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Critical Issues in Mathematics Education Series, Volume 5 May 2009 Workshop Report Prepared by Yvonne Lai Teaching Undergraduates Mathematics Workshop Organizing Committee

Critical Issues in Mathematics Education Series, Volume 5
May 2009 Workshop Report
Prepared by Yvonne Lai
A Workshop Publication Sponsored by MSRI with Generous Support from
National Science Foundation
Math for America
Noyce Foundation
Workshop Organizing Committee
William McCallum Deborah Loewenberg Ball
Rikki Blair David Bressoud
Amy Cohen-Corwin Don Goldberg
Jim Lewis Robert Megginson
Robert Moses James Donaldson
Mathematical Sciences Research Institute
17 Gauss Way, Berkeley, CA 94720-5070
510-642-0143 FAX 510-642-8609
Preface iii
List of Mathematics and Physics Problems v
List of Figures vi
List of Tables vii
1 Demands balanced in mathematics teaching 1
1.1 Articulation between high school and college 1
1.2 Relationships with other disciplines 6
Summary and further reading 13
2 What does the literature suggest? 15
2.1 Describing mathematical understanding 15
2.2 Describing problem solving 19
2.3 Knowledge for teaching 23
Summary and further reading 27
3 Portraits of teaching and mathematics 29
3.1 Teaching with inquiry, action, and consequence 29
3.2 Advanced mathematics from an elementary standpoint 31
3.3 Inquiry-oriented differential equations 33
Summary and further reading 35
4 Assessments 37
4.1 Force Concept Inventory and related diagnostic tests 38
4.2 Basic Skills Diagnostic Test and Calculus Concept Inventory 41
4.3 The Good Questions Project 45
Summary and further reading 48
5 End Notes 49
Bibliography 51
This booklet, based upon a May 2009 workshop
at MSRI titled Teaching Undergraduates Mathe-
matics, is written for undergraduate mathemat-
ics instructors who are curious what resources
and research may support the philosophy and
practice of their mathematics teaching.
Ten years ago the AMS report Towards Excel-
lence argued that “to ensure their institution’s
commitment to excellence in mathematics re-
search, doctoral departments must pursue ex-
cellence in their instructional programs.” Math-
ematicians in all collegiate institutions share
the common mission of teaching mathematics
to undergraduate students, and the common
problem that transitions from high school to
college and from 2-year to 4-year college are
challenging for many students. The success of
a mathematics program depends on habits of
learning and quality of instruction.
The following questions guided the workshop:
Research. What does research tell us about
how undergraduate students learn mathemat-
ics? Are we listening to and learning from that
Curriculum. How do considerations of de-
sign and assessment of courses and programs
enhance the success of our teaching? What
works at different types of institution (com-
munity colleges, four-year liberal arts colleges,
comprehensive universities, and research in-
tensive universities) and different student au-
diences (mathematics majors, engineers, scien-
tists, elementary teachers, business majors)?
Pedagogy. How does the way we teach influ-
ence our ability to recruit students to mathe-
matically intensive disciplines or to retain the
students we have? Can research experiences
play an important role in exciting students to
learn mathematics? How can technology be
harnessed to help undergraduates learn mathe-
matics and to help departments deliver instruc-
tion efficiently?
Articulation with High Schools. What math-
ematical knowledge, ability, and habits does a
high school graduate need for success in mathe-
matics in college? Do AP and concurrent enroll-
ment courses lead to the same learning as their
traditional on-campus counterparts? Is there
a need for greater articulation of high school
and collegiate mathematics? What mathemat-
ical and cultural problems do students have in
their transition from high school to college, and
what programs should colleges have that ad-
dress these problems?
The audience for the workshop included math-
ematicians, mathematics educators, classroom
teachers and education researchers who are
concerned with improving the teaching and
learning of mathematics in our undergraduate
classrooms. The workshop showcased courses,
programs and materials whose goal is to in-
crease students’ knowledge of mathematics,
with an emphasis on those that show promise
of being broadly replicable.
I thank the organizers for putting together a
lively workshop, and the numerous presenters
for sharing their work. I also thank David
Bressoud, Ed Dubinsky, Jerome Epstein, Karen
Marrongelle, and William McCallum for pro-
viding insightful feedback, and David Auckly
and Amy Cohen-Corwin for support in prepar-
ing this for publication. In writing this re-
port, I have drawn significantly from work-
shop conversations and events, especially the
presentations of David Bressoud, Marilyn Carl-
son, Bill Crombie, Wade Ellis, Jerome Epstein,
Deborah Hughes-Hallett, John Jungck, Karen
Rhea, Natasha Speer, Maria Terrell, and Joseph
Wagner. I gratefully acknowledge the work-
shop presenters and participants for inspiring
the contents of this report.
Structure of this booklet
Chapter 1discusses demands balanced in math-
ematics teaching, in particular, the articulation
between high school and college mathematics
teaching and learning, and the relationship of
mathematics to partner disciplines. The content
focus of this chapter is calculus, which plays a
central role in many collegiate programs.
Chapter 2summarizes some findings from the
mathematics education literature: ways to ob-
serve mathematical understanding, phases of
mathematical problem solving, and what is en-
tailed in teaching mathematics in the K-20 set-
ting. This chapter showcases ideas that help
us describe teaching and learning activities, so
that we can better see and hear our students and
Chapter 3provides snapshots of teaching: us-
ing inquiry to understand algebraic concepts,
teaching calculus concepts well before a for-
mal introduction to calculus, and using inquiry
to structure an ordinary differential equations
class. The purpose of this chapter is to provide
glimpses of teaching in action.
Chapter 4discusses several assessment projects:
the Force Concept Inventory and related diag-
nostic tests from physics, which inspired the
creation of the Basic Skills Diagnostic Test and
Calculus Concept Inventory, as well as the
Good Questions Project. This chapter highlights
findings from studies using these instruments.
It is often easier to understand ideas through
examples. Throughout this booklet are sam-
ple problems from the projects and assessments
discussed. The following page contains a list of
these mathematics and physics problems.
List of Mathematics and Physics
2.1 Bottle Problem ............................................ 19
2.2 Temperature Problem ........................................ 19
2.3 The Ladder Problem ........................................ 19
2.4 Paper Folding Problem ....................................... 21
2.5 The Mirror Number Problem ................................... 22
2.6 P´
olya Problem ............................................ 22
2.7 Car Problem ............................................. 22
2.8 Slicing with Planes ......................................... 22
2.9 Motivating Induction ........................................ 23
2.10 Mathematical Knowledge for Teaching Subtraction ...................... 23
3.1 Area Problem ............................................ 32
4.1 Mechanics Diagnostic Problem .................................. 38
4.2 ConcepTest (Blood Platelets) .................................... 41
4.3 Place Value .............................................. 42
4.4 Proportional Reasoning (Piaget’s Shadow Problem) ...................... 43
4.5 Exponential Reasoning ....................................... 43
4.6 Proportional Reasoning (Numbers Close to Zero) ....................... 44
4.7 Height Problem ........................................... 46
4.8 Repeating 9’s ............................................. 47
4.9 Adding Irrationals ......................................... 47
List of Figures
1.1 Timeline of AP Calculus events, 1950-1987. ........................... 2
1.2 High school mathematics course enrollments, 1982-2004. ................... 3
1.3 Fall mathematics course enrollments in 2-year colleges, 1985-2005. .............. 5
1.4 Percentage of students in 4-year programs enrolled in calculus or above, 1985-2005. .... 5
1.5 Fall mathematics course enrollments in 4-year programs, 1985-2005. ............. 5
1.6 Fall enrollments in Calculus I vs. AP Calculus exams taken, 1979-2009. ........... 5
1.7 Fall enrollment in Calculus II, 1990-2005. ............................ 6
1.8 Fall enrollment in Calculus III and IV, 1990-2005. ........................ 6
1.9 Intended and actual engineering majors, 1980-2008. ...................... 7
1.10 Prospective engineers vs. total fall calculus enrollments, 1980-2008. ............. 7
1.11 Intended and actual majors in selected STEM fields, 1980-2006. ................ 8
1.14 Quantitative concepts for undergraduate biology students. .................. 12
2.2 A geometric representation of inverse as process. ........................ 16
3.1 Set-up for inquiry questions on slope. .............................. 30
3.2 Set-up for inquiry questions about angle. ............................ 31
3.3 Diagram for the Area Problem. .................................. 32
3.4 Defining problems of elementary calculus. ........................... 32
3.5 Sequence of regions used to solve the Area Problem. ...................... 32
List of Tables
1.12 Some biological phenomena and their associated curves. ................... 10
1.13 Biological phenomena associated to graphs. ........................... 10
2.1 Action and process understandings of function, with examples. ............... 17
2.3 Mental actions during covariational reasoning. ......................... 20
2.4 Questioning strategies for the mental actions composing the covariational framework. . . 20
2.5 Approaches to the Paper Folding Problem. ........................... 21
4.1 Average Mechanics Diagnostic Test results by course and professor. ............. 40
4.2 Force Concept Inventory pre- and post-test data. ........................ 40
4.3 Peer Instruction results using the Force Concept Inventory and Mechanics Baseline Test . 42
4.4 Data from Basic Skills Diagnostic Test .............................. 44
4.5 Results of the Good Questions Project. .............................. 47
Who we teach & what we teach:
Demands balanced in
mathematics teaching
Teaching entails many demands. Good instruc-
tion, in addition to conveying mathematics with
integrity, also
responds to students’ mathematical back-
grounds, and
serves students well for their future, inside and
outside of mathematics.
