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Critical Issues in Mathematics Education Series, Volume 5

May 2009 Workshop Report

Prepared by Yvonne Lai

Teaching

Undergraduates

Mathematics

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 ◦www.msri.org

Contents

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

i

Preface

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

research?

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.

Acknowledgements

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

ourselves.

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

Problems

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

v

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

vi

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

vii

CHAPTER 1

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

courses.

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

students.

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?

1

2 CHAPTER 1. DEMANDS BALANCED IN MATHEMATICS TEACHING

1952!

ETS contracted to administer exams

for experimental high school program.

1953!

Birth of “Advanced Placement”:

285 students take CAAS exams in

10 subjects including math.

1956!1965!1969!1982!

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.

1984!

Joint statement by

NCTM and MAA

concerned for high

school preparation

for college calculus

AP program launches Calculus AB

1986!

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.

1987!

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-

periences.

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

classes.

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

3 1.1. ARTICULATION BETWEEN HIGH SCHOOL AND COLLEGE

0% 50% 100%

1982

1992

2004

Mathematics completed by

high school graduates

No math or low academic

math

Algebra I/Plane Geometry

Algebra II

Algebra II/Trigonometry/

Analytic Geometry

Precalculus

Calculus

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

(AP).

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

class.)

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.

4 CHAPTER 1. DEMANDS BALANCED IN MATHEMATICS TEACHING

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.

5 1.1. ARTICULATION BETWEEN HIGH SCHOOL AND COLLEGE

0

200000

400000

600000

800000

1000000

1200000

1985

1990

1995

2000

2005

Fall enrollments in 2-year

colleges

precollege

introductory

calc and advanced

Figure 1.3. Data compiled by Bres-

soud from CBMS data.

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

1985

1990

1995

2000

2005

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.

0

100000

200000

300000

400000

500000

600000

700000

800000

1985

1990

1995

2000

2005

Fall enrollment in 4-year

programs

precollege

introductory

calculus level

advanced

Figure 1.5. Data compiled by Bres-

soud from CBMS data.

0

50,000

100,000

150,000

200,000

250,000

300,000

350,000

1980

1985

1990

1995

2000

2005

2010

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.

6 CHAPTER 1. DEMANDS BALANCED IN MATHEMATICS TEACHING

0

50

100

150

1980

1985

1990

1995

2000

2005

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.

0

50

100

150

1980

1985

1990

1995

2000

2005

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

disciplines

“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

7 1.2. RELATIONSHIPS WITH OTHER DISCIPLINES

0

20,000

40,000

60,000

80,000

100,000

120,000

140,000

160,000

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

Engineering majors

actual majors

intended majors

1985

2005

2000

1990

1995

Figure 1.9. Data compiled by Bressoud for the workshop from The American

Freshman and NCES data.

150000

175000

200000

225000

250000

80,000 90,000 100,000 110,000 120,000

Number of prospective engineers vs. total fall calculus enrollments in research

universities

prospective engineers

1995 2000

1990 2005

1985

total fall calculus enrollment

R2 = 0.98146

Figure 1.10. Data compiled by Bressoud for the workshop from CBMS and CIRP

data.

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

8 CHAPTER 1. DEMANDS BALANCED IN MATHEMATICS TEACHING

0.0%

2.0%

4.0%

6.0%

8.0%

10.0%

12.0%

14.0%

1980

1984

1988

1992

1996

2000

2004

2008

Fraction of incoming freshmen intending to major in ...

engineering

bio science

physical science

computer science

mathematics

Source: The

American Freshman

0.0%

2.0%

4.0%

6.0%

8.0%

10.0%

12.0%

14.0%

1980

1984

1988

1992

1996

2000

2004

2008

Fraction of graduating seniors who majored in ...

engineering

bio science

physical science

computer science

mathematics

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-

9 1.2. RELATIONSHIPS WITH OTHER DISCIPLINES

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

years.

•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

structure.

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

disciplines.

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”

(physics),

•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-

ment),

•“summarize data, describe it in logical terms, to

draw inferences, and to make predictions” (bi-

ology),

•“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

engineering).

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-

tions.

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

School:

“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 :R2→R,

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

versa.

10 CHAPTER 1. DEMANDS BALANCED IN MATHEMATICS TEACHING

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-

11 1.2. RELATIONSHIPS WITH OTHER DISCIPLINES

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

this.

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

down

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-

texts.

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-

12 CHAPTER 1. DEMANDS BALANCED IN MATHEMATICS TEACHING

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

Stability

•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

Visualizing

•there are diverse methods to display data

•Simple line and bar graphs are often not suf-

ﬁcient.

