was necessary when I was a child.
No one calculates that way any
more! What they (we) need in today’s world is a deeper understanding of how and why Hindu-Arabicarithmetic works,” he said.
The two dialogues have arisen indifferent realms: The technologicalshifts pertain to undergraduate education, while the learning demands
relate more directly to how students
are prepared for college, career, andDegrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

5

general citizenship. They converge
in broader conversations about how
colleges’ expectations shape what
students need to learn in K–12
schools—as well as in community
colleges, which send hundreds ofthousands of transfer students annually to four-year institutions.
Will universities, for example, admit high school students who took
a data science class in lieu of Algebra 2? Will community college
students taking the Carnegie Foundation’s Statway sequence insteadof a traditional math sequence beable to transfer to the university of
their choice? If so, what majors will
be open to them? Will the biology
faculty interest in statistics training
ultimately influence quantitative expectations for incoming freshmen?
And how will the new Common
Core State Standards being implemented in most states affect thosechoices?
These decisions and the interactions
among them will have a major impact on the academic opportunities
of millions of students nationally.
And yet, discussions about them
have occurred largely in isolation,with each segment of the education
system bringing different priorities
to bear. Interestingly, the increasing engagement of researchers andpolicymakers in improving student
outcomes in community collegesis helping to make the connections

6

more explicit. That’s because community colleges sit at the nexus of
the two conversations, offering remedial courses that review K–12
material as well as lower-divisioncourses that transfer to four-year institutions.
L e a r n i n g Wo rk s’ 2 0 1 3 re p o r t ,Changing Equations: How Community Colleges Are Re-Thinking College
Readiness in Math, focused on experiments with alternative curricula
at two-year colleges. That report
noted that those efforts were notin sync with math requirements inhigh schools and four-year universities. Given that that these, too, arein flux, LearningWorks commissioned an additional series of reports to analyze the changing statusof math requirements throughout
the education system, its implications for students’ progress fromhigh school through college, and therole of schools, community colleges,
and universities in supporting suchprogress.
The series, Degrees of Freedom, isbased on dozens of interviews andextensive reviews of literature anddocuments related to U.S. highschool and college math curricula
since the mid-1900s. This reportwill examine the move toward differentiated math pathways linked
to students’ academic majors, highlighting some obstacles to implementing them and some principles

for addressing those obstacles.
The rest of the series will focus
more on math-related transitions in
California’s education system andsurface more specific policy recommendations for improving them.
The second report will analyze how
the distinct missions of high schools,
community colleges, and four-year
universities create tensions for articulating math curricula across segments, and the resulting challenges
for alternative math pathways as aroute to college readiness. The third
report will analyze more specifically
how college and university policies
for placing students in remedial
math courses create additional barriers for students in completing a
degree or transferring to a four-year
university.
Degrees of Freedom is being disseminated by LearningWorks and
PACE (Policy Analysis for California Education) with a goal of fostering needed conversations spanning
K–16 education to consider strategies to enhance student success inmath and improve curricular coherence for students as they progress
through the educational system.

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

linking knowledge, policy and practiceTechnological
Tectonics andMomentum for
ChangeUndergraduate math curricula areundergoing change nationally. College math faculties are facing pressure to transform their course offerings and make them responsive
to the needs of other disciplines.
In 2014, the Joint Policy Board forMathematics (an umbrella organization of several math and statistics
societies) issued a statement “callingupon the entire mathematical sciences community to achieve muchneeded change in undergraduate education.” And the National ResearchCouncil’s 2013 report, Mathematical
Sciences in 2025, noted, “The needto create a truly compelling menu
of creatively taught lower-divisioncourses in the mathematical sciences
tailored to the needs of twenty-first
century students is pressing.” (SeeSidebar, National Associations Urge
Change.)Many of the changes remain experimental and peripheral, if notcontroversial. While there is broad
agreement that new ways of teaching math are needed, some experts,
including some mathematicians, arecautious about deviating from conventional math content. Leaders inhigher education, wary of creating
pathways that could be consideredsecond-tier, tend to defer to mathdepartments when it comes to setting standards. More fundamental
doubts exist about how much weight
fields such as statistics and computerscience deserve. “A lot of mathematicians don’t consider statistics a reasonable part of mathematics,” notes
Jeremy Kilpatrick, a math education
expert at the University of Georgia.
“For them, it’s not really mathematics. It’s an application of mathematics, but not mathematics itself.”
Controversy aside, there is clear
momentum for diversifying students’ quantitative preparation. If it
continues, the revised courses could
grow to constitute a dominant
proportion of pre-collegiate andcollege-level math courses in years
to come. They are already gaining
a foothold in some areas of the undergraduate curriculum. Since mathrequirements are typically intended
to prepare students for later coursework, this examination of the shifts
begins by looking at the evolutionof math requirements for undergraduate majors.

National Associations
Urge Change
In the last five years, a host ofmath and science associations
have urged changes in undergraduate quantitative preparation:
2014—Joint Policy Board
for Mathematics, Meeting the
Challenges of Improved PostSecondary Education in the
Mathematical Sciences“We call upon all mathematicalscientists in academia to renew
their focus on post-secondary
mathematics education. We
challenge department chairs toincentivize innovation for thesake of their students and thehealth of our discipline. We encourage mathematics faculty toreach out to colleagues in mathematics-intensive disciplines inorder to heighten the relevanceof their courses to the careersof their students. And we urge
depart­ments as a whole to investigate with an open mind newteaching methodologies andtechnologies, keeping in mind
the need to retain and motivate
students.”
Continued on page 8.

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

7

National Associations
Urge Change (Cont.)
2013—National ResearchCouncil, Mathematical Sciencesin 2025“The educational offerings oftypical departments in the mathematical sciences have not kept
pace with the large and rapidchanges in how the mathematical sciences are used in science,engineering, medicine, finance,social science, and society at large.
This diversification entails a needfor new courses, new majors, newprograms, and new educational ...
partnerships ... Different pathways
are needed for students who may
go on to work in bioinformatics,
ecology, medicine, computing,and so on. It is not enough to rearrange existing courses to create
alternative curricula.”Life Sciences and Math
Requirements
Take the re-design of math requirements in some biology departments.
At both UC Berkeley and UCLA,
the traditional one-year calculus
course may eventually be replaced
by a course combining calculus and8