Integrity, responsiveness, and service are com-
peting principles. For example, what consti-
tutes a good mathematical explanation depends
on a students’ background the most econom-
ical or elegant explanation is not always the
most accessible. Instructors of prerequisite ser-
vice courses may need to negotiate mathemati-
cal coherence with the skills, habits, and dispo-
sitions needed by their students’ for their future
Because of its place in the curriculum, calcu-
lus is central to discussions about knowledge of
students and the mathematics they know. Cal-
culus is both an area with rich mathematical
foundations as well as a course prerequisite to
a host of disciplines: a calculus instructor must
balance integrity, responsiveness, and service.
Section 1.1 proposes a possible agenda for
improving calculus instruction. Investigating
what happens in high school calculus class-
rooms, as well as the motivation for taking cal-
culus, will give perspective on the mathemat-
ical background and needs of entering college
Among the students we teach are future rep-
resentatives of various disciplines. Section 1.2
discusses ways that mathematics and mathe-
matics classes interact with partner disciplines
in science, technology, and engineering.
1.1 Articulation between high
school and college teaching
and learning
Workshop presenter David Bressoud, then pres-
ident of the Mathematical Association of Amer-
ica, proposed in [5] that to serve their students
better, the mathematics community must:
Get more and better information about stu-
dents who study calculus in high school: What
leads high school students to take calculus, and
what are the benefits and risks to future mathe-
matical success of having taken high school cal-
culus classes?
ETS contracted to administer exams
for experimental high school program.
Birth of “Advanced Placement”:
285 students take CAAS exams in
10 subjects including math.
College Admission with Advanced Standing (CAAS) Study Committee on
Mathematics chaired by Professor Heinrich Brinkmann of Swarthmore
College. Representatives to committee come from Bowdoin, Brown,
Carleton, Haverford, MIT, Middlebury, Oberlin, Swarthmore, Wabash,
Wesleyan, and Williams.
Gov. Richard Riley (SC) passes
Education Improvement Act,
mandating AP access in all schools.
Joint statement by
concerned for high
school preparation
for college calculus
AP program launches Calculus AB
Total number of AP
Calculus exams taken
surpasses 50,000.
Year in Jaime Escalante’s
calculus class profiled by
Stand and Deliver.
Calculus is most commonly a
sophomore-level college course,
preceded by precalculus and
analytic geometry
NSF announces Calculus
Curriculum Development
Program, overseen by
DUE and DMS.
MAA CUPM report lays out undergraduate
curriculum, with calculus as centerpiece of
introductory mathematics; recommends that
“Mathematics 0” should be taught in high school.
Figure 1.1. Timeline of AP Calculus events, 1950-1987. For more details about
these events, see Bressoud’s articles [2][3][5][4][6][7].
Play a role in the design, support, and enforce-
ment of guidelines for high-school programs
offering calculus: High school calculus classes
must be designed to give students a solid math-
ematical preparation for college mathematics.
Re-examine first-year college mathematics:
There must be appropriate next courses that
work with and build upon the skills and knowl-
edge that students carry with them to college,
whether or not each student is ready for college
freshman calculus.
This section is based upon Bressoud’s articles
[2][4][5][6][7], which analyze the history of cal-
culus as a course in this country.
Section 1.1.1 summarizes how accountability,
along with two complementary and at times
conflicting ideals individual enrichment and
wide access contributed to the disarticulation
between high school Advanced Placement (AP)
and college calculus classes.
From a demographic perspective, high school
calculus enrollments have risen exponentially
since the first Advanced Placement Calculus
exam more than 50 years ago, while college cal-
culus enrollments have remained steady. Sec-
tion 1.1.2 discusses two NSF-sponsored stud-
ies, one with Bressoud as a Principal Inves-
tigator, which address Bressoud’s above pro-
posed agenda by identifying features of suc-
cessful high school and college mathematics ex-
Many students take AP Calculus more than
300,000 as of 2009; and calculus is foundational
in college curricula. Knowing more about AP
Calculus experiences and their impact on col-
lege learning are promising ways to understand
better the mathematical backgrounds and needs
of entering college students.
1.1.1 Disarticulation between high school
and college calculus
Enrollments in high school and college calcu-
lus courses are expressions of three ideals: ac-
countability, enrichment, and access. The ten-
sions across these ideals have contributed to the
disarticulation between high school and college
History and enrollment of calculus courses.
The Advanced Placement (AP) programs be-
gan more than fifty years ago, when calculus
was typically a college course for sophomores.
At this time, some leading collegiate institu-
tions formed the College Admission with Ad-
vanced Standing (CAAS) committee, which pi-
loted year-long programs aimed to enrich stu-
dents in selected strong high schools. The pro-
gram included end-of-year exams written by
what is now known as the College Board and
administered through the Educational Testing
Service. These programs eventually became
0% 50% 100%
Mathematics completed by
high school graduates
No math or low academic
Algebra I/Plane Geometry
Algebra II/Trigonometry/
Analytic Geometry
Figure 1.2. Percentage of high
school graduates who completed dif-
ferent levels of mathematics courses in
1982, 1992, 2004. Note that in 2004,
more than 75% of graduates had com-
pleted Algebra II or a more advanced
course, and more than 33% of grad-
uates had completed Precalculus or
a more advanced course. Data from
Dalton, Ingels, Downing, and Bozick
[14, p. 13].
what is now known as Advanced Placement
Both exam taking and mathematics course
enrollment have increased (see Figure 1.2).
Since its inception, the number of AP Calculus
exams taken has increased by several magni-
tudes of order. Given this dramatic shift, college
mathematics course enrollments are strangely
close to stagnant (see Figures 1.3-1.6) and may
potentially drop (as Section 1.2 discusses).
In 2-year programs, total mathematics en-
rollment during the fall term has remained at
roughly 25% of total enrollment in these col-
leges. But the percentage of mathematics enroll-
ment in precollege mathematics has increased
from 48% in 1980 to 57% in 2005 while the per-
centage of mathematics enrollment in calculus
and above has decreased from 9% to 6%. In 4-
year undergraduate programs, total mathemat-
ics enrollment during the fall term has dropped
from 20% of total undergraduate enrollment in
1980 to 15% in 2005. In 1980, 10% of all students
were taking a mathematics course at the level of
calculus or above in the fall term. By 2005, that
was down to 6%.
