:0;-97

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 FoundationMath for AmericaNoyce Foundation

Workshop Organizing Committee

William McCallum Deborah Loewenberg BallRikki Blair David Bressoud

Amy Cohen-Corwin Don GoldbergJim Lewis Robert Megginson

Robert Moses James Donaldson

Mathematical Sciences Research Institute

17 Gauss Way, Berkeley, CA 94720-5070510-642-0143 ◦ 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 11.1 Articulation between high school and college 1

1.2 Relationships with other disciplines 6Summary and further reading 13

2 What does the literature suggest? 152.1 Describing mathematical understanding 15

2.2 Describing problem solving 192.3 Knowledge for teaching 23Summary and further reading 27

3 Portraits of teaching and mathematics 293.1 Teaching with inquiry, action, and consequence 29

3.2 Advanced mathematics from an elementary standpoint 313.3 Inquiry-oriented differential equations 33

Summary and further reading 35

4 Assessments 374.1 Force Concept Inventory and related diagnostic tests 38

4.2 Basic Skills Diagnostic Test and Calculus Concept Inventory 414.3 The Good Questions Project 45

Summary and further reading 48

5 End Notes 49

Bibliography 51

i

PrefaceThis booklet, based upon a May 2009 workshopat MSRI titled Teaching Undergraduates Mathe-matics, is written for undergraduate mathemat-ics instructors who are curious what resourcesand research may support the philosophy andpractice of their mathematics teaching.

Ten years ago the AMS report Towards Excel-lence argued that “to ensure their institution’scommitment to excellence in mathematics re-search, doctoral departments must pursue ex-cellence in their instructional programs.” Math-ematicians in all collegiate institutions sharethe common mission of teaching mathematicsto undergraduate students, and the commonproblem that transitions from high school tocollege and from 2-year to 4-year college arechallenging for many students. The success ofa mathematics program depends on habits oflearning and quality of instruction.

◦ } ◦The following questions guided the workshop:

Research. What does research tell us abouthow undergraduate students learn mathemat-ics? Are we listening to and learning from thatresearch?

Curriculum. How do considerations of de-sign and assessment of courses and programsenhance the success of our teaching? Whatworks 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 thestudents we have? Can research experiencesplay an important role in exciting students tolearn mathematics? How can technology beharnessed 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 ahigh school graduate need for success in mathe-matics in college? Do AP and concurrent enroll-ment courses lead to the same learning as theirtraditional on-campus counterparts? Is therea need for greater articulation of high schooland collegiate mathematics? What mathemat-ical and cultural problems do students have intheir transition from high school to college, andwhat programs should colleges have that ad-dress these problems?

◦ } ◦The audience for the workshop included math-ematicians, mathematics educators, classroomteachers and education researchers who areconcerned with improving the teaching andlearning of mathematics in our undergraduateclassrooms. The workshop showcased courses,programs and materials whose goal is to in-crease students’ knowledge of mathematics,with an emphasis on those that show promiseof being broadly replicable.

AcknowledgementsI thank the organizers for putting together alively workshop, and the numerous presentersfor sharing their work. I also thank DavidBressoud, Ed Dubinsky, Jerome Epstein, KarenMarrongelle, and William McCallum for pro-viding insightful feedback, and David Aucklyand 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 thepresentations of David Bressoud, Marilyn Carl-son, Bill Crombie, Wade Ellis, Jerome Epstein,Deborah Hughes-Hallett, John Jungck, KarenRhea, Natasha Speer, Maria Terrell, and JosephWagner. I gratefully acknowledge the work-shop presenters and participants for inspiringthe contents of this report.

Structure of this bookletChapter 1 discusses demands balanced in math-ematics teaching, in particular, the articulationbetween high school and college mathematicsteaching and learning, and the relationship ofmathematics to partner disciplines. The contentfocus of this chapter is calculus, which plays acentral role in many collegiate programs.

Chapter 2 summarizes some findings from themathematics education literature: ways to ob-serve mathematical understanding, phases ofmathematical problem solving, and what is en-tailed in teaching mathematics in the K-20 set-ting. This chapter showcases ideas that helpus describe teaching and learning activities, sothat we can better see and hear our students andourselves.

Chapter 3 provides snapshots of teaching: us-ing inquiry to understand algebraic concepts,teaching calculus concepts well before a for-mal introduction to calculus, and using inquiryto structure an ordinary differential equationsclass. The purpose of this chapter is to provideglimpses of teaching in action.

Chapter 4 discusses several assessment projects:the Force Concept Inventory and related diag-nostic tests from physics, which inspired thecreation of the Basic Skills Diagnostic Test andCalculus Concept Inventory, as well as theGood Questions Project. This chapter highlightsfindings from studies using these instruments.

It is often easier to understand ideas throughexamples. Throughout this booklet are sam-ple problems from the projects and assessmentsdiscussed. The following page contains a list ofthese mathematics and physics problems.

List of Mathematics (and Physics)Problems

2.1 Bottle Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Temperature Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 The Ladder Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Paper Folding Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5 The Mirror Number Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.6 Polya Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.7 Car Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.8 Slicing with Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.9 Motivating Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.10 Mathematical Knowledge for Teaching Subtraction . . . . . . . . . . . . . . . . . . . . . . 233.1 Area Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.1 Mechanics Diagnostic Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2 ConcepTest (Blood Platelets) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3 Place Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.4 Proportional Reasoning (Piaget’s Shadow Problem) . . . . . . . . . . . . . . . . . . . . . . 434.5 Exponential Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.6 Proportional Reasoning (Numbers Close to Zero) . . . . . . . . . . . . . . . . . . . . . . . 444.7 Height Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.8 Repeating 9’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.9 Adding Irrationals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

v

List of Figures

1.1 Timeline of AP Calculus events, 1950-1987. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 High school mathematics course enrollments, 1982-2004. . . . . . . . . . . . . . . . . . . . 31.3 Fall mathematics course enrollments in 2-year colleges, 1985-2005. . . . . . . . . . . . . . . 51.4 Percentage of students in 4-year programs enrolled in calculus or above, 1985-2005. . . . . 51.5 Fall mathematics course enrollments in 4-year programs, 1985-2005. . . . . . . . . . . . . . 51.6 Fall enrollments in Calculus I vs. AP Calculus exams taken, 1979-2009. . . . . . . . . . . . 51.7 Fall enrollment in Calculus II, 1990-2005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.8 Fall enrollment in Calculus III and IV, 1990-2005. . . . . . . . . . . . . . . . . . . . . . . . . 61.9 Intended and actual engineering majors, 1980-2008. . . . . . . . . . . . . . . . . . . . . . . 71.10 Prospective engineers vs. total fall calculus enrollments, 1980-2008. . . . . . . . . . . . . . 71.11 Intended and actual majors in selected STEM fields, 1980-2006. . . . . . . . . . . . . . . . . 81.14 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Set-up for inquiry questions about angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3 Diagram for the Area Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4 Defining problems of elementary calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.5 Sequence of regions used to solve the Area Problem. . . . . . . . . . . . . . . . . . . . . . . 32