2012—President’s Councilof Advisors on Science andTechnology, Engage to Excel“Employers in the private sector,
government, and military frequently cite that they cannot findenough employees with needed
levels of mathematics skills. This
lack of preparation imposes alarge burden on higher educationand employers.
Higher education alone spends
at least $2 billion per year on developmental education to compensate for deficiencies. Also, introductory mathematics courses
often leave students with the impression that all STEM fields aredull and unimaginative, whichhas particularly harmful effects
for students who later become
K–12 teachers. Reducing or eliminating the mathematics-preparation gap is one of the most urgent challenges—and promising
opportunities—in preparing theworkforce of the 21st century.”
statistics as the required math course
for biology majors. At Berkeley, the
new class—entitled Methods ofMathematics: Calculus, Statistics
and Combinatorics—was createdby a committee of mathematicians
and biologists. While some mathematicians were initially skeptical,
it helped that the biology professor

behind the class had a joint appointment in mathematics. He ensured
his colleagues that students would
still be studying advanced math.
“Molecular biology in particular has
become a mathematical science,”
said the professor, Lior Pachter.
“The real radical change is happening with DNA sequencing. There’s a
lot of data collection and statistics.”
His course now enrolls about 300
students, primarily biology majors.
At UCLA, the process has been
more controversial. About four years
ago, Life Sciences faculty dissatisfied
with their students’ math preparation created the specifications for a
new course. While de-emphasizing
some typical introductory calculus
topics, Mathematics for Life Scientists stresses topics such as statistics,
dynamic modeling, and differential
equations. It also includes computational labs and increased teachingassistant involvement. Unlike thetraditional sequence of three trimesters of Calculus plus a fourth of
Statistics, the new sequence consistsof three trimesters total. The mathematics department, however, did
not find the math content acceptable, so the College of Life Sciences
began its own pilot in 2013, supported by a $2.3 million National
Science Foundation grant to develop and evaluate the course.
Changes in these and other biol-

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

linking knowledge, policy and practiceogy departments correspond witha shift starting to happen at medical schools. For decades, Harvard
Medical School, for example, asked
incoming students to have at leasta year of calculus under their belts.
But 2015 will be the final year thata standard Calculus course will be
accepted. The school’s revised requirement applies greater emphasis
on statistics than on calculus, which
it says often focuses on the “derivation of biologically low-relevancetheorems.” Besides a “familiarity with calculus,” it encourages a“broader and more flexible range ofrequirements,” adding that, “given
the importance of statistics for understanding the literature of scienceand medicine, adequate groundingin statistics is required.”
Other leading medical schools suchas Johns Hopkins and Stanford are
moving in a similar direction, stressing grounding in statistics, epidemiology, and the capacity to evaluate scientific research reports. Suchchanges echo recommendations theAmerican Association of Medical
Colleges made in 2009.
Since some critiques of the traditional math curriculum have comefrom individuals in the humanities
and social sciences, their proposals
are sometimes eyed suspiciously as
sacrificing quantitative rigor. The
involvement of prestigious biology
departments and medical schools

in re-thinking math requirements
could lend legitimacy to alternative
approaches and give rise to a new
definition of mathematical rigor. Its
goal would be to ensure that U.S. institutions foster technological prow-

ess and attract talented students totechnical fields while simultaneously
developing the quantitative abilitiesof students pursuing non-technicalfields. A tall order, to be sure.

Leading Medical Organizations Revise Their MathPrerequisites for Physicians
2010—Harvard Medical School:
A full year of calculus focusing on the derivation of biologically low-relevancetheorems is less important than mastery of more relevant algebraic and trigonometric quantitative skills. To prepare adequately for the quantitative reasoning demands of the contemporary medical curriculum and certain medical
specialties, to provide analytic perspective and to appreciate the uncertainties
in evaluation of biological systems, students are required to have familiarity
with calculus. A broader and more flexible range of requirements is encouraged, however, and given the importance of statistics for understanding the literature of science and medicine, adequate grounding in statistics is required.
2009—Scientific Foundations for Future Physicians—Entering
Medical Student Expectations
Learning Objectives:
1. Demonstrate quantitative numeracy and facility with the language ofmathematics.
2. Interpret data sets and communicate those interpretations using visual andother appropriate tools.
3. Make statistical inferences from data sets.
4. Extract relevant information from large data sets.
5. Make inferences about natural phenomena using mathematical models.
6. Apply algorithmic approaches and principles of logic (including the distinction between cause/effect and association) to problem solving.
7. Quantify and interpret changes in dynamical systems.

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

9

Quantitative Graduation
Requirements and the Growth of
Statistics
Elsewhere in the undergraduate
curriculum, students’ math enrollments are generally dictated either
by the demands of their intended
major or, for those whose majors
don’t specify math expectations, byuniversity-wide general education
requirements. University-wide mathexpectations weren’t always a fixtureof graduation requirements. In the
1960s, according to a Mathematical Association of America (MAA)
report, the organization approached
the idea of quantitative literacy for
all students in a “gingerly manner”(MAA, 1994). At the time, a low
point for general education, a small
minority of institutions required
math for all students. In the early
1970s, the vast majority of mathdepartment chairs surveyed by the
Conference Board of the Mathematical Sciences (CBMS) did notthink institution-wide requirements
were even necessary.
The requirements began gaining
prevalence in the 1970s, along withthe expansion of general education,
leading to math’s current “gatekeeper” status within the general education curriculum. This growth coincided with enrollment expansion
and demographic change. But there
was a tension between the mathematicians’ view of their discipline10

and the idea of general education.
“When it came to having general
education requirements, history
and psychology departments developed service courses for non-history
and non-psychology majors, for example. Mathematics didn’t do that.
They said, ‘We’ll teach them thesame thing we’re teaching now,’”
noted Bernard Madison, a University of Arkansas mathematician andmath educator. “Mathematicians
never really came to grips with using mathematics in service of general education.”
Meanwhile, a parallel shift was occurring: As overall enrollment inmathematical sciences was increasing significantly, much of the growth
was in computer science and statistics courses, not mathematics itself.
Upper division math enrollments
were actually dropping—falling by
32 percent from 1970 to 1975, forexample. “The clear overall impression from course enrollment data isa shift toward mathematical science
courses that are applicable as preparation for specific post-college careers,” noted the CBMS in its 1980
survey.
At the same time, remedial enrollments were jumping dramatically,
more than doubling in a decade atboth two-year and four-year colleges. While noting that the increase
matched professors’ impressions of