Thus, across all students, enrollment increase
in calculus and above has seen a modest in-
crease, but it is close to the increase in total col-
lege enrollments.
Accountability, enrichment, and access.
What might explain the simultaneous sec-
ondary expansion and tertiary stagnation?
The CAAS formed the Advanced Placement
program in the 1950’s to enrich students in high
schools already known for intellectual strength.
But, starting approximately twenty years later,
the public perceived the AP program as a vehi-
cle to find and help talented students regardless
of background. (The 1982 blockbuster Stand and
Deliver profiled Jaime Escalante’s AP Calculus
In 1986, the National Council of Teachers of
Mathematics (NCTM) and MAA issued a joint
statement warning students against taking cal-
culus in high school with the expectation of re-
taking it in college, entreating them instead to
spend time mastering the prerequisites of calcu-
lus. Whether the NCTM and MAA interpreted
the data accurately in the 1980’s, there seems to
be little effect from AP Calculus exam taking on
college mathematics enrollments.
One possible explanation for this contrast is
that accountability exacerbated the tension be-
tween enrichment and access. It is certainly
desirable to improve access to challenging, in-
teresting mathematics. However, AP Calculus
was not designed for mass expansion. Based on
conversations with students, Bressoud suspects
that many students take AP Calculus and col-
lege calculus not for the mathematics, but as a
step toward future employment. This suggests
that calculus is viewed as a course culminating
in a one-time test, rather than an opportunity
for mathematics to influence lifetime learning.
Some of Bressoud’s students arrived unpre-
pared for college-level calculus and its applica-
tions. Some remaining students, despite con-
tent mastery, arrived with visceral distaste for
mathematical study. Both cases are problematic.
The AP Calculus program strives to articulate
with college calculus. As part of regular mainte-
nance of the AP curriculum, the College Board
periodically surveys the calculus curricula of
the 300 tertiary institutions receiving the most
AP Calculus scores. However, history suggests
that topic lists alone cannot effect preparedness
in or appreciation of mathematics.
1.1.2 Articulation between high school
and college mathematics
Two studies, currently underway, support Bres-
soud’s proposed agenda (see the beginning of
Section 1.1). The Characteristics of Successful
College Calculus Programs (CSCCP), an NSF-
sponsored project headed by Bressoud, Mari-
lyn Carlson, Michael Pearson, and Chris Ras-
mussen will examine collegiate data via a sur-
vey conducted in Fall 2010; and Factors Influ-
encing College Success in Mathematics (FICS-
Math), a study out of Harvard, will examine
secondary data collected in Fall 2009.
Knowing students better. College mathe-
matics instructors must help students overcome
distaste and mischaracterization of mathemati-
cal study. A dangerous temptation is to treat
students as blank slates. However, personal
dispositions are not easily dislodged, even af-
ter hearing the statement of a better alternative
(e.g., Confrey [10]).
Instructional interventions must be finely tar-
geted, addressing clearly described problems
with well-defined goals. The CSCCP and FICS-
Math studies will give insight into college math-
ematics students as a whole. However, individ-
ual instructors should still engage in conversa-
tion with their own students about their mo-
tivations and background. Knowing their stu-
dents better will help instructors support math-
ematical learning, therefore supporting stu-
dents’ mathematical trajectory through college.
Guidelines for calculus. History suggests
that successful articulation between high school
and college calculus must go beyond lists of
topics. After all, instruction does not consist of
a collection of topics: it also includes interac-
tions between students and the topics, as well
as between the students and the teachers. The
CSCCP and FICS-Math studies will shed light
on these interactions, and how these may in-
form worthwhile guidelines for the design of
calculus in college and high school.
Re-examining first-year college mathemat-
ics. College calculus is where mathematics de-
partments interact with the most number and
variety of students. Moreover, it is most com-
monly a foundation for future study or a cap-
stone. In both cases, calculus should be an op-
portunity to influence the mathematical knowl-
edge and dispositions of undergraduate stu-
dents. To do so, instructors must better know
their students, and the content must also be bet-
ter suited to the mathematical backgrounds and
needs of the students. The CSCCP and FICS-
Math studies can inform the design of courses
to supplement or build upon calculus that will
be mathematically profitable for students.
Fall enrollments in 2-year
calc and advanced
Figure 1.3. Data compiled by Bres-
soud from CBMS data.
Percentage enrollment in
calculus or above
percentage of students in 4-year
undergraduate programs enrolled in
calculus or above
Figure 1.4. Slight drop in advanced
course taking. Data compiled by Bres-
soud from CBMS data.
Fall enrollment in 4-year
calculus level
Figure 1.5. Data compiled by Bres-
soud from CBMS data.
Fall enrollments in Calculus I
vs. AP Calculus exams taken
total AP exams
4-year colleges
2-year colleges
Figure 1.6. Nearly constant enroll-
ments vs. approximately exponential
exam taking. Data compiled by Bres-
soud from CBMS and ETS data.
Fall enrollments in Calculus II
4-year programs
2-year colleges
Figure 1.7. Fall enrollment in Calculus II, 1990-2005. Since 1995, there has
been a 22% decrease in the number of students taking Calculus II in the Fall term
in 2-year and 4-year programs.
Fall enrollments in Calculus III and Calculus IV
4-year programs
2-year colleges
Figure 1.8. Fall enrollment in Calculus III and IV, 1990-2005.
1.2 The role of mathematics
courses: relationships with
mathematics and other
“Most of our students,” Deborah Hughes-
Hallett opened her presentation, “will not go on
in mathematics. Most of our students are in our
classes because someone sent them there usu-
ally not themselves.”
Calculus is a pre-requisite for the STEM fields
of Engineering, Physics, Chemistry, and Mathe-
matics. It is sometimes a pre-requisite for for
Computer Science, and occasionally for Eco-
nomics and Biology. The data strongly sug-
gest that the number of prospective engineering
majors predicts fall calculus enrollments (see
Figures 1.9 and 1.10), and this population is
percentage-wise on the decline. If this trend
continues, the mathematics community should
expect dropping calculus enrollment.
At the same time, over the past twenty years,
prospective biological sciences majors are on
the rise (see Figure 1.11). Biology undergradu-
ate programs do not consistently require math-
ematics classes beyond calculus I for their ma-
jors, even though biological work uses mathe-
matics found in Calculus I, Calculus II, and Or-
dinary Differential Equations.
In serving the needs of other disciplines,
mathematics instructors face a disadvantage.
The majority of our students are in their first
two years of college, before they have taken
the courses that apply the mathematics found
in our courses, leaving our mathematics con-
textless. The students in our classes may not
be able to provide feedback on how to accom-
plish this mission. However, by conversing
with professors of their future courses, we may
be able to find out more. We highlight two
Engineering majors
actual majors
intended majors
Figure 1.9. Data compiled by Bressoud for the workshop from The American
Freshman and NCES data.
80,000 90,000 100,000 110,000 120,000
Number of prospective engineers vs. total fall calculus enrollments in research
prospective engineers
1995 2000
1990 2005
total fall calculus enrollment
R2 = 0.98146
Figure 1.10. Data compiled by Bressoud for the workshop from CBMS and CIRP
talks, one by Deborah Hughes-Hallett on the
MAA-CRAFTY (Curriculum Renewal Across
the First Two Years) project, discussed by Deb-
orah Hughes-Hallett; and one on curriculum
reform efforts, by John Jungck, one of the
founders of the BioQUEST Curriculum Consor-
tium (Quality Undergraduate Education Simu-
lations and Tools).