vi

List of Tables

1.12 Some biological phenomena and their associated curves. . . . . . . . . . . . . . . . . . . . 101.13 Biological phenomena associated to graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1 Action and process understandings of function, with examples. . . . . . . . . . . . . . . . 172.3 Mental actions during covariational reasoning. . . . . . . . . . . . . . . . . . . . . . . . . . 202.4 Questioning strategies for the mental actions composing the covariational framework. . . 202.5 Approaches to the Paper Folding Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.1 Average Mechanics Diagnostic Test results by course and professor. . . . . . . . . . . . . . 404.2 Force Concept Inventory pre- and post-test data. . . . . . . . . . . . . . . . . . . . . . . . . 404.3 Peer Instruction results using the Force Concept Inventory and Mechanics Baseline Test . 424.4 Data from Basic Skills Diagnostic Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.5 Results of the Good Questions Project. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

vii

CHAPTER 1Who we teach & what we teach:

Demands balanced inmathematics teaching

Teaching entails many demands. Good instruc-tion, in addition to conveying mathematics withintegrity, also

• responds to students’ mathematical back-grounds, and

• serves students well for their future, inside andoutside of mathematics.

Integrity, responsiveness, and service are com-peting principles. For example, what consti-tutes a good mathematical explanation dependson a students’ background – the most econom-ical or elegant explanation is not always themost 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 futurecourses.

Because of its place in the curriculum, calcu-lus is central to discussions about knowledge ofstudents and the mathematics they know. Cal-culus is both an area with rich mathematicalfoundations as well as a course prerequisite toa host of disciplines: a calculus instructor mustbalance integrity, responsiveness, and service.

Section 1.1 proposes a possible agenda forimproving 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 collegestudents.

Among the students we teach are future rep-resentatives of various disciplines. Section 1.2discusses ways that mathematics and mathe-matics classes interact with partner disciplinesin science, technology, and engineering.

1.1 Articulation between highschool and college teachingand learning

Workshop presenter David Bressoud, then pres-ident of the Mathematical Association of Amer-ica, proposed in [5] that to serve their studentsbetter, the mathematics community must:

• Get more and better information about stu-dents who study calculus in high school: Whatleads high school students to take calculus, andwhat 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 aboutthese 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 programsoffering calculus: High school calculus classesmust 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 thatwork with and build upon the skills and knowl-edge that students carry with them to college,whether or not each student is ready for collegefreshman 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 timesconflicting ideals – individual enrichment andwide access – contributed to the disarticulationbetween high school Advanced Placement (AP)and college calculus classes.

From a demographic perspective, high schoolcalculus enrollments have risen exponentiallysince the first Advanced Placement Calculusexam 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 than300,000 as of 2009; and calculus is foundationalin college curricula. Knowing more about APCalculus experiences and their impact on col-lege learning are promising ways to understandbetter the mathematical backgrounds and needsof entering college students.

1.1.1 Disarticulation between high schooland 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 thedisarticulation between high school and collegeclasses.

History and enrollment of calculus courses.The Advanced Placement (AP) programs be-gan more than fifty years ago, when calculuswas 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 bywhat is now known as the College Board andadministered through the Educational TestingService. 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 highschool graduates who completed dif-ferent levels of mathematics courses in1982, 1992, 2004. Note that in 2004,more than 75% of graduates had com-pleted Algebra II or a more advancedcourse, and more than 33% of grad-uates had completed Precalculus ora more advanced course. Data fromDalton, Ingels, Downing, and Bozick[14, p. 13].

what is now known as Advanced Placement(AP).

Both exam taking and mathematics courseenrollment have increased (see Figure 1.2).Since its inception, the number of AP Calculusexams taken has increased by several magni-tudes of order. Given this dramatic shift, collegemathematics course enrollments are strangelyclose to stagnant (see Figures 1.3-1.6) and maypotentially drop (as Section 1.2 discusses).

In 2-year programs, total mathematics en-rollment during the fall term has remained atroughly 25% of total enrollment in these col-leges. But the percentage of mathematics enroll-ment in precollege mathematics has increasedfrom 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 droppedfrom 20% of total undergraduate enrollment in1980 to 15% in 2005. In 1980, 10% of all studentswere taking a mathematics course at the level ofcalculus or above in the fall term. By 2005, thatwas down to 6%.

Thus, across all students, enrollment increasein 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 Placementprogram in the 1950’s to enrich students in highschools 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 regardlessof background. (The 1982 blockbuster Stand andDeliver profiled Jaime Escalante’s AP Calculusclass.)

In 1986, the National Council of Teachers ofMathematics (NCTM) and MAA issued a jointstatement warning students against taking cal-culus in high school with the expectation of re-taking it in college, entreating them instead tospend time mastering the prerequisites of calcu-lus. Whether the NCTM and MAA interpretedthe data accurately in the 1980’s, there seems tobe little effect from AP Calculus exam taking oncollege mathematics enrollments.

One possible explanation for this contrast isthat accountability exacerbated the tension be-tween enrichment and access. It is certainlydesirable to improve access to challenging, in-teresting mathematics. However, AP Calculuswas not designed for mass expansion. Based onconversations with students, Bressoud suspectsthat many students take AP Calculus and col-lege calculus not for the mathematics, but as astep toward future employment. This suggeststhat calculus is viewed as a course culminatingin a one-time test, rather than an opportunityfor 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 formathematical study. Both cases are problematic.

The AP Calculus program strives to articulatewith college calculus. As part of regular mainte-nance of the AP curriculum, the College Boardperiodically surveys the calculus curricula ofthe 300 tertiary institutions receiving the mostAP Calculus scores. However, history suggeststhat topic lists alone cannot effect preparednessin or appreciation of mathematics.

1.1.2 Articulation between high schooland college mathematics

Two studies, currently underway, support Bres-soud’s proposed agenda (see the beginning ofSection 1.1). The Characteristics of SuccessfulCollege 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 examinesecondary data collected in Fall 2009.

Knowing students better. College mathe-matics instructors must help students overcomedistaste and mischaracterization of mathemati-cal study. A dangerous temptation is to treatstudents as blank slates. However, personaldispositions 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 problemswith 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 suggeststhat successful articulation between high school

and college calculus must go beyond lists oftopics. After all, instruction does not consist ofa collection of topics: it also includes interac-tions between students and the topics, as wellas between the students and the teachers. TheCSCCP and FICS-Math studies will shed lighton these interactions, and how these may in-form worthwhile guidelines for the design ofcalculus in college and high school.