declining preparation levels among
students, the CBMS reports do notdiscuss the likelihood that quantitative graduation requirements (pairedwith overall enrollment growth inhigher education) were responsible
for bringing more under-preparedstudents into math departments.
Math requirements, then, became ade facto filter that determined students’ readiness for college.
In more recent years, discussions
have centered not on whether tohave quantitative graduation requirements—most higher education institutions do—but on how tomake them relevant and engaging
for students. Today the majority ofstudents appear to meet them witha course situated in between Algebra
2 and Calculus—most often College
Algebra or Pre-Calculus. (Whilethe definitions of these courses differ among institutions, and there issignificant overlap between them,typically Pre-Calculus includes trigonometry.)
But other mathematical sciences
have seen dramatic increases in enrollment—particularly statistics. In
1960, the first year for which data
is available, statistics represented
about three percent of all mathematical science enrollments atfour-year institutions. By 2010, 19
percent of introductory math enrollments at four-year schools werein statistics courses, according to theDegrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

linking knowledge, policy and practice

CBMS survey. College Algebra andsimilar courses, while still dominant,constituted a shrinking proportionof introductory enrollments from1995 to 2000, even as enrollment inother introductory courses (including Liberal Arts Math, Finite Math,Business Math, and Mathematics
for Teachers) grew by 37 percent.
A different 2010 survey of math departments found that about threequarters of colleges allowed students
to meet quantitative graduation requirements with Statistics, and halfallowed courses outside of mathdepartments. Seventeen percent allowed a course focused specificallyon quantitative reasoning or quantitative literacy (Schield, 2010).
Ironically, then, the overall expansion of math enrollments coincidedwith the adoption of quantitative
graduation requirements, but alternatives such as Statistics have growndisproportionately, as institutions
looked for courses that were palatable to students outside of mathintensive majors. Statistics is said toappeal to students with strong conceptual capacity whether or not they
excel at algebraic manipulation. The
increasing demand for “data crunchers” in fields ranging from biomedical research to consumer marketing
to policy analysis led Google’s chief
economist to call data crunchinga “sexy job.” And AP Statistics hasgrown from 7,000 exams in 1997

(compared with 110,000 in Calculus) to 180,000 exams in 2014
(compared with 280,000 in Calculus).
Statistics also plays a key role inmany social sciences. Researchersin psychology, sociology, and political science increasingly employ
data collected by hospitals, schools,
social service agencies, and otherpublic entities. Many social science
departments require statistics courses. Some offer one for their majors,presenting statistical techniques
within the context of the disciplinein order to familiarize students withdata sources and research methodologies common in their fields. Suchcourses fulfill general education requirements at some institutions, butothers require College Algebra as a
general education course and Statistics as an additional, upper-divisioncourse.
Interdisciplinary Discussions andMath Requirements
Aware of math’s gatekeeper role,
the MAA has actively cultivated the
involvement of faculty from the social sciences and other disciplines
in conversations about students’quantitative preparation. This, inturn, has led to shifts in priorities.
In 2007, when the organization
brought social scientists and mathematicians together to develop recommendations, the participants feltthat college students ideally should
study both advanced algebra andstatistics. But noting the impracticality of a two-course sequence, they
concluded that it is “imperative thatstudents come away with an appreciation for statistics, and that thispreparation takes priority over more
typical introductory mathematics
courses” (Ganter & Haver, 2011).
With the decline in math majors,
and an increasing number of quantitative requirements being fulfilled
in other departments, some mathleaders are concerned about staying relevant. “I think that it needs
to be realized that mathematics, astaught by mathematicians, is seen as
a separate and arcane discipline by alarge fraction of the academic community and that this is a problem
for our profession,” wrote one mathchair in responding to a 2010 survey
about math requirements (Schield,2010).
In fact, cultivating mathematicians’
understanding of how math is usedoutside of math departments has
been seen as vital to the future ofmathematics. “It’s a real problem,
because many mathematicians andteachers of mathematics are notaware of all the new and different
ways in which math is now being
used,” noted Mark Green, a retired
UCLA math professor and lead author of the Mathematical Sciences in2025 report.

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11

Devlin recalls coming to Stanford
in 1987, expecting to benefit fromthe institution’s prowess in technology research and discovering something broader. “What changed me
far more was coming up against theworld-class expertise in the humansciences that Stanford also has,” hetold Forbes. “It was after working
with (actually, mostly listening to)leading human scientists and philosophers who consider those issues
that I was forced to reassess the rolethat mathematics can play in thehuman sciences.”
This increasing involvement of other disciplines in discussions aboutmath may be eroding mathematicians’ traditional veto power overquantitative requirements. “People
in other departments are really questioning the learning goals, and saying, ‘Those shouldn’t be our learning goals.’ That suddenly is taking
the power away from the mathdepartments,” notes James Stigler,a UCLA psychologist who studies
math learning. “The questioning of
the idea that the math department
knows what math all students needto know is new.”
College Algebra and Curricular
ChangeDespite gains by Statistics and otheralternatives, the traditional courses
still represent the majority of introductory math enrollments, largely

12

due to general education requirements. College Algebra courses
enroll close to a million students
per year in two-year and four-yearcolleges combined—with two-year
enrollment having quadrupled overthe last 40 years. Such large enrollments have historically been a sourceof math departments’ influence.
College Algebra’s role as a stapleof the undergraduate curriculum
is sometimes compared to Latin,
which fell out of favor in the lastcentury. “Many would skip College
Algebra if they did not have to passit to get the degree they need to enter their chosen career,” according
to a 2002 address by Arnold Packer,
a former Department of Labor official who directed a federal commission on 21st century skills in theworkforce, “For too many students
it looks like—and is—a painful experience that they would prefer toskip” (Packer, 2002). Packer was
speaking at an MAA conference
about College Algebra. Conference
participants cited failure rates above
40 percent as evidence that thecourse needed reform, particularly
since the educational purpose of thelarge enrollment is unclear: ThoughCollege Algebra is intended to bea prerequisite for Calculus, fewer
than 10 percent of College Algebra
students ultimately take Calculus(Small, 2002).
Some of the interdisciplinary dis-cussions fostered by the MAA havefocused on ways of modernizing thecourse. A key reform is an emphasis
on modeling, which an MAA report
says students find more relevant
and engaging than the standard version. “Generally, attendance andthe classroom environment improvedramatically. Most importantly, in
our view, students develop habits ofmind necessary for life-long learning while learning the mathematics
that will be most important to themin future endeavors,” the report said
(Ganter & Haver, 2011).
As reform efforts accelerate, it’s become clear that a reason College
Algebra enrollments remained highin some places has been the absenceof other options for meeting quantitative requirements. “It’s about
traditional views of rigor and not somuch about relevance,” notes Bruce
Vandal, vice president of Complete
College America, an advocacy groupthat promotes alternative mathpathways as a means of increasingcollege completion rates.
That is beginning to change, however. For example, by 2010, ArizonaState University had removed College Algebra from the list of coursesthat fulfill its quantitative graduation
requirement. “The department has
taken this action as it believes students requiring only one mathematics course in their college experienceshould be introduced to mathemat-

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

linking knowledge, policy and practice

ics that is more applied in nature,”according to a survey response. “We
further believe any student taking
College Algebra should have every
intention of taking another mathematics course” (Schield, 2010).

courses including Making Sense:
Language, Logic, and Communication; Deductive Logic; Nutrition
and Health; and Myths, Paradigms,
and Science. (See Box, Quantitative
Course Selection, on p. 14.)