1.2.1 CRAFTY: Reports of conversations
with partner disciplines
An “asymmetry” lies between mathematics and
other disciplines. Math majors may have
taken a chemistry or physics course or two in
high school, but students in these fields may
well have been required to take two or more
semesters of mathematics courses in college.
In general, math majors are not required to take
more courses in any other particular scientific
field than members of that field are required to
take of mathematics courses. Thus, whether or
not other disciplines have an understanding of
mathematics in a way that we would character-
ize as accurate, it remains that they know our
courses in a way that we do not know theirs.
Reflecting upon conversations with col-
leagues, Hughes-Hallett recommends, “The
Fraction of incoming freshmen intending to major in ...
bio science
physical science
computer science
Source: The
American Freshman
Fraction of graduating seniors who majored in ...
bio science
physical science
computer science
Source: National
Center for
Education Statistics
Figure 1.11. Data compiled by Bressoud for the workshop from The American
Freshman and NCES data.
thing I have found most helpful is not whether
they need to know this topic or that topic,
because that shifts over time. Instead, what
is more helpful as a common thread is to ask
them what is useful about how they think about
mathematics.” The MAA CRAFTY project,
“Voices from the Partner Disciplines” [17], com-
piled reports from faculty in other disciplines
on what they would like to see in mathematics
courses their students take during the first two
years of college.
A few salient themes from the MAA-
CRAFTY project are stances on graphing calcu-
lators and conceptual understanding.
Our partner disciplines would like to see
our courses place more emphasis on approxi-
mation and estimation, and advocate spread-
sheet modeling rather than graphing calcu-
Excerpt from the CRAFTY Summary Rec-
ommendations for Understanding, Skills,
and Problem Solving [17]:
Emphasize conceptual understanding:
Focus on understanding broad concepts and ideas
in all mathematics courses during the first two
Emphasize development of precise, logical think-
ing. Require students to reason deductively from
a set of assumptions to a valid conclusion.
Present formal proofs only when they enhance
understanding. Use informal arguments and
well-chosen examples to illustrate mathematical
There is a common belief among mathematicians that
the users of mathematics (engineers, economists, etc.)
care primarily about computational and manipulative
skills, forcing mathematicians to cram courses full of
algorithms and calculations to keep “them” happy.
Perhaps the most encouraging discovery from the
Curriculum Foundations Project is that this stereo-
type is largely false. Though there are certainly in-
dividuals from the partner disciplines who hold the
more strict algorithmic view of mathematics, the dis-
ciplinary representatives at the Curriculum Founda-
tions workshops were unanimous in their emphasis on
the overriding need to develop in students a conceptual
understanding of the basic mathematical tools.
lators, which are rarely used in, for example,
physics, chemistry, biology, business, engineer-
ing, or information technology. In her conversa-
tions, Hughes-Hallett has heard repeatedly that
spreadsheets are consistently the “second best”
technology for working on a problem, and in
this way are fundamental to the toolkit of many
As far as conceptual understanding, the skills
regarded as essential by most partner disci-
plines include the concept of function, graphi-
cal reasoning, approximation and estimation of
scale and size, basic algebraic skills, and numer-
ical methods. For example, partner disciplines
would like students to:
“become very comfortable with the use of sym-
bols and naming of quantities and variables”
have an “understanding that many quantitative
problems are ambiguous and uncertain” and be
“comfortable taking a problem and casting it
in mathematical terms” (business and manage-
“summarize data, describe it in logical terms, to
draw inferences, and to make predictions” (bi-
“formulate the model and identify variables,
knowns and unknowns”, “select an appropri-
ate solution technique and develop appropriate
equations; apply the solution technique (solve
the problem); and validate the solution” (civil
Thus our mathematics courses should nurture
conceptual understanding, mathematical mod-
eling, facility with applications, and fluency
with symbols and graphs as a language tool.
On solution methods, almost all disciplines
broached the importance of fluency in numer-
ical solutions rather than analytical solutions.
However, more intricate problems in engineer-
ing may require understanding analytical solu-
tions so as to be able to validate numerical solu-
Partner disciplines value computational
skills. But, without a strong conceptual under-
standing, the computational skills become im-
potent. To understand this assertion, Hughes-
Hallett offered a quote from her colleague
Nolan Miller, a microeconomist at the Kennedy
“While much of the time in calculus courses is spent
learning rules of differentiation and integration, what is
more important for us is not that the students can take
complicated derivatives, but rather that they are able to
work with the abstract concept of ‘the derivative’ and un-
derstand that it represents the slope, that if u :R2R,
then u1/u2is the slope of a level surface of the function
in space.”
It may at first seem striking to separate the abil-
ity to do difficult derivatives from the ability to
capture a definition as a geometric object. How-
ever, these abilities are in fact distinct. One can
be quite skillful at “complicated derivatives”
while lacking the ability to verbalize concep-
tual understanding in a precise way and vice
Table 1.12. Some biological phenomena and their associated curves. Prepared
by John Jungck for this workshop.
Curve Biological phenomena
linear fat intake vs. cancer
log-linear log survival vs. dose radiation
log-log allometry
positively exponential exercise curve vs. O2
negatively exponential Newton’s law of cooling
gaussian variation
sinusoidal heart rhythm
logistic r,K
chaotic tribolium
rectangular hyperbolic Michaelis-Menten
elliptical phase predator-prey, PV loop
hysteresis DNA melting
Table 1.13. Biological phenomena associated to graphs. Prepared by John
Jungck for this workshop.
food webs brain circuits metabolic pathways
pedigrees phylogenies fate maps
interactomes microarray clusters linkage maps
restriction maps complementation maps nucleotide sequences
protein sequences 3-D protein backbones (or HP lattices)
1.2.2 Mathematics curricula and the
biological sciences
Biological research and mathematics. As
Jungck argued in his presentation, the num-
ber of biological science majors is on the rise,
and mathematics and biology faculty stand to
benefit from each others’ expertise. Biologi-
cal research depends on mathematical know-
how, and mathematicians can engage students
through mathematical modeling content.
Classically, understanding the dynamics of
biological phenomena required understanding
functions for example, linear, exponential,
chaotic, logistic functions. (See Table 1.12 for
examples.) “Part of the literacy for my biol-
ogy students,” Jungck observed, “is that when
they see their graphs of their data coming out in
these kind of forms, that they can begin to de-
velop a simple kind of intuition. We’re not ask-
ing them to remember the equation. But these
are familiar objects, an alphabet for thinking
about modeling. For many mathematical biolo-
gists, having this kind of repertoire of biological
examples that fit these kinds of things is kind of
like a beginning kind of language.”
More recently, the mathematics relevant to bi-
ological research has had more to do with rela-
tions than functions, and more to do with topol-
ogy than dynamics. Drawing a comparison with
families of functions, Jungck proposed, “We can
have a similar set of topologies of simple graphs
that almost every biologist would immediately
recognize, whether it’s a food web or a pedigree
or a phylogenetic tree or a metabolic pathway,
that these kinds of things are there. You have,
again, an advantage. You already know our
language, you already understand the topol-
ogy of these kinds of systems.” (See Table 1.13
for examples.) These systems deal with rela-
tions because they often feature many-to-one
and one-to-many maps, simultaneously. It is
in part due to mathematics that biologists can
work with this data; behind meaningful inter-
pretations of biological phenomena such as ro-
bustness or fragility are mathematics.
As early as 1996, Lou Gross proposed the
option of teaching the relevant mathematics
through biology departments:
It is unrealistic to expect many math faculty to have
any strong desire to really learn significant applications
of math that students will readily connect to their other
course work, though there is a core group who might do
So what do we do to enhance quantitative understand-
ing across disciplines? Below is what I say to life science
faculty: Who can foster change in the quantitative skill of
life science students? Only you, the biologists can do this!