•Non-linear transformations can yield new

insights.

Figure 1.14. Quantitative concepts

used in biology (adapted from Gross

[18]).

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].

13 1.2. RELATIONSHIPS WITH OTHER DISCIPLINES

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.

Articles

•Ten Equations that Changed Biology: Mathematics

in Problem-Solving Biology Curricula. (Article by

John Jungck.)

http://papa.indstate.edu/amcbt/volume_

23/

•Meeting the Challenge of High School Calculus.

(Series by David Bressoud, as part of his online

column, Launchings from the CUPM Curricu-

lum Guide)

http://www.macalester.edu/˜bressoud/pub/

launchings/

Reports

•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.)

http://www.nap.edu/catalog.php?record_

id=10497

•Curriculum Foundations Project: Voices of the Part-

ner Disciplines. (CRAFTY report.)

http://www.maa.org/cupm/crafty/

•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.)

http://www.maa.org/mtc/Quant-Bio-report.

pdf

CHAPTER 2

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

P´

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-

15

16 CHAPTER 2. WHAT DOES THE LITERATURE SUGGEST?

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-

cess.

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

teaching.

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

x

f (y) = x

-1

process for f

process for f

-1

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.

17 2.1. DESCRIBING MATHEMATICAL UNDERSTANDING

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

understanding:

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

7−x, ﬁnd f−1(x).

Given the graph of f(x), sketch a graph of

f−1(x).

Find the domain and range of f(x) = √1+x

x+3.

If the graph of an invertible function is con-

tained in the fourth quadrant, what quadrant

is the graph of its inverse function contained

in?

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

understanding:

Express (f◦g)−1as a composition of the

functions f−1and g−1.

Simplify cos(arcsin t)using the notion that

an inverse “undoes”.

A sunﬂower plant is measured every day t,

for t≥0. The height, h(t)centimeters, of the

plant can be modeled with

h(t) = 260

1+24(0.9)t.

What is the domain of this function? What

is the range? What does this tell you about

the sunﬂower’s growth? Explain your

reasoning.∗

Use the ﬁgures below to graph the func-

tions f(g(x)),g(f(x)),f(f(x)),g(g(x)).∗

-2

0

f(x)

x

-2

2-1 10

-1

1

2

-2

0

g(x)

x

-2

2-1 10

-1

1

2

∗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.

18 CHAPTER 2. WHAT DOES THE LITERATURE SUGGEST?

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,h∈S4not 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 (f◦g)−1equals f−1◦g−1or

g−1◦f−1.

(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) =

4x−3.

(b) Ask students to ﬁnd the preimage of an in-

terval in the context of the deﬁnition of limit or

continuity.

•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:

43=4×4×4

=the number that is the product of

3 factors of 4

bx=b×b×b×...×b(xtimes)

=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.

b2b4=b×b×b×b×b×b=b6

The product of 2 factors of band

4 factors of bis 6 factors of b.

bbx=b×(b×b×b×...×b

| {z }

xtimes

)=bx+1

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,t3−1)as expressing a relationship

19 2.2. DESCRIBING PROBLEM SOLVING

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

time

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-

2.4.

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

20 CHAPTER 2. WHAT DOES THE LITERATURE SUGGEST?

Table 2.3. Mental actions during covariational reasoning (adapted from [28, p.

163]). Behaviors are those observed in students working on the Bottle Problem.

mental

action

description of mental action behaviors

Mental

Action 1

(MA1)

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)

Mental

Action 2

(MA2)

Coordinating the direction of change of

one variable with changes in the other

variable.

Constructing a monotonic graph

Verbalizing an awareness of the direc-

tion of change of the output while con-

sidering changes in the input.

Mental

Action 3

(MA3)

Coordinating the amount of change in

one variable with changes in the other

variable.

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.

Mental

Action 4

(MA4)

Coordinating the average rate-of-

change of the function with uniform

increments of change in the input

variable.

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.

Mental

Action 5

(MA5)

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?

21 2.2. DESCRIBING PROBLEM SOLVING

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

2x2,

and the area of the lower white

region is 3 −x2. The areas of the

white and black regions are equal, so

3−x2=1

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 2√2. 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

line?

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

22 CHAPTER 2. WHAT DOES THE LITERATURE SUGGEST?

[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

P´

olya’s teaching.

In his presentation on problem solving,

David Bressoud commented on materials by

P´

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.

23 2.3. KNOWLEDGE FOR TEACHING

Many students guess that 4 planes could cre-

ate 16 regions – rather than the maximum of 15.

Expectations are overthrown. At this moment,

P´

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

k=1n2=n(n+1)(2n+1)

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

forms.

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