Re-examining first-year college mathemat-ics. College calculus is where mathematics de-partments interact with the most number andvariety 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 knowtheir students, and the content must also be bet-ter suited to the mathematical backgrounds andneeds of the students. The CSCCP and FICS-Math studies can inform the design of coursesto supplement or build upon calculus that willbe 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 advancedcourse 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 exponentialexam taking. Data compiled by Bres-soud from CBMS and ETS data.


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 hasbeen a 22% decrease in the number of students taking Calculus II in the Fall termin 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 mathematicscourses: relationships withmathematics and otherdisciplines

“Most of our students,” Deborah Hughes-Hallett opened her presentation, “will not go onin mathematics. Most of our students are in ourclasses because someone sent them there – usu-ally not themselves.”

Calculus is a pre-requisite for the STEM fieldsof Engineering, Physics, Chemistry, and Mathe-matics. It is sometimes a pre-requisite for forComputer Science, and occasionally for Eco-nomics and Biology. The data strongly sug-gest that the number of prospective engineeringmajors predicts fall calculus enrollments (seeFigures 1.9 and 1.10), and this population ispercentage-wise on the decline. If this trend

continues, the mathematics community shouldexpect dropping calculus enrollment.

At the same time, over the past twenty years,prospective biological sciences majors are onthe 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 firsttwo years of college, before they have takenthe courses that apply the mathematics foundin our courses, leaving our mathematics con-textless. The students in our classes may notbe able to provide feedback on how to accom-plish this mission. However, by conversingwith professors of their future courses, we maybe 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

19

81

1982

1983

1984

19

85

1986

1987

1988

19

89

1990

19

91

1992

1993

1994

19

95

1996

1997

1998

19

99

2000

20

01

2002

2003

2004

20

05

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 AmericanFreshman 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

tota

l fa

ll c

alc

ulu

s e

nro

llme

nt

R2 = 0.98146

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

talks, one by Deborah Hughes-Hallett on theMAA-CRAFTY (Curriculum Renewal Acrossthe First Two Years) project, discussed by Deb-orah Hughes-Hallett; and one on curriculumreform efforts, by John Jungck, one of thefounders of the BioQUEST Curriculum Consor-tium (Quality Undergraduate Education Simu-lations and Tools).

1.2.1 CRAFTY: Reports of conversationswith partner disciplines

An “asymmetry” lies between mathematics andother disciplines. Math majors may have

taken a chemistry or physics course or two inhigh school, but students in these fields maywell have been required to take two or moresemesters of mathematics courses – in college.In general, math majors are not required to takemore courses in any other particular scientificfield than members of that field are required totake of mathematics courses. Thus, whether ornot other disciplines have an understanding ofmathematics in a way that we would character-ize as accurate, it remains that they know ourcourses in a way that we do not know theirs.

Reflecting upon conversations with col-leagues, Hughes-Hallett recommends, “The


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 AmericanFreshman and NCES data.

thing I have found most helpful is not whetherthey need to know this topic or that topic,because that shifts over time. Instead, whatis more helpful as a common thread is to askthem what is useful about how they think aboutmathematics.” The MAA CRAFTY project,“Voices from the Partner Disciplines” [17], com-piled reports from faculty in other disciplineson what they would like to see in mathematics

courses their students take during the first twoyears 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 seeour 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 ideasin all mathematics courses during the first twoyears.

• Emphasize development of precise, logical think-ing. Require students to reason deductively froma set of assumptions to a valid conclusion.

• Present formal proofs only when they enhanceunderstanding. Use informal arguments andwell-chosen examples to illustrate mathematicalstructure.

There is a common belief among mathematicians thatthe users of mathematics (engineers, economists, etc.)care primarily about computational and manipulativeskills, forcing mathematicians to cram courses full ofalgorithms and calculations to keep “them” happy.

Perhaps the most encouraging discovery from theCurriculum Foundations Project is that this stereo-type is largely false. Though there are certainly in-dividuals from the partner disciplines who hold themore strict algorithmic view of mathematics, the dis-ciplinary representatives at the Curriculum Founda-tions workshops were unanimous in their emphasis onthe overriding need to develop in students a conceptualunderstanding 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 thatspreadsheets are consistently the “second best”technology for working on a problem, and inthis way are fundamental to the toolkit of manydisciplines.

As far as conceptual understanding, the skillsregarded as essential by most partner disci-plines include the concept of function, graphi-cal reasoning, approximation and estimation ofscale and size, basic algebraic skills, and numer-ical methods. For example, partner disciplineswould like students to:

• “become very comfortable with the use of sym-bols and naming of quantities and variables”(physics),

• have an “understanding that many quantitativeproblems are ambiguous and uncertain” and be“comfortable taking a problem and casting itin mathematical terms” (business and manage-ment),

• “summarize data, describe it in logical terms, todraw inferences, and to make predictions” (bi-ology),

• “formulate the model and identify variables,knowns and unknowns”, “select an appropri-ate solution technique and develop appropriateequations; apply the solution technique (solvethe problem); and validate the solution” (civilengineering).

Thus our mathematics courses should nurtureconceptual understanding, mathematical mod-eling, facility with applications, and fluencywith symbols and graphs as a language tool.

On solution methods, almost all disciplinesbroached 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 computationalskills. But, without a strong conceptual under-standing, the computational skills become im-potent. To understand this assertion, Hughes-Hallett offered a quote from her colleagueNolan Miller, a microeconomist at the KennedySchool:

“While much of the time in calculus courses is spentlearning rules of differentiation and integration, what ismore important for us is not that the students can takecomplicated derivatives, but rather that they are able towork with the abstract concept of ‘the derivative’ and un-derstand that it represents the slope, that if u : R2 → R,then −u1/u2 is the slope of a level surface of the functionin space.”

It may at first seem striking to separate the abil-ity to do difficult derivatives from the ability tocapture a definition as a geometric object. How-ever, these abilities are in fact distinct. One canbe quite skillful at “complicated derivatives”while lacking the ability to verbalize concep-tual understanding in a precise way – and viceversa.


Table 1.12. Some biological phenomena and their associated curves. Preparedby John Jungck for this workshop.