Two routes:
1. Convince the math faculty that they’re letting you
2. Teach the courses yourself.
Gross [19], as quoted in Jungck [26]
The disappointment launched at mathematics
faculty resulted from lack of immediate rele-
vance of mathematics coursework to biological
applications. Even if a student could in the-
ory derive the mathematics from starting prin-
ciples, it is not the ability to use basic principles
that is the most critical it is the ability to apply
the mathematics after the derivations have fin-
ished. Application and derivation are distinct
areas of mathematical fluency, and teaching one
does not ensure expertise in the other.
BioQUEST and lessons learned. The relation-
ship between mathematics, computer science,
and biological research motivated the found-
ing of BioQUEST (Quality Undergraduate Ed-
ucation Simulations and Tools), which sought
deep reform of the undergraduate biology pro-
gram. The BioQUEST curriculum consortium
began as a collection of mathematicians, com-
puter scientists, philosophers of science, science
and math educators, biology educators, and
biology researchers. In 2005, BioQUEST con-
vened kindred programs who sought to effect
change in undergraduate education, includ-
ing the Harvard Calculus Consortium, Work-
shop Mathematics Project, Project CALC, and
C*ODE*E (Consortium of ODE Experiments).
At the workshop Investigating Interdisciplinary
Interactions: Collaboration, Community, & Con-
nections, these programs met with others from
biological sciences, computer science, statistics,
and physics, among other disciplines.
John Jungck, one of the initial founders of
BioQUEST, has found that discussions about an
individual course or an individual department
tend to be ineffective for the reform-oriented.
“Frankly, if you want to change the culture to
a more learner-centered student achievement,
you may find your best ally in someone in a cog-
nate discipline, and they may already be con-
nected to a national curricular initiative. I urge
you to expand your community to beyond the
peers in the next-door office.” He pointed out
that as partner disciplines, we write and read
one another’s grants, retentions, promotions,
and awards. In our academic environment, we
rely on each other; our curricula and teaching
should reflect this.
To borrow an idea from anthropology, pop-
ularized by Silicon Valley, we need to “cross
the chasm.” Jungck advocates looking for al-
lies in other disciplines and other schools, and
to maintain a broad view. Enthusiasts must
be able to work with, convince, and talk to
many departments in schools of a variety of per-
suasions community colleges, Research-I, lib-
eral arts, small state schools, historically black
schools, predominantly undergraduate institu-
tions. To go beyond the “early adopters” of nu-
clear, local projects, and reach a national or in-
ternational perspective, the earlier enthusiasts
must demonstrate success in a variety of con-
Principles for Biology classes. Biology depart-
ments require mathematics courses, yet their
coursework may not use mathematics. The Na-
tional Research Council [12] supports the inclu-
sion of more mathematics in biology courses:
Given the profound changes in the nature of biology
and how biological research is performed and communi-
cated, each institution of higher education should reexam-
ine its current courses and teaching approaches to see if
they meet the needs of today’s undergraduate biology stu-
dents. Those selecting the new approaches should consider
the importance of building a strong foundation in mathe-
Quantitative concepts for undergraduate
biology students (Lou Gross)
Rate of change
specific (e.g. per capita) and total
discrete - as in difference equations
continuous - calculus-based
Notion of a perturbation and system re-
sponse to this.
Alternative definitions exist including not
just whether a a system returns to equilib-
rium but how it does so.
Multiple stable states can exist - initial con-
ditions and the nature of perturbations (his-
tory) can affect long-term dynamics
there are diverse methods to display data
Simple line and bar graphs are often not suf-
Non-linear transformations can yield new
Figure 1.14. Quantitative concepts
used in biology (adapted from Gross
matics, physical, and information sciences to prepare stu-
dents for research that is increasingly interdisciplinary in
character. The implementation of new approaches should
be accompanied by a parallel process of assessment, to ver-
ify that progress is being made toward the institutional
goal of student learning.” (p. 44)
“Concepts, examples, and techniques from mathemat-
ics, and the physical and information sciences should be
included in biology courses, and biological concepts and
examples should be included in other science courses. Fac-
ulty in biology, mathematics, and physical sciences must
work collaboratively to find ways of integrating mathe-
matics and physical sciences into life science courses as
well as providing avenues for incorporating life science
examples that reflect the emerging nature of the discipline
into courses taught in mathematics and physical sciences.”
(pp. 47-48)
Some quantitative concepts, compiled by Lou
Gross, are shown in Figure 1.14.
If the average grade of a pre-med student in
a calculus class is an A, then biology classes
from lower-division to upper-division courses
should use calculus. Jungck has written that the
“exclusion of equations in [biological] textbooks
has three unfortunate consequences; namely, a
lack of respect for, consistency with, and em-
powerment of students” [26, p. 13]. Without
more mathematics, biology classes are guilty
of the same. Using the mathematics shows
respect for the discipline of mathematics as
well as students’ intellectual capabilities. Cur-
rently, only upper-division courses use calculus.
The lack of consistency between lower-division
courses and upper-division courses causes de-
skilling and frustration in students. One form
of empowerment is economic access, and lack
of mathematics “has differential career conse-
quences” [26, p. 13]. There is a strong, positive
correlation between the amount of mathematics
and computer sciences that biologists have had
and their professional career opportunities and
advancement (e.g., Gross [19]).
We end with a quote from Jungck.
Go to your library and open a variety of biological
journals; the diversity and richness of mathematics therein
may surprise you. Why shouldn’t this literature be acces-
sible to far more of our students?
John Jungck , in [26].
Summary and further reading
Mathematics plays a variety of roles in the pur-
suit of disciplinary knowledge: it gives ways to
express quantities and concepts, to approximate
and estimate, to model and predict real-life phe-
nomena, to prove, to derive, and to problem
solve. Each of these domains is distinct from the
rest, and expertise in one area does not guaran-
tee expertise in the rest. Mathematics and our
partner disciplines would like service courses to
nurture fluency in all these domains.
Those who have been heavily invested in
teaching mathematics in service courses have
found that relevance and respect can help over-
come mathematical fears and dislikes. Rel-
evant material can interest students; relevant
skills align with applications to the majors we
serve. Respecting students must include build-
ing upon students’ prior knowledge and ex-
periences rather than ignoring or denying that
students come in with ideas about content and
what it means to do mathematics; respect also
includes supporting a variety of future course-
work in as direct a manner as possible. To re-
spect students and teach relevant material, indi-
viduals of the mathematics community need to
find out more about their students’ experiences
in high school, and to interact with partner dis-
ciplines at local institutions.
References and readings by presenters or rec-
ommended by presenters include the following.
Experiences in engaging students
The Algebra Project. A national, nonprofit orga-
nization that uses mathematics as an organizing
tool to ensure quality public education for every
child in America.
The Young People’s Project. Uses math and me-
dia literacy to build a network of young peo-
ple who are better equipped to navigate lifes
circumstances, are active in their communities,
and advocate for education reform in America.
Mathematics and Theoretical Biology Institute. The
efforts of this institute has significantly in-
creased the national rate of production of U.S.
Ph.D.’s since the inception of the institute, and
recognizes the need for programmatic change
and scholarly environments which support and
enhance underrepresented minority success in
the mathematical sciences.
BioQUEST. This project supports undergradu-
ate biology education through collaborative de-
velopment of open curricula in which students
pose problems, solve problems, and engage in
peer review.
MathForLife. An innovative one semester ter-
minal mathematics course intended to replace
existing core or terminal courses ranging from
”math-for-poets” to Finite Math whose primary
audience is the undergraduate majoring in the
humanities or social sciences.
Ten Equations that Changed Biology: Mathematics
in Problem-Solving Biology Curricula. (Article by
John Jungck.)
Meeting the Challenge of High School Calculus.