In Georgia, a task force including
respected mathematicians reacheda similar conclusion in 2013: “Thepractice of using College Algebra as
a proxy for general quantitative ability or to ensure that students canlater transfer to a STEM major mustend,” it wrote. “System data suggestthat placement in College Algebra
for the above reasons is a major contributor to student failure. Furthermore, the broad audience in College Algebra makes it challenging
to organize this important course
as a true stepping stone to Calculus” (University System of Georgia,
2013). The task force recommended
that students who aren’t majoring in
STEM fields should be allowed tocomplete their math requirementswith courses in quantitative skills ormathematical modeling.

As with the innovations occurringin biology departments and elsewhere in universities, however, there
are few avenues for sharing and disseminating information about thesecourses. An unpublished survey in2010 appears to be the best overview
available of how institutions treat
their quantitative requirements.
Therefore, it is unknown how many
institutions offer such a broad range
of options, and for those that do,to what extent students transferringfrom two-year colleges can use themto meet math requirements.

Interestingly, diversity in math requirements appears to be morecommon at selective institutions,with broader access universities andcolleges tending to have narrower
offerings. Examples of acceptablecourses at Stanford University include: Feeding Nine Billion, Remote
Sensing of the Oceans, and Cancer
Epidemiology. Harvard accepts

the current state of experimentationand innovation in the general education curriculum.

“Almost everywhere that people havecapacity and are actively engaged
in mathematical research and havepeople who care about undergraduate education, there’s experimentation going on,” notes Treisman.
“The problem isn’t a lack of experimentation. It’s that the experimentation takes place in a fog of collective amnesia.” Treisman, Green,and other leading mathematicians
have initiated a project called TPSE
Math (Transforming Postsecondary
Education in Math) focused on themodernization of the undergraduate math curriculum. Undoubtedly,
one of their tasks will be to examine

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

13

Quantitative Course Selection: Courses Fulfilling Quantitative Reasoning
Requirements at Six Universities
Stanford• Cancer Epidemiology• Riding the Data Wave
• Electric Automobiles and Aircraft• Experimental Economics• Remote Sensing of the Oceans• Feeding Nine Billion
Harvard• Making Sense: Language, Logic,
and Communication• Analyzing Politics• Deductive Logic• Nutrition and Health  • Myths, Paradigms, and Science• Great Ideas in Computer Science
UCLA• Biostatistics• Computing• Inductive Logic (Philosophy)• Introduction to Data Analysis (Political Science)• Statistics

San Francisco State University• Calculus with Business Applications• Data Analysis in Education• Calculus or Business Calculus• Elementary Statistics• Pre-Calculus• Quantitative Reasoning in Psychology
Cal State University- Northridge• College Algebra• Mathematical Methods for Business• Pre-Calculus• Mathematical Ideas• Introductory Statistics• Calculus for the Life Sciences
CUNY- College of Staten Island• Mathematics for Liberal Arts
• Finite Mathematics• Probability and Statistics• College Algebra and Trigonometry• Pre-Calculus

Source: Institutions’ websites. With the exception of San Francisco State University, the courses listed are a subset of those accepted tomeet quantitative reasoning requirements.

14

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

linking knowledge, policy and practice

Low Math Achievement
and the Demand for
Deeper Learning
The momentum for diversifying
postsecondary math curricula stems
from the quantitative and statisticaldemands of various academic disciplines and industries and the ineffectiveness of courses such as College
Algebra in preparing students forthese demands. Discussions about
improvements to pre-college math
instruction are commonly driven
by alarm about U.S. students’ weakachievement in math. As such, they
are often disconnected from discussions about the undergraduate curriculum for which they are intended
to prepare students. Yet these discussions, too, have begun to focus
on the possibility that alternative
math content could actually provide a better foundation for somestudents.
The reasons for poor math achievement have been explored in debatesthat unfortunately are best known
for their lack of consensus. The evidence—test scores—has been lesscontroversial. On the most recent
PISA (Programme for International
Student Assessment) round, for example, U.S. students ranked 26thout of 34 countries in math. Even
against domestic standards, U.S.
students don’t look much better,with only seven percent of 17-yearolds reaching the highest level ofproficiency on standardized tests.

way mathematics is taught.

In recent years, though, the mostglaring evidence of students’ mathdeficiencies has been the large proportion who are sent to remedial
math courses upon taking college
placement exams. The vision of
millions of college students spending time and money on high school
material is an unsettling one topolicymakers, parents, and students
alike—even more so as research has
revealed that these courses have no
positive effects in terms of student
success.

“The major problem is that the way
math has been cut up, chopped up,
and delivered makes it hard to getsense-making,” notes Alan Schoenfeld, a professor of education at UC
Berkeley. “What kids get, by andlarge, is a diet of facts and formulas, which they can remember withcramming for the final exam, andby the time September rolls around,they’ve lost half of it.”Nationally, about $2 billion is spent
on remedial math education (President’s Council, 2012). About 35
percent of students attending fouryear universities require a remedial course in English or math, andamong entering community college
students, around 59 percent test into
remedial math. The vast majority of
those students never earn a college
degree (Hodara, 2013).
“The developmental math problem is an indictment of our whole
education system,” notes Stigler of
UCLA. There is general agreementthat vast improvement in instruction is needed in both K12 andremedial math classes. In the matheducation community, there is an
emerging consensus that students
lack conceptual understanding,which is directly attributable to the

At one Southern California community college, Stigler and colleagues
found that only 22 percent of students who took the math placement
exam could successfully order four
numbers consisting of two simple
fractions and two decimals. After
converting the fractions to decimals,
sixty percent of students orderedthem by the number of digits.
“These errors reveal that rather thanusing number sense, students rely
on a memorized procedure, only
to carry out the procedure incorrectly or inappropriately,” they said
(Stigler et al., 2010). The findings
echoed Stigler’s earlier researchshowing that, compared to students
in higher performing nations, U.S.
students tended to learn math as aset of formulas without understanding how to think mathematically,
and that this won’t change without
different teaching methods (Stigler& Hiebert, 1999).