Curve Biological phenomenalinear fat intake vs. cancer

log-linear log survival vs. dose radiationlog-log allometry

positively exponential exercise curve vs. O2negatively exponential Newton’s law of cooling

gaussian variationsinusoidal heart rhythm

logistic r, Kchaotic tribolium

rectangular hyperbolic Michaelis-Mentenelliptical phase predator-prey, PV loop

hysteresis DNA melting

Table 1.13. Biological phenomena associated to graphs. Prepared by JohnJungck for this workshop.

food webs brain circuits metabolic pathwayspedigrees phylogenies fate maps

interactomes microarray clusters linkage mapsrestriction maps complementation maps nucleotide sequences

protein sequences 3-D protein backbones (or HP lattices)

1.2.2 Mathematics curricula and thebiological sciences

Biological research and mathematics. AsJungck argued in his presentation, the num-ber of biological science majors is on the rise,and mathematics and biology faculty stand tobenefit from each others’ expertise. Biologi-cal research depends on mathematical know-how, and mathematicians can engage studentsthrough mathematical modeling content.

Classically, understanding the dynamics ofbiological phenomena required understandingfunctions – for example, linear, exponential,chaotic, logistic functions. (See Table 1.12 forexamples.) “Part of the literacy for my biol-ogy students,” Jungck observed, “is that whenthey see their graphs of their data coming out inthese 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 theseare familiar objects, an alphabet for thinkingabout modeling. For many mathematical biolo-

gists, having this kind of repertoire of biologicalexamples that fit these kinds of things is kind oflike 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 withfamilies of functions, Jungck proposed, “We canhave a similar set of topologies of simple graphsthat almost every biologist would immediatelyrecognize, whether it’s a food web or a pedigreeor a phylogenetic tree or a metabolic pathway,that these kinds of things are there. You have,again, an advantage. You already know ourlanguage, you already understand the topol-ogy of these kinds of systems.” (See Table 1.13for examples.) These systems deal with rela-tions because they often feature many-to-oneand one-to-many maps, simultaneously. It isin part due to mathematics that biologists canwork 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 theoption of teaching the relevant mathematicsthrough biology departments:

It is unrealistic to expect many math faculty to haveany strong desire to really learn significant applicationsof math that students will readily connect to their othercourse work, though there is a core group who might dothis.

So what do we do to enhance quantitative understand-ing across disciplines? Below is what I say to life sciencefaculty: Who can foster change in the quantitative skill oflife science students? Only you, the biologists can do this!Two routes:

1. Convince the math faculty that they’re letting youdown

2. Teach the courses yourself.Gross [19], as quoted in Jungck [26]

The disappointment launched at mathematicsfaculty resulted from lack of immediate rele-vance of mathematics coursework to biologicalapplications. Even if a student could in the-ory derive the mathematics from starting prin-ciples, it is not the ability to use basic principlesthat is the most critical – it is the ability to applythe mathematics after the derivations have fin-ished. Application and derivation are distinctareas of mathematical fluency, and teaching onedoes 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 soughtdeep reform of the undergraduate biology pro-gram. The BioQUEST curriculum consortiumbegan as a collection of mathematicians, com-puter scientists, philosophers of science, scienceand math educators, biology educators, andbiology researchers. In 2005, BioQUEST con-vened kindred programs who sought to effectchange in undergraduate education, includ-ing the Harvard Calculus Consortium, Work-shop Mathematics Project, Project CALC, andC*ODE*E (Consortium of ODE Experiments).At the workshop Investigating Interdisciplinary

Interactions: Collaboration, Community, & Con-nections, these programs met with others frombiological sciences, computer science, statistics,and physics, among other disciplines.

John Jungck, one of the initial founders ofBioQUEST, has found that discussions about anindividual course or an individual departmenttend to be ineffective for the reform-oriented.“Frankly, if you want to change the culture toa 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 urgeyou to expand your community to beyond thepeers in the next-door office.” He pointed outthat as partner disciplines, we write and readone another’s grants, retentions, promotions,and awards. In our academic environment, werely on each other; our curricula and teachingshould reflect this.

To borrow an idea from anthropology, pop-ularized by Silicon Valley, we need to “crossthe chasm.” Jungck advocates looking for al-lies in other disciplines and other schools, andto maintain a broad view. Enthusiasts mustbe able to work with, convince, and talk tomany departments in schools of a variety of per-suasions – community colleges, Research-I, lib-eral arts, small state schools, historically blackschools, 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 enthusiastsmust demonstrate success in a variety of con-texts.

Principles for Biology classes. Biology depart-ments require mathematics courses, yet theircoursework 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 biologyand how biological research is performed and communi-cated, each institution of higher education should reexam-ine its current courses and teaching approaches to see ifthey meet the needs of today’s undergraduate biology stu-dents. Those selecting the new approaches should considerthe importance of building a strong foundation in mathe-


Quantitative concepts for undergraduatebiology 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 notjust 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 newinsights.

Figure 1.14. Quantitative conceptsused in biology (adapted from Gross[18]).

matics, physical, and information sciences to prepare stu-dents for research that is increasingly interdisciplinary incharacter. The implementation of new approaches shouldbe accompanied by a parallel process of assessment, to ver-ify that progress is being made toward the institutionalgoal of student learning.” (p. 44)

“Concepts, examples, and techniques from mathemat-ics, and the physical and information sciences should beincluded in biology courses, and biological concepts andexamples should be included in other science courses. Fac-ulty in biology, mathematics, and physical sciences mustwork collaboratively to find ways of integrating mathe-matics and physical sciences into life science courses aswell as providing avenues for incorporating life scienceexamples that reflect the emerging nature of the disciplineinto courses taught in mathematics and physical sciences.”(pp. 47-48)

Some quantitative concepts, compiled by LouGross, are shown in Figure 1.14.

If the average grade of a pre-med student ina 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] textbookshas three unfortunate consequences; namely, alack of respect for, consistency with, and em-powerment of students” [26, p. 13]. Withoutmore mathematics, biology classes are guiltyof the same. Using the mathematics showsrespect for the discipline of mathematics aswell as students’ intellectual capabilities. Cur-rently, only upper-division courses use calculus.The lack of consistency between lower-divisioncourses and upper-division courses causes de-skilling and frustration in students. One formof empowerment is economic access, and lackof mathematics “has differential career conse-quences” [26, p. 13]. There is a strong, positivecorrelation between the amount of mathematicsand computer sciences that biologists have hadand their professional career opportunities andadvancement (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 thereinmay surprise you. Why shouldn’t this literature be acces-sible to far more of our students?

– John Jungck , in [26].


Summary and further readingMathematics plays a variety of roles in the pur-suit of disciplinary knowledge: it gives ways toexpress quantities and concepts, to approximateand estimate, to model and predict real-life phe-nomena, to prove, to derive, and to problemsolve. Each of these domains is distinct from therest, and expertise in one area does not guaran-tee expertise in the rest. Mathematics and ourpartner disciplines would like service courses tonurture fluency in all these domains.