(Series by David Bressoud, as part of his online
column, Launchings from the CUPM Curricu-
lum Guide)˜bressoud/pub/
BIO2010: Transforming Undergraduate Educa-
tion for Future Research Biologists. (Report by
the National Research Council Committee on
Undergraduate Biology Education to Prepare
Research Scientists for the 21st Century.)
Curriculum Foundations Project: Voices of the Part-
ner Disciplines. (CRAFTY report.)
Math & Bio 2010: Linking Undergraduate Disci-
plines. (MAA publication, edited by Lynn Steen)
Quantitative Biology for the 21st Century. (Gives
concrete examples, with references, of biological
research strongly influenced by mathematical
and statistical sciences. Report by Alan Hast-
ings, Peter Arzberger, Ben Bolker, Tony Ives,
Norman Johnson, Margaret Palmer.)
Teaching problem solving and
understanding: What does the
literature suggest?
“Procedural knowledge” versus “conceptual
learning”, “teacher-directed instruction” versus
“student-centered discovery”: these debates
distract the community with false dichotomies
and vague premises.
With this opening, Marilyn Carlson called at-
tention back to foundational questions:
What does it mean for students to understand a
mathematical idea?
What are problem solving abilities and processes for
mathematics learners?
What is the nature of the knowledge that teachers
need to have?
This chapter summarizes and elaborates upon
Carlson’s presentation.
A challenge to common ground on “under-
standing” is that many topics in mathemat-
ics have no widely accepted specification on
what it means “to understand”. Promisingly,
there are key topics of secondary and tertiary
mathematics whose learning has been exam-
ined in detail. One such topic is (real) functions.
This chapter discusses two alternative charac-
terizations of understanding functions, Action-
Process-Object-Schema (APOS) Theory and Co-
variation. In its treatment of APOS Theory, this
chapter focuses on Action and Process.
With respect to the second question, vari-
ous researchers and mathematicians have stud-
ied the teaching and learning of problem solv-
ing. To support problem solving in mathemat-
ics classes, this chapter describes stages of prob-
lem solving as examined by Carlson and her
colleagues. This work builds upon literature by
olya and Schoenfeld among others.
Finally, there is currently no broad consen-
sus on the nature of the knowledge needed for
teaching, which is problematic for TA training
programs as well as K-12 teacher preparation
programs. We discuss research on the mathe-
matical knowledge entailed in teaching, includ-
ing research on tertiary instruction presented by
Natasha Speer and Joe Wagner.
2.1 Describing mathematical
understanding: Functions
Algebra is a gateway class: completing mathe-
matics beyond the level of Algebra II correlates
significantly with enrollment in a four-year col-
lege and graduation from college (e.g., National
Mathematics Advisory Panel, [29, p. 4]).
At the heart of school algebra are func-
tions, especially linear, quadratic, and expo-
nential functions. Two characterizations of un-
derstanding functions prevalent in the litera-
ture on undergraduate mathematics are Action-
Process-Object-Schema (APOS) Theory and Co-
variation. Mathematicians may be interested in
these ideas as ways to help observe and assess
their students’ thinking.
2.1.1 Action and process understandings
“Action” and “process” are part of Dubinsky’s
APOS Theory (Action-Process-Object-Schema;
see Dubinsky and McDonald [15] for an intro-
duction). There are four stages to Dubinsky’s
theory, inspired by Piaget’s developmental the-
ories on children’s learning; this section concen-
trates on the first two stages, Action and Pro-
Although this chapter as a whole focuses on
algebraic concepts, Section 2.1.1 provides exam-
ples from exponential expressions and group
theory as well, intending that a greater vari-
ety of examples will provide more leverage for
readers to apply APOS Theory to their own
Examples regarding functions in the text and
the tables are from Oehrtman, Carlson, and
Thompson [28] and Connally, Hughes-Hallett,
Gleason, et al. [11]. Examples regarding expo-
nential expressions are from Weber [39]. Ex-
amples regarding group theory and the descrip-
tions of action and process stages are from Du-
binsky and McDonald [15].
f(x) = y
f (y) = x
process for f
process for f
Figure 2.2. A geometric representa-
tion of inverse as process.
Action. An action on a set of mathematical
objects is a step-by-step transformation of the
objects to make another mathematical object or
objects. A student in the action stage of un-
derstanding an object can likely, for instance,
perform algorithmic computations on those ob-
jects. The student also likely needs prompting
to take the action.
For example, in the action stage of under-
standing a particular function for gexpressed
in terms of x, students can likely evaluate f(x)
or even g(f(x)) for given x. However, students
may not be able compose functions whose data
is given to them only through tables and graphs
(e.g., see Table 2.1). As well, the understanding
of functions as primarily step-by-step manipu-
lations comes with implications for understand-
ing of graphs, inverses, and domain and range.
In the case of exponential expressions, stu-
dents can view 23as repeated multiplication of
2, but may not be able to make sense of non-
integral exponents or logarithms.
In the case of group theory, students can com-
pute the left cosets of {0, 4, 8, 12, 16}in Z/20Z
by adding elements of the whole group to el-
ements of subgroup. However, such students
may encounter difficulty with more intricate
structures, such as for cosets of D4, the sym-
metry group of a square within a permutation
group such as S4. Students may be able to
compute through brute force, but would not be
likely to find efficient, holistic techniques.
Process. When a student repeats an action
and reflects upon it, they internalize the ac-
tion into a process, which may no longer need
external prompting to perform. “An individ-
ual can think of performing a process with-
out actually doing it, and therefore can think
about reversing it and composing it with other
processes”[15, p. 276].
In the process stage of understanding func-
tions, students can likely find simple composi-
tions from tables and graphs; as well, the con-
cepts of injectivity, inverse function, and do-
main and range are more accessible. Examples
are provided in Tables 2.1, and a geometric rep-
resentation of inverse as process is provided in
Figure 2.2.
Table 2.1. Action and process understandings of function (adapted from Oehrt-
man, Carlson, and Thompson [28]). Each understanding is followed by examples of
the types of problems (adapted from [28] and Connally, Hughes-Hallett, Gleason,
et al. [11]) that a student in that stage could likely complete.
Action understanding Process understanding
Working with functions requires the comple-
tion of specific rules and computations.
Inverse is about algebraic manipulation,
for example, solving for yafter switching
yand x; or it is about reflecting across a
diagonal line.
Finding the domain and range is at most
an algebraic manipulation problem, for ex-
ample, solving for when the denominator is
zero, or when radicands are negative.
Examples of problems solvable with an action
Find h(y), where h(y) = y2, and y=5.
Find f(g(x)) for f(x) = 4x3,g(x) = x+1,
and x=2.
Given f(x) = 2x+1
7x, find f1(x).
Given the graph of f(x), sketch a graph of
Find the domain and range of f(x) = 1+x
If the graph of an invertible function is con-
tained in the fourth quadrant, what quadrant
is the graph of its inverse function contained
Working with functions involves mapping a
set of input values to a set of output values;
it is possible to work with a space of inputs
rather than just specific values.
Inverse is the reversal of a process that de-
fines a mapping from a set of output values
to a set of input values.
Domain and range are produced by op-
erating and thinking about the set of all
possible inputs and outputs.
Examples of problems solvable with process
Express (fg)1as a composition of the
functions f1and g1.
Simplify cos(arcsin t)using the notion that
an inverse “undoes”.
A sunflower plant is measured every day t,
for t0. The height, h(t)centimeters, of the
plant can be modeled with
h(t) = 260
What is the domain of this function? What
is the range? What does this tell you about
the sunflower’s growth? Explain your
Use the figures below to graph the func-
tions f(g(x)),g(f(x)),f(f(x)),g(g(x)).
2-1 10
2-1 10
These problems are adapted from Connally, Hughes-Hallett, Gleason, et al., Functions Modeling Change:
A Preparation for Calculus,§2.2: Example 3, and §8.1: Problems 27-30, c
2006, John Wiley & Sons, Inc.