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

15

Emerging Strategies and Tension
Over Algebra 2
The conclusion of this body of research is that students must movebeyond discrete skills and formulas
used to answer test questions toward a conceptual understandingin which they are able to reason andsolve problems and nimbly apply
those skills in other contexts. Froma content perspective, efforts to address this challenge lie on a continuum whose extremes are:
• New approaches to existing
content: These entail finding
ways to teach the traditional content more effectively so that more
students succeed. Many efforts
have adopted this approach, particularly because of the imperative of preparing a subset of students to succeed in STEM fields.
To date, there has been limited
success, however. High failure
rates in high school as well as college remedial math classes persist,
impeding some students from
earning college degrees. These
approaches are most common inhigh school, where the philosophy has been to keep students’
options open.
• New math content: Another approach is to develop math content that may be more relevant
for students, particularly those
who are not pursuing STEM

16

fields, and that better supports
their acquisition of quantitative
reasoning skills. Diversificationin the college math courses taken
by non-STEM majors lends morecredibility to this approach. This
work is being pursued mainly incommunity colleges.
Few efforts lie at one end or theother. Indeed, there is a strong argument that both improved instructional approaches as well as updated
curricula are needed. However, thedominant K–12 and community
college math reforms are moving
in opposite directions in their treatment of Algebra 2. While K–12
schools implement Common Corealigned curricula that retain an algebra emphasis, alternative remedial
pathways that downplay second-year
algebra are gaining steam in community colleges.
Many agree about the value of Algebra 1 for a broad population of
students. But whether Algebra 2
should be required for all students,
regardless of educational or career
aspirations, has long been debated.
The course became a fixture of highschool curricula due to Ivy League
entrance requirements in the 1950s
and a drive to offer it to all students
grew from studies correlating thecourse with college success (AchieveInc., 2004).
Algebra 1 (often called Elementary

Algebra at the remedial level) introduces the basic concept of using
variables and abstract notation torepresent quantities one wants tomanipulate to solve problems. Algebra 2 (known as Intermediate Algebra at the college level) builds on
those concepts, but requires greateruse of specific procedural formulas
such as solving quadratic equations
or manipulations of logarithmic
functions.
“Algebra 2 for all students is a very
unrealistic course, and always has
been,” says Phil Daro, one of theauthors of the Common Core State
Standards for Math and former director of the California Math Project. “Real teachers can’t stand up
there for weeks at a time with the
majority of kids not understanding
a thing. Teachers have had to accommodate reality in various ways.
The name Algebra 2 is slapped ona wide variety of courses, students
pass the courses, but the vast majority of college-going kids learn very
little of what’s in the syllabus.”
Indeed, analyses of national tests aswell as studies of high school transcripts have found a growing gap between the titles of courses and theircontent (Loveless, 2013). This partially explains large college remedial
enrollments, even among students
who pass high school math courses.
California State University admits,for example, have passed Algebra 2

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

linking knowledge, policy and practicewith at least a C, but a third don’t
pass the placement exam for college-level courses, despite earning amean high school GPA of 3.17.
Concerns about college readinesshave led some states to require Algebra 2 for high school graduation. Minnesota and Connecticut,
for example, have recently added
it. National initiatives such as theAmerican Diploma Project, led by
the organization Achieve (which
now runs one of the new Common
Core-aligned testing consortia),
have promoted this emphasis to ensure more students have the mathbackground that many colleges expect. By 2012, about three quartersof U.S. students, including a greater
proportion of minority students,
were taking Algebra 2 in high school(Loveless, 2013).
On the other hand, the argumentthat algebra-intensive requirements
crowd out other courses that might
be more beneficial for many students is gaining traction. In an analysis of the math demands of firstyear community college courses, theNational Center on Education andthe Economy found that students
need to use statistics, data analysis,
and applied geometry—topics thatare rarely taught in schools—more
often than Algebra 2. To ensurethat students acquire these skills,
the report recommended ending
the movement to require Algebra

2 for all students. “To require thesecourses in high school is to deny tomany students the opportunity tograduate high school because they
have not mastered a sequence ofmathematics courses they will neverneed,” the report said.
A few states have also dropped
the course from their high school
graduation requirements for all students. These include Texas, which
originated the movement to require
Algebra 2. In 2013, the Texas legislature reversed the policy, allowing
students to opt out of Algebra 2 infavor of courses in statistics and algebraic reasoning. The move mainly
affects students planning to attend
community colleges (or no college),as most universities still require
two years of algebra for admission.
The situation is similar in California, where students need two yearsof math (not including Algebra 2)
to graduate, but at least three years(through Algebra 2) to enter public
universities.
Common Core Implementationin K–12. The Common Core standards being implemented in California and most other states retain the
traditional emphasis on two yearsof algebra and a year of geometry
in keeping with conventional standards. Yet, they represent a fresh approach to math content within traditional topic lists because of theiremphasis on practice standards.

These include math competencies
such as problem solving, adaptive
reasoning, and procedural fluency.
The standards also acknowledge
that certain high school topics areneeded only by students pursuing
STEM fields. (See Box, A Fresh Approach, on p. 18)
Organized to promote what Schoenfeld calls “sense-making,” not justcontent knowledge, the new standards are designed to provide students with a stronger groundingin basic arithmetic and pre-algebra
topics in the early grades to support
eventual success with high school
content. They also constitute a
break from the prior standards’ reliance on discrete skills, in favor ofa more holistic approach to conceptual understanding.
The standards were developed toaddress deficits of the previous standards, including the charge thatthey were “a mile wide and an inch
deep.” In practice, though, when
it came to high school mathematics, despite adding some probability
and statistics standards, the authors
kept most of Algebra 2 in order toensure that the standards would be
deemed rigorous.
Some observers are optimistic thatthese standards can lead to instructional improvements that will help
more students proceed further intraditional math topics. Schoen-

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation17

feld is among them. “If we make asignificant investment in the mathematical understanding of teachers
and in teacher preparation, a lot
more people would get over the algebra hump than currently do,” hesaid. “Most kids would be able to go
further in the math curriculum because it would make sense.”
There was interest among somedevelopers of the standards in introducing a second pathway in the
junior year similar to Texas’ move,but time constraints and politi-

cal realities limited their ability tomake such major shifts. Instead,they attempted pragmatic reforms
to Algebra 2, said Daro. “We tried
to make it a more reasonable course
for regular people. We did that. We
could have done better. We were up
against the same thing that everybody else trying to reform this is up
against: the mentality that we needto know who to flunk,” he recalled.

ultimately be placed on specifictopics versus competency in thepractice standards. And it will takemany more years before the successof Common Core will be known.
Most observers say the standards
will succeed in sparking significant
change only if they are accompanied
by robust investments in teacher
training and professional development, and it is far from clear thatthese will materialize.