Those who have been heavily invested inteaching mathematics in service courses havefound that relevance and respect can help over-come mathematical fears and dislikes. Rel-evant material can interest students; relevantskills align with applications to the majors weserve. Respecting students must include build-ing upon students’ prior knowledge and ex-periences rather than ignoring or denying thatstudents come in with ideas about content andwhat it means to do mathematics; respect alsoincludes 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 tofind out more about their students’ experiencesin 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 organizingtool to ensure quality public education for everychild 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 lifescircumstances, are active in their communities,and advocate for education reform in America.

• Mathematics and Theoretical Biology Institute. Theefforts of this institute has significantly in-creased the national rate of production of U.S.Ph.D.’s since the inception of the institute, andrecognizes the need for programmatic change

and scholarly environments which support andenhance underrepresented minority success inthe mathematical sciences.

• BioQUEST. This project supports undergradu-ate biology education through collaborative de-velopment of open curricula in which studentspose problems, solve problems, and engage inpeer review.

• MathForLife. An innovative one semester ter-minal mathematics course intended to replaceexisting core or terminal courses ranging from”math-for-poets” to Finite Math whose primaryaudience is the undergraduate majoring in thehumanities or social sciences.

Articles• Ten Equations that Changed Biology: Mathematics

in Problem-Solving Biology Curricula. (Article byJohn Jungck.)http://papa.indstate.edu/amcbt/volume_23/

• Meeting the Challenge of High School Calculus.(Series by David Bressoud, as part of his onlinecolumn, 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 bythe National Research Council Committee onUndergraduate Biology Education to PrepareResearch 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. (Givesconcrete examples, with references, of biologicalresearch strongly influenced by mathematicaland 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

http://papa.indstate.edu/amcbt/volume_23/

http://papa.indstate.edu/amcbt/volume_23/

http://www.macalester.edu/~bressoud/pub/launchings/

http://www.macalester.edu/~bressoud/pub/launchings/

http://www.nap.edu/catalog.php?record_id=10497

http://www.nap.edu/catalog.php?record_id=10497

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

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

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

CHAPTER 2Teaching problem solving andunderstanding: What does the

literature suggest?

“Procedural knowledge” versus “conceptuallearning”, “teacher-directed instruction” versus“student-centered discovery”: these debatesdistract the community with false dichotomiesand vague premises.

With this opening, Marilyn Carlson called at-tention back to foundational questions:

• What does it mean for students to understand amathematical idea?

• What are problem solving abilities and processes formathematics learners?

• What is the nature of the knowledge that teachersneed to have?

This chapter summarizes and elaborates uponCarlson’s presentation.

A challenge to common ground on “under-standing” is that many topics in mathemat-ics have no widely accepted specification onwhat it means “to understand”. Promisingly,there are key topics of secondary and tertiarymathematics 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, thischapter 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 hercolleagues. This work builds upon literature byPolya and Schoenfeld among others.

Finally, there is currently no broad consen-sus on the nature of the knowledge needed forteaching, which is problematic for TA trainingprograms as well as K-12 teacher preparationprograms. We discuss research on the mathe-matical knowledge entailed in teaching, includ-ing research on tertiary instruction presented byNatasha Speer and Joe Wagner.

2.1 Describing mathematicalunderstanding: Functions

Algebra is a gateway class: completing mathe-matics beyond the level of Algebra II correlatessignificantly with enrollment in a four-year col-

15

16 CHAPTER 2. WHAT DOES THE LITERATURE SUGGEST?

lege and graduation from college (e.g., NationalMathematics 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 inthese ideas as ways to help observe and assesstheir students’ thinking.

2.1.1 Action and process understandings“Action” and “process” are part of Dubinsky’sAPOS Theory (Action-Process-Object-Schema;see Dubinsky and McDonald [15] for an intro-duction). There are four stages to Dubinsky’stheory, 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 onalgebraic concepts, Section 2.1.1 provides exam-ples from exponential expressions and grouptheory as well, intending that a greater vari-ety of examples will provide more leverage forreaders to apply APOS Theory to their ownteaching.

Examples regarding functions in the text andthe tables are from Oehrtman, Carlson, andThompson [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

xf (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 mathematicalobjects is a step-by-step transformation of theobjects to make another mathematical object orobjects. 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 promptingto take the action.

For example, in the action stage of under-standing a particular function f or g expressedin terms of x, students can likely evaluate f (x)or even g( f (x)) for given x. However, studentsmay not be able compose functions whose datais given to them only through tables and graphs(e.g., see Table 2.1). As well, the understandingof 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 23 as repeated multiplication of2, 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 studentsmay encounter difficulty with more intricatestructures, such as for cosets of D4, the sym-metry group of a square within a permutationgroup such as S4. Students may be able tocompute through brute force, but would not belikely to find efficient, holistic techniques.

Process. When a student repeats an actionand reflects upon it, they internalize the ac-tion into a process, which may no longer needexternal prompting to perform. “An individ-ual can think of performing a process with-out actually doing it, and therefore can thinkabout reversing it and composing it with otherprocesses”[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. Examplesare provided in Tables 2.1, and a geometric rep-resentation of inverse as process is provided inFigure 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 ofthe 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 understandingWorking with functions requires the comple-tion of specific rules and computations.

Inverse is about algebraic manipulation,for example, solving for y after switchingy and x; or it is about reflecting across adiagonal line.

Finding the domain and range is at mostan algebraic manipulation problem, for ex-ample, solving for when the denominator iszero, or when radicands are negative.

◦ } ◦

Examples of problems solvable with an actionunderstanding:

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+17−x , find f−1(x).

Given the graph of f (x), sketch a graph off−1(x).

Find the domain and range of f (x) =√

1+xx+3 .

If the graph of an invertible function is con-tained in the fourth quadrant, what quadrantis the graph of its inverse function containedin?

Working with functions involves mapping aset of input values to a set of output values;it is possible to work with a space of inputsrather than just specific values.

Inverse is the reversal of a process that de-fines a mapping from a set of output valuesto a set of input values.

Domain and range are produced by op-erating and thinking about the set of allpossible inputs and outputs.

◦ } ◦

Examples of problems solvable with processunderstanding:

Express ( f ◦ g)−1 as a composition of thefunctions f−1 and g−1.

Simplify cos(arcsin t) using the notion thatan inverse “undoes”.

A sunflower plant is measured every day t,for t ≥ 0. The height, h(t) centimeters, of theplant can be modeled with

h(t) =260

1 + 24(0.9)t .

What is the domain of this function? Whatis the range? What does this tell you aboutthe sunflower’s growth? Explain yourreasoning.∗

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.


In the case of exponential expressions, a stu-dent can likely interpret bx as “the number thatis the product of x factors of b” and logb m as“the number of factors of b that are in the num-ber m” [39].