This material is reproduced with permission of John Wiley & Sons, Inc.
In the case of exponential expressions, a stu-
dent can likely interpret bxas “the number that
is the product of xfactors of b and logbmas
“the number of factors of bthat are in the num-
ber m [39].
In the case of left cosets, the student can likely
find at least two elements g,hS4not in the
subgroup D4and so that gand hrepresent dis-
tinct left cosets.
Applying the notions of action and process
understandings to teaching. APOS Theory
can guide in-class activities, exam problems, or
homework. Below are several recommenda-
tions to help students advance from action un-
derstanding to process understanding. Sugges-
tions on teaching functions are taken from [28]
unless otherwise noted.
Ask students to explain basic function facts in
terms of input and output.
Examples. (a) Ask students to explain their rea-
soning for whether (fg)1equals f1g1or
(b) In addition to questions such as “Solve for
xwhere f(x) = 6”, ask students to “find the
input value(s) for which the output of fis 6”,
both algebraically and from a labelled graph of
the function, and to explain their reasoning.
Ask about the behavior of functions on entire in-
tervals in addition to single points.
Examples. (a) Ask students to find the image
of a function applied to an infinite-cardinality
set (such as an interval), e.g., find the length of
f(g([1, 2]), where f(x) = 2x+1 and g(x) =
(b) Ask students to find the preimage of an in-
terval in the context of the definition of limit or
Ask students to make and compare judgements
about functions across multiple representations,
that is, how a function is introduced or what in-
formation students are given about the function.
Ask students to describe symbols as mathemat-
ical objects.
Examples from [39], with desired student re-
sponses given in bold. (a) Describe each of the
exponential expressions in terms of a product
and in terms of words:
=the number that is the product of
3 factors of 4
=the number that is the product of
xfactors of b
(b) Simplify each of the expressions below by
writing each exponential term as a product.
Summarize each simplification in words.
The product of 2 factors of band
4 factors of bis 6 factors of b.
| {z }
The product of band xfactors of b
is (x+1) factors of b.
Incorporate computer software packages that
help students visualize or experiment with
mathematical concepts, and use computer pro-
gramming to help students reflect upon actions.
A description of a number of studies in which
computer software and programming aided
student learning can be found in Dubinsky and
Tall [16]; in fact, the examples on exponen-
tial expressions, from [39], are part of a study
which included MAPLE programming activities
for the students.
2.1.2 Covariational reasoning
The Oxford English Dictionary defines covari-
ant as, “Changing in such a way that interre-
lations with another simultaneously changing
quantity or set of quantities remain unchanged;
correlated.” In studying students’ learning of
functions, Carlson has focused on helping stu-
dents relate dependent quantities. This section
presents some of her findings, especially from
Carlson, Jacobs, Coe, Larsen, and Hsu [9] and
Oerhtman, Carlson, and Thompson [28].
In [9], covariational reasoning is described as
the “cognitive activities involved in coordinat-
ing two varying quantities while attending to
the ways in which they change in relation to
each other”(p. 354), for example, viewing
(x,y)=(t,t31)as expressing a relationship
where xand ycan both change over time, and
changes in xmay come with changes in y.
Covariational reasoning means attending to
co-varying quantities in contexts such as para-
metric equations, physical phenomena, graphs,
and rates of change. Precalculus, calculus, mul-
tivariable calculus and differential equations all
feature simultaneously varying quantities.
The following are three examples of prob-
lems, from [9] and [28], whose solutions entail
covariational reasoning.
Problem 2.1: Bottle Problem.
Imagine this bottle filling with water. Sketch a graph
of the height as a function of the amount of water that
is in the bottle.
In this case, quantities to attend to are height
and volume of water. Covariation appears
through applying concepts related to rate of
change and convexity.
Problem 2.2: Temperature Problem.
Given the graph of the rate of change of the temper-
ature over an 8-hour time period, construct a rough
sketch of the graph of the temperature over the 8-hour
time period. Assume the temperature at time t =0
is zero degrees Celsius.
rate of change
of temperature
2 4 6 8
Here, quantities to attend to are the rate of
change and the original function. Covariation
appears through the interpretation of critical
points, positive slopes, and negative slopes.
Problem 2.3: The Ladder Problem.
From a vertical position against a wall, a ladder is
pulled away at the bottom at a constant rate. De-
scribe the speed of the top of the ladder as it slides
down the wall. Justify your claim.
Here, quantities to attend to are the speed of
the top of the ladder and the placement of the
bottom of the ladder.
One way that studies in math education can
serve mathematics instructors is elaborating
what it means to “understand”, and how stu-
dents arrive at understanding. Observations of
students working on problems similar to the
above suggest that covariational reasoning de-
composes into five kinds of mental action; this
led Carlson, Jacobs, Coe, Larsen, and Hsu to de-
velop interventions that improved calculus stu-
dents’ covariational reasoning abilities [9]. The
mental actions are summarized in Tables 2.3-
Ways suggested in [9] to enhance students’
covariational reasoning may include:
Ask for clarification of rate of change informa-
tion in various contexts and representations.
For example, ask students to provide interpre-
tations about rates in real-world contexts, given
algebraic or graphical information. Probe fur-
ther if students do not incorporate all variables
in their explanation, and the relationship be-
tween the variables. If students use phrases
such as “increases at a decreasing rate”, ask
them to explain what this means in more detail.
Ask questions associated with each of the men-
tal actions. Questioning strategies are found
in Table 2.4 for discussing rates of changes, a
concept foundational to calculus and differen-
tial equations.
2.2 Describing problem solving
Mathematics instructors often would like their
students to be problem solvers: to celebrate
mathematical tasks that are not immediately
Table 2.3. Mental actions during covariational reasoning (adapted from [28, p.
163]). Behaviors are those observed in students working on the Bottle Problem.
description of mental action behaviors
Action 1
Coordinating the dependence of one
variable on another variable
Labeling axes, verbally indicating the
dependence of variables on each other
(e.g., ychanges with changes in x)
Action 2
Coordinating the direction of change of
one variable with changes in the other
Constructing a monotonic graph
Verbalizing an awareness of the direc-
tion of change of the output while con-
sidering changes in the input.
Action 3
Coordinating the amount of change in
one variable with changes in the other
Marking particular coordinates on the
graph and/or constructing secant lines.
Verbalizing an awareness of the rate of
change of the output while considering
changes in the input.
Action 4
Coordinating the average rate-of-
change of the function with uniform
increments of change in the input
Marking particular coordinates on the
graph and/or constructing secant lines.
Verbalizing an awareness of the rate of
change of the output (with respect to the
input) while considering uniform incre-
ments of the input.
Action 5
Coordinating the instantaneous rate-of-
change of the function with continuous
changes in the independent variable for
the entire domain of the function.
Constructing a smooth curve with clear
indications of concavity changes
Verbalizing an awareness of the instan-
taneous changes in the rate-of-change
for the entire domain of the function
(direction of concavities and inflection
points are correct).
Table 2.4. Questioning strategies for the mental actions composing the covari-
ational framework (adapted from [28, p. 164]).
mental action questioning strategy
MA1 What values are changing? What variables influence the quantity of interest?
MA2 Does the function increase or decrease if the independent variable is increased
(or decreased)?
MA3 What do you think happens when the independent variable changes in con-
stant increments? Can you draw a picture of what happens (at intervals) near
this input? Can you represent that algebraically? Can you interpret this in
terms of the rate of change in this problem?
MA4 Can you compute several example average rates of change, possibly using the
picture to help you? What units are you working with? What is the meaning
of those units?
MA5 Can you describe the rate of change of a function event as the independent
variable continuously varies through the domain? Where are the inflection
points? What events do they correspond to in real-world situations? How
could these points be interpreted in terms of changing rate of change?
Table 2.5. Two approaches to the Paper Folding Problem.