Until the Common Core-based assessments are implemented, it will
be hard to know what weight will

A Fresh Approach: While the Common Core State Standards cover the traditional math topics at the high schoollevel, they embrace the idea of alternative pathways in several ways:• Notably, they include a set of“practice standards” that transcendmath content and address specificcapacities—such as problem solving, adaptive reasoning, conceptual understanding, and procedural fluency—that students needin order to have a mature grasp ofmathematics.
• The standards are explicitly neutral
on curriculum or course names,
telling instructors that various
pathways can be used to reach theoutcomes. While reaching a certain benchmark on the tests will
designate a student as ready tostart college without taking remedial math, there are several versionsof how the student can get there.
The authors give two examples of18

curricular pathways for teaching thestandards, and acknowledge roomfor others. The traditional sequence
consists of Algebra 1 in ninth grade,
Geometry in 10th grade, and Algebra 2 in 11th grade, while the integrated sequence weaves those topics
together at each grade level.
• There are basic math standards, designed for all students, as well as anadditional set of “plus” standards forstudents intending to pursue STEM
majors or attend more selective universities.
• Though the standards call for four
years of math, the assessments will
be based on three years of material,
so some students who are not interested in STEM fields or don’t want
to apply to selective colleges may opt

out of the fourth year.
• Other students who wish to takeCalculus or another college-levelclass while they are still in highschool (for example, to boost theirchances of admission to selective
colleges or majors) can take a compressed or accelerated version of
the curriculum, effectively covering five years’ worth of material infour years.
• To deal with the difficulty thatsolving exponential equations
presents for many students, theemphasis was placed on modelingwith exponential functions, not on
manipulating expressions.

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

linking knowledge, policy and practice

Alternative Remedial Pathways in
Community Colleges. If the Common Core standards are successful in motivating improvements inmath instruction, the need for college remedial courses could, in time,
dramatically decrease even withoutchanges in colleges’ math instruction. However, some are skepticalthat will happen, noting that similar
claims were made when the 1989
math standards were introduced.
“That’s asking for a very long-termchange in the whole culture of teaching and learning and our whole viewof what it means to do math,” notes
Stigler. “That’s a very long process.”
Poor teaching has become so calcified in the traditional math pathway, he believes, that new standards
will not be enough to change howteachers teach.
Even if the new standards succeed, it
will take many years before they are
fully integrated into K–12 schools,
and students steeped in them beginenrolling in college. So, at least for
now, remedial math education willremain a fixture of many public colleges. The idea of alternative pathways is gaining particular traction
as a reform strategy among leadersof efforts to improve remedial education and thereby increase graduation rates at community colleges.
Fueled by foundation support andpolicymaker concern about highfailure rates, experimentation is tak-ing place in states including Texas,
Colorado, Georgia, Ohio, and California.
In many states, the highest level remedial course is Intermediate Algebra, which corresponds loosely tohigh school Algebra 2. The course
is intended to prepare students forCollege Algebra, though neither
course has a very precise definition.
Intermediate Algebra courses arecoming under scrutiny, especiallygiven the backdrop of shifting undergraduate math requirements. As
colleges continue to adopt alternatives to Calculus and College Algebra for their credit-bearing classes,
some have begun to question the
purpose of prerequisites like Intermediate Algebra.
In 2014, the American Mathematics Association of Two-Year Colleges (AMATYC) passed a resolution
saying that Intermediate Algebra
should not be a “universal prerequisite for all college-level mathematics
courses. College‐level courses outside of the calculus‐based course of
study can be better served by otherprerequisite courses that are more
appropriate and relevant for preparing students for non‐STEM coursesof study. The content of a course,
as defined by the course description
and learning outcomes, should determine its mathematical level, prerequisites, and transferability.”

While acknowledging that an alternative like statistics, too, can betaught in an ineffective, procedural
fashion, advocates think it stands a
better chance of being taught wellfor several reasons. First, they say
the challenge of changing the way
teachers teach math may be easier
to address with fresh content. Secondly, statistics is more immediately
useful to students for things thatinterest them like analyzing sports
data, following political polls, orunderstanding medical research.
And for college students, it’s lesslikely to be a course that they took
(and struggled with) in high school.
Though there isn’t much evidence(aside from some very recent experiments that will be detailed in thesecond report), advocates are hopeful that new curricula will address
the problem highlighted in 2003 by
Carnevale and Desrochers:
Too many students get bogged
down in the abstract proceduresthat remain the focus of muchof the current mathematics curriculum. Others know the formulas and procedures but do
not understand what they know
well enough to use mathematics outside mathematics class….
Ultimately we will need a curriculum that teaches these higher
level quantitative reasoning skills
in a more applied and accessible
context in which the goal is bothknowledge and understanding.

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

19

The experiments are beginning toreport surprisingly strong results.
In its second year, working with 23
institutions in five states, the Carnegie Foundation’s Statway program
tripled remedial students’ passage ofcollege-level gateway math courses
in half the time. An evaluation ofeight early adopter colleges in the
California Acceleration Project saw
success in gatekeeper math courses
improve more than fourfold compared to the standard sequence. A
somewhat newer companion effort
in Texas, the New Mathways Project,reports similar early findings, but its
first evaluation was not yet available
when this report was completed.
Other approaches are being implemented in states including Georgia,
Colorado, and Massachusetts. The
promise of these experiments as well
as obstacles to their success will be
examined further in the second report in Degrees of Freedom.
Diversification and Its Dilemmas
Given the traditional prominenceof math requirements in the undergraduate curriculum, the implementation of alternatives faces ahost of challenges. Some obstacles,
such as tension over academic turf,
may be political in nature. But others are more substantive, such as the
economic need to encourage more
students to pursue the STEM disciplines that rely on training in algebra and calculus. The prospect for