In the case of left cosets, the student can likelyfind at least two elements g, h ∈ S4 not in thesubgroup D4 and so that g and h represent dis-tinct left cosets.

Applying the notions of action and processunderstandings to teaching. APOS Theorycan guide in-class activities, exam problems, orhomework. 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 interms of input and output.Examples. (a) Ask students to explain their rea-soning for whether ( f ◦ g)−1 equals f−1 ◦ g−1 org−1 ◦ f−1.(b) In addition to questions such as “Solve forx where f (x) = 6”, ask students to “find theinput value(s) for which the output of f is 6”,both algebraically and from a labelled graph ofthe 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 imageof a function applied to an infinite-cardinalityset (such as an interval), e.g., find the length off (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 orcontinuity.

• Ask students to make and compare judgementsabout 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 theexponential expressions in terms of a product

and in terms of words:

43 = 4 × 4 × 4

= the number that is the product of3 factors of 4

bx = b × b × b × . . .× b (x times)

= the number that is the product ofx factors of b

(b) Simplify each of the expressions below bywriting 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 b and4 factors of b is 6 factors of b.

bbx = b× ( b × b × b × . . .× b︸︷︷︸x times

) = bx+1

The product of b and x factors of bis (x+1) factors of b.

• Incorporate computer software packages thathelp students visualize or experiment withmathematical concepts, and use computer pro-gramming to help students reflect upon actions.A description of a number of studies in whichcomputer software and programming aidedstudent learning can be found in Dubinsky andTall [16]; in fact, the examples on exponen-tial expressions, from [39], are part of a studywhich included MAPLE programming activitiesfor the students.

2.1.2 Covariational reasoningThe Oxford English Dictionary defines covari-ant as, “Changing in such a way that interre-lations with another simultaneously changingquantity or set of quantities remain unchanged;correlated.” In studying students’ learning offunctions, Carlson has focused on helping stu-dents relate dependent quantities. This sectionpresents some of her findings, especially fromCarlson, Jacobs, Coe, Larsen, and Hsu [9] andOerhtman, Carlson, and Thompson [28].

In [9], covariational reasoning is described asthe “cognitive activities involved in coordinat-ing two varying quantities while attending tothe ways in which they change in relation toeach other”(p. 354), for example, viewing(x, y) = (t, t3 − 1) as expressing a relationship

19 2.2. DESCRIBING PROBLEM SOLVING

where x and y can both change over time, andchanges in x may come with changes in y.

Covariational reasoning means attending toco-varying quantities in contexts such as para-metric equations, physical phenomena, graphs,and rates of change. Precalculus, calculus, mul-tivariable calculus and differential equations allfeature simultaneously varying quantities.

The following are three examples of prob-lems, from [9] and [28], whose solutions entailcovariational reasoning.

Problem 2.1: Bottle Problem.Imagine this bottle filling with water. Sketch a graphof the height as a function of the amount of water thatis in the bottle.

In this case, quantities to attend to are heightand volume of water. Covariation appearsthrough applying concepts related to rate ofchange 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 roughsketch of the graph of the temperature over the 8-hourtime period. Assume the temperature at time t = 0is zero degrees Celsius.

rate

of

chan

ge

of t

emp

erat

ure

time2 4 6 8

Here, quantities to attend to are the rate ofchange and the original function. Covariation

appears through the interpretation of criticalpoints, positive slopes, and negative slopes.

Problem 2.3: The Ladder Problem.From a vertical position against a wall, a ladder ispulled away at the bottom at a constant rate. De-scribe the speed of the top of the ladder as it slidesdown the wall. Justify your claim.

Here, quantities to attend to are the speed ofthe top of the ladder and the placement of thebottom of the ladder.

One way that studies in math education canserve mathematics instructors is elaboratingwhat it means to “understand”, and how stu-dents arrive at understanding. Observations ofstudents working on problems similar to theabove suggest that covariational reasoning de-composes into five kinds of mental action; thisled Carlson, Jacobs, Coe, Larsen, and Hsu to de-velop interventions that improved calculus stu-dents’ covariational reasoning abilities [9]. Themental 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, givenalgebraic or graphical information. Probe fur-ther if students do not incorporate all variablesin their explanation, and the relationship be-tween the variables. If students use phrasessuch as “increases at a decreasing rate”, askthem to explain what this means in more detail.

• Ask questions associated with each of the men-tal actions. Questioning strategies are foundin Table 2.4 for discussing rates of changes, aconcept foundational to calculus and differen-tial equations.

2.2 Describing problem solvingMathematics instructors often would like theirstudents to be problem solvers: to celebratemathematical 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.

mentalaction

description of mental action behaviors

MentalAction 1(MA1)

Coordinating the dependence of onevariable on another variable

Labeling axes, verbally indicating thedependence of variables on each other(e.g., y changes with changes in x)

MentalAction 2(MA2)

Coordinating the direction of change ofone variable with changes in the othervariable.

Constructing a monotonic graphVerbalizing an awareness of the direc-tion of change of the output while con-sidering changes in the input.

MentalAction 3(MA3)

Coordinating the amount of change inone variable with changes in the othervariable.

Marking particular coordinates on thegraph and/or constructing secant lines.Verbalizing an awareness of the rate ofchange of the output while consideringchanges in the input.

MentalAction 4(MA4)

Coordinating the average rate-of-change of the function with uniformincrements of change in the inputvariable.

Marking particular coordinates on thegraph and/or constructing secant lines.Verbalizing an awareness of the rate ofchange of the output (with respect to theinput) while considering uniform incre-ments of the input.

MentalAction 5(MA5)

Coordinating the instantaneous rate-of-change of the function with continuouschanges in the independent variable forthe entire domain of the function.

Constructing a smooth curve with clearindications of concavity changesVerbalizing an awareness of the instan-taneous changes in the rate-of-changefor the entire domain of the function(direction of concavities and inflectionpoints are correct).

Table 2.4. Questioning strategies for the mental actions composing the covari-ational framework (adapted from [28, p. 164]).

mental action questioning strategyMA1 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) nearthis input? Can you represent that algebraically? Can you interpret this interms of the rate of change in this problem?

MA4 Can you compute several example average rates of change, possibly using thepicture to help you? What units are you working with? What is the meaningof those units?

MA5 Can you describe the rate of change of a function event as the independentvariable continuously varies through the domain? Where are the inflectionpoints? What events do they correspond to in real-world situations? Howcould 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 2Let x be the interior altitude of theblack triangle. Then its area is 1

2 x2,and the area of the lower whiteregion is 3 − x2. The areas of thewhite and black regions are equal, so3− x2 = 1

2 x2. It follows that x =√

2.