Approach 1 Approach 2
Let xbe the interior altitude of the
black triangle. Then its area is 1
and the area of the lower white
region is 3 x2. The areas of the
white and black regions are equal, so
2x2. It follows that x=2.
The three regions in the figure below
have equal area; since the square has
total area 3, the black triangle and
white triangle form a square of area
2. Hence the diagonal of this square
has length 22. The length sought is
half the diagonal, or 2.
solvable, to confront these tasks with a clear
head, and to think carefully and completely.
The mathematician P´
olya, in his famous How
to Solve It [30], characterized four stages:
understanding the problem “we have to see
clearly what is required”
developing a plan “we have to see how the
various items are connected, how the unknown
is linked to the data, in order to obtain an idea
of the solution, to make a plan.”
carrying out the plan
looking back “we look back at the completed
solution, we review it and discuss it.”
Almost all research on mathematical problem
solving traces back to P´
olya’s insight.
Though P´
olya’s stages have much face va-
lidity to mathematically-savvy problem solvers,
carrying them out in practice has remained
mathematically and affectively difficult for stu-
dents in general. One of the key players in
mathematical problem solving research is Alan
Schoenfeld, who developed a course that im-
proved undergraduates’ problem solving abil-
ities by focusing on metacognitive processes,
especially the relationship between students’
beliefs and their practices. The emphasis on
metacognitive processes was inspired by emer-
gent artificial intelligence research of the time,
and has continued to shape research since.
The previous section in this chapter, Sec-
tion 2.1, addressed conceptual knowledge; the
present section discusses the relationship be-
tween conceptual knowledge, affect, and prob-
lem solving, as suggested by Carlson and
Bloom [8]. The purpose of this section, as with
Section 2.1, is to provide description that can
help instructors sharpen observations of stu-
dent thinking, and to provide language that fa-
cilitates conversation among colleagues around
teaching and learning.
2.2.1 Conceptual knowledge and emotion
Carlson and Bloom observed a dozen mathe-
maticians and PhD candidates solving a collec-
tion of elementary mathematics problems de-
signed to evoke problem solving processes. The
collection included the bottle problem (from
Section 2.1.2), and the Paper Folding Problem.
Problem 2.4: Paper Folding Problem.
A square piece of paper ABCD is white on the
frontside and black on the backside and has an area of
3in2. Corner A is folded over to point A0which lies
on the diagonal AC such that the total visible area is
half white and half black. How far is A0from the fold
Two approaches to the problem are summa-
rized in Table 2.5.
Carlson and Bloom found that P´
olya’s stages
could delineate stages of behavior observed in
their participants. So, within each stage, Carl-
son and Bloom examined how problem solvers
used conceptual knowledge, applied heuris-
tics, exhibited motivation and emotion, and re-
flected upon their own work.
Conceptual knowledge and problem solv-
ing. Mathematicians do powerful planning
when problem solving. Carlson noted in her
presentation that the mathematicians studied in
[8] often exhibited rapid 20-minute cycles, stop-
ping to ask themselves about a particular line of
reasoning: “Hmm ... should I do that? Maybe
I should plug in some numbers. If I do that,
then I will get this relationship between the tri-
angle and the square . . . This sort of construc-
tion and evaluation expertly applies conceptual
knowledge of area and triangles. When the
problem solver who used Approach 2 initially
went off track, his knowledge of area allowed
him to get unstuck.
The detail and organization in planning re-
minded Carlson of mathematicians discussing
precalculus material. Carlson observed in her
presentation, “It’s one thing to say ‘distance’.
It’s another thing to say, dis the number of
miles that it takes that it takes to drive from
Phoenix and Tucson.’ Whatever the mathe-
matical level, problem solving entails thought-
ful, non-linear processes that draw upon careful
connections to conceptual knowledge.
Conceptual knowledge was also used to ver-
ify answers. The successful problem solvers in
[8] used their understanding of areas and geom-
etry to check results and computations, for ex-
ample, by making certain that the areas of the
regions totaled to 3.
Emotion and problem solving. Carlson and
Bloom were struck by the intimacy of the solv-
ing process. In their participants, they ob-
served frustration, joy, and the pursuit of ele-
gance. Successful problem solving entails effec-
tive management of emotion, especially to per-
sist through many false attempts.
2.2.2 Sample problems
For readers’ interest, the following were other
tasks used in the study reported in [8].
Problem 2.5: The Mirror Number Problem.
Two numbers are “mirrors” if one can be obtained by
reversing the order of the digits (i.e., 123 and 321 are
mirrors). Can you find: (a) Two mirrors whose prod-
uct is 9256? (b) Two mirrors whose sum is 8768?
Problem 2.6: P´olya Problem.
Each side of the figure below is of equal length. One
can cut this figure along a straight line into two
pieces, then cut one of the pieces along a straight line
into two pieces. The resulting three pieces can be fit
together to make two identical side-by-side squares,
that is a rectangle whose length is twice its width.
Find the two necessary cuts.
Problem 2.7: Car Problem.
If 42% of all the vehicles on the road last year were
sports-utility vehicles, and 73% of all single car
rollover accidents involved sports-utility vehicles,
how much more likely was it for a sports-utility ve-
hicle to have such an accident than another vehicle?
2.2.3 Intellectual need as motivation
We end the section on problem solving with an
argument for “intellectual need” and a nod to
olya’s teaching.
In his presentation on problem solving,
David Bressoud commented on materials by
olya and Guershon Harel introducing induc-
tion. A classic problem used by P´
olya in Let us
teach guessing (now a DVD, originally on film in
1966) is the following.
Problem 2.8: Slicing with Planes.
How many regions that one plane can slice R3into?
2planes? 3planes? n planes?
In this situation,
1 plane creates 2 regions
2 planes can create 4 regions
3 planes can create 8 regions.
Many students guess that 4 planes could cre-
ate 16 regions rather than the maximum of 15.
Expectations are overthrown. At this moment,
olya argues, learning can happen.
The “intellectual need” motivating genuine
mathematical reasoning has also been dis-
cussed by Guershon Harel. In a recent
MAA workshop, Harel characterized back-
wards teaching, which begins by generic out-
lines of techniques rather than a situation to mo-
tivate utility. For example, backwards teach-
ing of induction might begin by discussing row
of dominoes, and how knocking the first cas-
cades the rest down. However, this theoreti-
cal description will be meaningless to someone
who has never needed proof by induction. Only
providing examples where the induction state-
ment is explicitly in the problem statement, e.g.,
showing n
6, exacerbates
the situation. When taught this way, students
tend to look for an n in the problem, and miss
opportunities to use proof by induction when
there is no apparent n”, even in contexts where
induction provides a productive approach.
Harel proposes opening with problems with
implicit inductive statements, for example:
Problem 2.9: Motivating Induction.
Find an upper bound for the following sequence.
2, q2+2, r2+q2+2, . . .
The idea of building upon a previous case is
something that students understand intuitively;
the domino analogy only tells them something
they feel they already know, without anchors
to any mathematics. But providing a problem
where the students must focus on how to build
rather than merely the fact that something is
being built serves students’ needs. This prob-
lem provides intellectual need to articulate pat-
terns in terms of previous patterns, motivating
the use of an index variable as well as recursive
2.3 Knowledge for teaching
Mathematics teachers teach mathematics; and
teaching entails skills beyond the content aimed
to students. For example, consider this ques-
tion, about subtraction in Thames [37]:
Problem 2.10: Mathematical Knowledge for
Teaching Subtraction.
Order these subtraction problems from easiest to
hardest for students learning the standard subtrac-
tion algorithm, and explain the reasons for your or-
dering: (a) 322 115 (b) 302 115 (c) 329 115.
Engaging in this sort of reasoning un-
prompted is an instance of mathematical think-
ing that skillful teaching requires, and which we
do not requir