20

scaling the alternative approaches,particularly at the pre-college level,
will depend largely on how the dilemmas are resolved.
Rigor and requirements. For one
thing, the alternative approaches
face scrutiny about their level ofrigor. There is little uniformity, especially when looking across educational segments. Some Introductory Statistics courses are actuallyquite advanced, assuming a background in calculus. These tend tobe the type that more senior mathprofessors are familiar with. But infact, such courses constitute only 11
percent of introductory statistics enrollments (CBMS, 2010).
Technology use has made statistics more accessible by eliminatingfrom introductory courses a great
deal of the computation on which
it is based. “Calculus lays the foundation of statistical theory, but the
majority of students don’t need tounderstand calculus to understandIntroductory Statistics,” says Rebecca Nichols, Director of Education for the American Statistics
Association. “They need to understand data, how to make decisions
based on data, and what questions
to ask before believing the results ofa study.”
Among AP courses, Statistics istreated as on a par with Calculus.
For transfer from community col-leges to public four-year universities
in many states, Statistics is generally
considered the equivalent of College Algebra. There is no data on
how many Statistics courses—or
which ones—require algebra, butthere is increasing evidence from experiments with alternative curriculathat many Algebra 2 topics are notrequired for students to learn statistics. “They need concepts from
Algebra 2 to do statistics, but notnecessarily the entire course,” said
Nichols.
A recent study at the City University of New York (CUNY) impliedthat Statistics students might noteven require Algebra 1, or Elementary Algebra. Students who placed
into elementary algebra were randomly assigned into one of two Elementary Algebra courses or into an
Introductory Statistics class with aweekly workshop. The study foundthat most high school graduates who
tested into Elementary Algebra but
skipped it were able to pass Statistics. It’s unclear whether this provesthat the students don’t need evenintroductory algebra. It could also
show that students retained enoughof high school algebra to do well in
Statistics, despite a low score on theplacement exam.
While the CUNY finding heartened
proponents of compressing students’
remedial sequences to speed theirentry into college-level courses, it

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

linking knowledge, policy and practicealso could make statistics vulnerable to being labeled insufficiently
rigorous to be a college-level course.
“They really do know that statisticalmaterial, but it’s not based upon astrong understanding of mathematics,” observed Wade Ellis, a retired
community college mathematician.
The rigor question cuts both ways.
More selective institutions that bydefinition cannot admit all students
often use rigor as a screen to eliminate students who haven’t passed
difficult courses. Broad-access institutions like community colleges arein a different situation. Under pressure to increase graduation rates,
many are focused on rooting out
unnecessary barriers and reject theidea of rigor for its own sake. Given
that some elite public institutions
enroll large numbers of community
college transfer students, there is an
inherent tension between the two
perspectives.
Articulation and alignment. A
related challenge for alternatives iswhether they will be accepted byfour-year universities. Though somefour-year institutions have embraced
alternatives to College Algebra suchas Statistics, most have been slower
to consider alternatives to Algebra
2 (for entering freshmen) or Intermediate Algebra (as a remedial prerequisite for the math courses community college students can use fortransferring).

Such policies present challenges togathering evidence on the effectiveness of the alternative sequences.
Despite their growth at community
colleges, many college leaders are
reluctant to expand the alternative
remedial programs without assurance that their students can transfer
smoothly. Challenges around alignment and transferability are the focus of the next report in this series.
Another dilemma is that the success, if it continues, of the remedialcourses could call into question highschools’ continued reliance on more
traditional math courses. Higher
education leaders have for years
been advocating that K–12 schools
focus more on college preparation,
but now, just as K–12 reforms are
moving beyond rhetoric to incorporate college readiness as an operating
principle, higher education’s definition of math readiness—on whichthose reforms have been based—isin flux.
The notion of eliminating Algebra 2
in high school appears to give educators the greatest pause. Years ofeffort have been expended ensuringthat more students have access tothe courses that are correlated withhigher education opportunities.
Access to algebra, in fact, has been
considered a civil rights issue. Thishistory leaves many cautious that alternative pathways would create separate but unequal tracks. “It’s an in-teresting idea, fraught with danger,”says Schoenfeld. “In the languageof the 1989 math standards, the socalled ‘non-college-intending’ tracks
all turned out to be dead ends.”
UCLA’s Green is only slightly lesswary. “One of the problems this effort faces is that, on the one hand,the need for math keeps going up
and getting more diverse, but at thesame time, students’ preparation
for it may not be keeping pace,” henoted. “You have a danger of people
being limited throughout their lives
by what math they got early on—or
didn’t. There’s a lot of stuff that uses
Algebra 2, and students who don’ttake it may be unaware that they are
limiting their options later on.”“On the other hand,” he acknowledges, “it’s much better to havesomeone who genuinely understands
modeling and quantitative reasoning and has a feeling for statistics
than someone who took an Algebra
2 class but is totally bewildered byit.” He believes that it’s incumbent
on schools offering alternatives toinform students that such courses
won’t prepare them for various scientific fields.
For now, a tentative consensus isemerging among some proponentsof alternatives that high school is
too early for students to opt out of
Algebra 2. Thus, most of the experiments are occurring at the college

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

21

remedial level. College students,
after all, are in a position to knowwhether they will be pursuing a fieldthat requires calculus. High school
students, arguably, are not.
“I love Intermediate Algebra,” said
Malcolm Adams, a University ofGeorgia mathematician who participated in his state’s task force toreform math curricula. “I believe
everybody should see that at least
once. I just don’t believe that everybody should be forced to be an expert at it.”Nevertheless, it’s not hard to predictthat if the early success of the community college alternatives is borneout in future experiments, discussion over alternatives in K–12 will
only accelerate. Some observers expect future revisions of high school
standards to incorporate the idea ofa second pathway. “It’s too early toback off the Algebra 2 expectation
for high schools,” said David Spence,president of the Southern Regional
Education Board, a regional association that promotes college readinessand other educational improvement
initiatives in the South. “I think it
will take four or five years for things
to coalesce around some different
pathways. The idea of less emphasison the upper reaches of Algebra 2
for college readiness probably will
creep into high schools.”

22

For now, K–12 leaders who think
students need more grounding inmath topics other than Algebra 2
are monitoring the outcomes of thecommunity college experiments—
plus a few taking place within
K–12.
The Los Angeles Unified School District (LAUSD), for example, is collaborating with researchers at UCLA
on a new data science class they describe as a “computation-based statistics and probability class.” Lastfall, 10 teachers began the first pilots.
According to the course description,
the prerequisites are the same as foran Algebra 2 class, and the class will
satisfy the math requirements for admission to UC and CSU in lieu of
Algebra 2. Students who complete
the course will be eligible to take AP
Statistics. The course is part of a larger effort funded by NSF to expose
more LAUSD students to statistics
and computer science.
California’s state legislature is also
growing impatient about students’high school preparation. It recently
passed two bills on high school computer science courses. One asks the
public universities to develop guidelines for computer science courses tofulfill math requirements for admission. The other will allow districts requiring more than two math courses
for graduation to count computerscience courses toward graduation.