The three regions in the figure belowhave equal area; since the square hastotal area 3, the black triangle andwhite triangle form a square of area2. Hence the diagonal of this squarehas length 2

√2. The length sought is

half the diagonal, or√

2.

solvable, to confront these tasks with a clearhead, and to think carefully and completely.

The mathematician Polya, in his famous Howto Solve It [30], characterized four stages:

• understanding the problem – “we have to seeclearly what is required”

• developing a plan – “we have to see how thevarious items are connected, how the unknownis linked to the data, in order to obtain an ideaof the solution, to make a plan.”

• carrying out the plan

• looking back – “we look back at the completedsolution, we review it and discuss it.”

Almost all research on mathematical problemsolving traces back to Polya’s insight.

Though Polya’s stages have much face va-lidity to mathematically-savvy problem solvers,carrying them out in practice has remainedmathematically and affectively difficult for stu-dents in general. One of the key players inmathematical problem solving research is AlanSchoenfeld, 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 onmetacognitive 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; thepresent section discusses the relationship be-tween conceptual knowledge, affect, and prob-lem solving, as suggested by Carlson andBloom [8]. The purpose of this section, as withSection 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 aroundteaching and learning.

2.2.1 Conceptual knowledge and emotionCarlson 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. Thecollection included the bottle problem (fromSection 2.1.2), and the Paper Folding Problem.

Problem 2.4: Paper Folding Problem.A square piece of paper ABCD is white on thefrontside and black on the backside and has an area of3 in2. Corner A is folded over to point A′ which lieson the diagonal AC such that the total visible area ishalf white and half black. How far is A′ from the foldline?

Two approaches to the problem are summa-rized in Table 2.5.

Carlson and Bloom found that Polya’s stagescould delineate stages of behavior observed intheir participants. So, within each stage, Carl-son and Bloom examined how problem solversused 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 planningwhen problem solving. Carlson noted in herpresentation that the mathematicians studied in


[8] often exhibited rapid 20-minute cycles, stop-ping to ask themselves about a particular line ofreasoning: “Hmm ... should I do that? MaybeI 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 conceptualknowledge of area and triangles. When theproblem solver who used Approach 2 initiallywent off track, his knowledge of area allowedhim to get unstuck.

The detail and organization in planning re-minded Carlson of mathematicians discussingprecalculus material. Carlson observed in herpresentation, “It’s one thing to say ‘distance’.It’s another thing to say, ‘d is the number ofmiles that it takes that it takes to drive fromPhoenix and Tucson.’” Whatever the mathe-matical level, problem solving entails thought-ful, non-linear processes that draw upon carefulconnections 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 theregions totaled to 3.

Emotion and problem solving. Carlson andBloom 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 problemsFor readers’ interest, the following were othertasks used in the study reported in [8].

Problem 2.5: The Mirror Number Problem.Two numbers are “mirrors” if one can be obtained byreversing the order of the digits (i.e., 123 and 321 aremirrors). Can you find: (a) Two mirrors whose prod-uct is 9256? (b) Two mirrors whose sum is 8768?

Problem 2.6: Polya Problem.Each side of the figure below is of equal length. Onecan cut this figure along a straight line into twopieces, then cut one of the pieces along a straight lineinto two pieces. The resulting three pieces can be fittogether 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 weresports-utility vehicles, and 73% of all single carrollover 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 motivationWe end the section on problem solving with anargument for “intellectual need” and a nod toPolya’s teaching.

In his presentation on problem solving,David Bressoud commented on materials byPolya and Guershon Harel introducing induc-tion. A classic problem used by Polya in Let usteach guessing (now a DVD, originally on film in1966) is the following.

Problem 2.8: Slicing with Planes.How many regions that one plane can slice R3 into?2 planes? 3 planes? 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,Polya argues, learning can happen.

The “intellectual need” motivating genuinemathematical reasoning has also been dis-cussed by Guershon Harel. In a recentMAA 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 rowof dominoes, and how knocking the first cas-cades the rest down. However, this theoreti-cal description will be meaningless to someonewho has never needed proof by induction. Onlyproviding examples where the induction state-ment is explicitly in the problem statement, e.g.,showing ∑n

k=1 n2 = n(n+1)(2n+1)6 , exacerbates

the situation. When taught this way, studentstend to look for an “n” in the problem, and missopportunities to use proof by induction whenthere is no apparent “n”, even in contexts whereinduction provides a productive approach.

Harel proposes opening with problems withimplicit inductive statements, for example:

Problem 2.9: Motivating Induction.Find an upper bound for the following sequence.

√2,

√2 +√

2,

√2 +

√2 +√

2, . . .

The idea of building upon a previous case issomething that students understand intuitively;the domino analogy only tells them somethingthey feel they already know, without anchorsto any mathematics. But providing a problemwhere the students must focus on how to build– rather than merely the fact that something isbeing built – serves students’ needs. This prob-lem provides intellectual need to articulate pat-terns in terms of previous patterns, motivatingthe use of an index variable as well as recursiveforms.

2.3 Knowledge for teachingMathematics teachers teach mathematics; andteaching entails skills beyond the content aimedto students. For example, consider this ques-tion, about subtraction in Thames [37]:

Problem 2.10: Mathematical Knowledge forTeaching Subtraction.Order these subtraction problems from easiest tohardest 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 wedo not require of students.

As an example from college algebra, from[37]: what difficulties might solving the follow-ing linear system present to a student?

y = 3x− 1y = −5x + 2.

Students often struggle with working with frac-tional expressions. If, as in this case, the twodistances are integral and relatively prime, frac-tions will arise. Here, the y-intercepts have dis-tance 3 from each other, and the slopes differ by8. The intersection point is(

38

,18

).

Because the difference between y-intercepts isrelatively prime to the difference between theslopes, the x-coordinate will not simplify, andso a student will need to combine fractions tosolve for the y-coordinate.

On the other hand, suppose the difference be-tween slopes is a factor of the distance betweenintercepts, as in

y = 3x− 7y = −5x + 9,

whose intersection is

(2,−1).


Solving for the y-coordinate will only involveintegral computation, which students gener-ally have less difficulty with. Knowing howthe relationship between distances between y-intercepts and differences slopes affects the al-gebraic manipulations is another instance of themathematical knowledge entailed in teaching.

Further examples of knowledge entailed incollege algebra teaching are suggested by thefollowing questions:

1. What kinds of of functions do students have dif-ficulty finding the inverse of?

2. What are the typical difficulties that studentsstumble into when trying to describe the mean-ings of expressions such as f−1(5), given theverbal description of f (e.g., f is the height ofa football in feet, t seconds after it was tossed)?