Bridges and barriers. Interestingly,
both sides of the debate on alternatives are grounded in a concern
about eliminating barriers for students: While proponents of alternative pathways view algebra-intensive
curricula as a potential barrier tostudents’ success, critics fear that an
education without advanced algebra
itself constitutes a barrier.
Take the LAUSD data science course.
There remains the possibility thatsome students taking it will discover
belatedly that they want to become
engineers or physicists and won’t
have the necessary math preparation.
That concern is one of the main arguments against alternatives. Proponents of alternatives say this risk
isn’t sufficient reason to require all
students to take the standard curriculum—or pre-emptively fail them.
“The sequence of abstract highschool mathematics courses that prepares students for advanced degrees
in mathematics and science is still
crucial to our advanced economy,
but moving the entire school-age
population through the academichierarchy as a sorting strategy for
producing elite mathematical talentrequired of a small share of college
majors and fewer than 5 percent of
the workforce does not match wellwith our more general needs for applied reasoning abilities and practical numeracy,” wrote Carnevale andDesrochers in 2003.

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

linking knowledge, policy and practiceOne solution is to make bridge
courses—effectively remedial courses—available for students who needto make up material. Such programs
are common in other fields: Students
who discover after college that theywant to become doctors, for example, can enroll in post-baccalaureate
premedical programs. The Carnegie
Foundation will design math bridge
courses for community college students who take a statistics or quantitative reasoning pathway only to
discover an interest in STEM. The
idea also makes sense for community college students in associate degree or certificate programs who later discover they want to major, say,in math. Some universities that are
addressing remediation challenges
are also interested in the idea.
“It seems to me that would be amore constructive approach,” says
Harry Hellenbrand, a provost at Cal
State Northridge. “Considering theamount of money we waste by people flunking out of these remedialcourses, I would guess that if we put
one tenth of that into preparing thebridge classes well, we would havemore money for the math programsin general.”Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

23

Looking Ahead
As educational leaders look to improve high school graduation rates
and streamline transfer, especiallyamong minority and low-income
students, math requirements are a
critical lever. Whether students areapplying to college, transferringfrom a two-year to a four-year university, choosing an undergraduate
major or applying to medical school,
math represents one of the greatest
keys to success. Unfortunately, for
too many students, the very courses
intended to support that success are
instead functioning as barriers.
Changing this will likely require a
multi-faceted strategy that includes
improvements to instruction at alllevels. Indeed, every segment of public education in California—from
K–12 schools to community colleges to the two university systems—isengaging in innovations to ensurethat students have essential mathpreparation. For now, however, eachis pursuing these innovations largely
in isolation. With little understanding of what the others are doing, theinevitable tensions among them are
not transparent, leaving insufficient
clarity about what preparation isneeded and by whom. Curriculum
standards and university admission requirements often stand in asmeans of communication across theeducational segments.

24

Making sure students are successful in required math courses entails
improving how those courses aretaught, but it also involves making
sure the requirements are appropriate, accurately communicated, sufficiently aligned across segments,
and relevantly assessed. As reports
two and three in this series will discuss, California has much work to
do in this area to ensure that mathrequirements can support student
success and advancement.
Undergirding the series of reports
are a few assumptions. While there
is unquestionably a need to improve
instruction beginning in elementary
school, as the Common Core State
Standards are designed to do, institutions of higher education cannot
afford to wait a dozen years untilcohorts of students fully exposed to
the new standards reach their doors.
California’s schools and colleges
need to consider practical approaches that will best serve the students
they have now, while simultaneously investing in and evaluating
longer-term strategies.

New policies and practices
should be driven by several principles:• Keep the focus on maximizing student success statewide;• At the college level, base remedial policies and other requirements on equipping students
for subsequent coursework,
career, and citizenship (rather
than abstract notions about
desired knowledge);• Increase communication sosegments can better understand the implications of
their policies on students inother segments and minimize
misalignment;• Apply greater scrutiny to academic requirements noted forhigh failure rates; and• Make room for experimentation and innovation whilegathering evidence.

The second and third reports inthis series will include specific recommendations about policies andpractices that do this. This reportconcludes with a set of principles
that will underscore those specificrecommendations.

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

linking knowledge, policy and practice

References
Achieve, Inc. (2004). The expectations gap: A 50-state review of highschool graduation requirements.
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1). Meeting the challenges of improved post-secondary education in
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Burdman, P. (2013). Changing
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Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

linking knowledge, policy and practiceDegrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

27

About the Author
Pamela Burdman is a Berkeleybased policy analyst on college access, readiness, and success and aformer program officer at the William and Flora Hewlett Foundation.
She previously served as a journalist.
Her writing has appeared in the San
Francisco Chronicle, The New YorkTimes, San Jose Mercury News, andother publications.
Acknowledgments
The research and writing of this report was funded by LearningWorks.
The views expressed are those of theauthor and do not necessarily reflect
the views of PACE, LearningWorks,
or their funders.

About Policy Analysis for
California Education (PACE)
Policy Analysis for California Education is an independent, nonpartisan research center based at
Stanford University, the University of California-Davis, and the
University of Southern California.
PACE seeks to define and sustain along-term strategy for comprehensive policy reform and continuous
improvement in performance at alllevels of California’s education system, from early childhood to postsecondary education and training.
PACE bridges the gap between
research and policy, working withscholars from California’s leading
universities and with state and local
policymakers to increase the impactof academic research on educational
policy in California.

About LearningWorks (LW)
LearningWorks aims to strengthen
student achievement in community
colleges. It does so by facilitating,disseminating and funding practitioner-informed recommendations
for changes at the system and classroom levels, infusing these strategies
with statewide and national insights.
LearningWorks has been supportedby the William and Flora Hewlett
Foundation, the Walter S. Johnson
Foundation and the James IrvineFoundation.
http://www.learningworksca.org

http://www.edpolicyinca.org

28

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation

Degrees of Freedom:
Diversifying Math
Requirements forCollege Readinessand Graduation(Report 1 of a
3-part series)
Pamela Burdman
April 2015

linking knowledge, policy and practice

http://www.edpolicyinca.org

linking knowledge, policy and practice

http://www.learningworksca.org