3. Given that students may have difficulty in ex-pressing the meaning of expressions such asf−1(5), what are ways of scaffolding the con-struction of such descriptions?

4. What ideas about inverse functions do each ofthe following notions of inverse function ob-scure or highlight: “undoing”, “swapped inputsand outputs”, “graph reflected across the liney = x”?

5. What counterexamples should be chosen todemonstrate non-invertible functions?

Considerations for these questions may involveknowledge about symbolic manipulation – forinstance, finding the inverse of f when f (x) =

2xx−1 poses algebraic difficulties for some stu-dents; using variables other than x and y, asstudents sometimes have difficulty conceivingof input and output in terms of non-standardvariables; and situations that call for expressingf−1 in terms of y if students have already solvedfor f−1 in terms of another variable such as x.As well, college algebra teaching entails decid-ing on representations: whether to use tables,verbal description, or graphs to communicatethe function; whether or not to use a continuousfunction; and being careful about trigonometricfunctions, which have defined inverse functionsonly after taking a restriction.

Skillfulness around potential minefields ispart of purposeful teaching: the more sensitiveteachers are to differences in the solution steps,the more effectively they can minimize distrac-

tions, choose examples to illustrate a point, ordecide when to initiate discussion over proce-dures that are difficult for students but inciden-tal to the main results.

2.3.1 Knowing a discipline for teaching:several perspectives

The notion that there is a distinction betweenthe ways that teachers must know their disci-pline and the ways that other practitioners mustknow the discipline can be traced back to LeeShulman.

Although he does not directly address math-ematics, his writings have influenced the workon mathematics teacher education, and hisideas have analogues within mathematics. Forexample, Shulman [33] discusses three views onbiology: as a science of molecules, as a scienceof ecological systems, and as a science of bio-logical organisms. He argues that the “well-prepared biology teacher will recognize theseand alternative forms of organization and thepedagogical grounds for selecting one undersome circumstances and others under differ-ent circumstances”(p. 9). Examples from col-lege algebra of specialized knowledge for teach-ing might include knowing how to demonstratethat a inverse function “undoes” an originalfunction in algebraic, tabular, and graphical ex-amples; or identifying the functions implicit ina verbal description of a physical phenomenon.

Shulman’s work was one inspiration for thework of Deborah Ball on Mathematical Knowl-edge for Teaching (MKT). In the past threedecades, Ball and her colleagues have con-ducted research on Mathematical Knowledgefor Teaching (MKT), the mathematics entailedin the work of teaching (for a more detailed de-scription of MKT, see, e.g., Ball, Hill, and Bass[1]). Ball has focused her work on the elemen-tary level, and proficiency with MKT has beendemonstrated to be measurable and correlatedwith student achievement gains for first andthird graders (see Hill, Rowan, and Ball [25]).One focus of MKT is distinctively mathematicalwork of teaching – for example, knowing a va-riety of ways a concept could be depicted math-ematically, analyzing student errors and non-


standard methods, or selecting and sequencingproblems to teach a particular idea.

Patrick Thompson, whose work has primar-ily examined secondary and collegiate teach-ing, has proposed a complementary line of re-search, focusing on mathematical understand-ings that carry through a sequence of mathe-matics courses. Thompson motivated this view-point with three observations: that improvingand sustaining students’ high-quality mathe-matical learning is the reason that mathematicseducators exist; that most students rarely expe-rience “significant” mathematics – “ideas thatcarry through an instructional sequence, thatare foundational for learning other ideas, andthat play into a network of ideas”; and thatmathematics teachers tend to spend too littletime attending to meaning, wanting to rush to“condense rich reasoning into translucent sym-bolism” and get to the methods an main resultsof the day (see Silverman and Thompson [34],or Thompson [38]).

As an example of significant mathematicalideas, Thompson used base ten numeration. Astudent who is fluent with the meaning behindbase ten and place value should be able to, with-out prior practice on similar problems, reasonthrough a question such as “How many hun-dreds are there in 35821?” and – whether theyarrive at 358 or 358.21 – explain their answer.

Thompson [38] argued, “The issue of coher-ence is always present in any discussion ofideas. Ideas entail meanings, meanings overlap,and incoherence in meanings quickly reveals it-self”(p. 46). The meaning of a mathematicalidea is not simply the definition – for example,explaining average speed as “distance divided bytime” to someone who does not already under-stand the concept may not help them recognizeaverage speed in context. The meaning of aver-age speed must include the following:

• It involves a complete trip or the anticipation ofa complete trip (i.e., having a start and an end).

• The trip takes or will take a path which involvesmoving a definite distance in a definite amountof time.

• The average speed for that trip is the constantspeed at which someone must travel to coverthe same distance in the same amount of time.

Understanding why this definition translates todistance “divided by time” requires powerfulunderstanding of constant rate. “When stu-dents understand the ideas of average rate ofchange and constant rate of change with themeanings described here they see immediatelythe relationships among average rate of change,constant rate of change, slope, secant to a graph,tangent to a graph, and the derivative of a func-tion. They are related by virtue of their commonreliance on meanings of average rate of changeand constant rate of change.”[38, p. 52].

2.3.2 Mathematical knowledge forleading collegiate mathematicaldiscussions

On the college level, Natasha Speer has devotedher research to the professional developmentof mathematics instructors and teaching assis-tants. Speer and Joseph Wagner presented re-cent research on the mathematical knowledgeentailed in leading discussion-oriented class-rooms, reported in their work [35]. Two typesof scaffolding that occur in these settings areare social scaffolding to keep a discussion mov-ing, and analytic scaffolding, which keeps a dis-cussion moving toward a mathematically pro-ductive direction. Ineffective instruction arisesfrom an inappropriate balance of the two types.A overemphasis on the social scaffolding canleave students buzzing with ideas, but notknowing where they have gone. On the otherhand, blind pursuit of a particular mathemat-ical direction can lead students to a correctanswer that they do not appreciate or under-stand. Speer and Wagner focused on the waysof knowing mathematics to carry out analyticscaffolding.

Speer and Wagner studied the question,“What does a teacher need to do and know to be ableto recognize a student’s contribution as productive?”The setting for their research was a universityclass of 21 students. The professor, who is amathematician, used the inquiry-oriented cur-riculum developed by Chris Rasmussen, whichemphasizes conceptual understanding throughgroup activities, problem solving, and extensivediscussion to formalize the methods and the-


ory in the class. The mathematician who taughtthe class has seen all the data and participatedin the analysis. The data collected includesvideo recordings of almost all classes and audiorecordings of de-briefing sessions held after al-most every class with the mathematician, Speer,and Wagner.

As an example of the sort of knowledgeneeded in practice, consider the followingepisode from the professor’s class. The class isdiscussing the equation

dPdt

= 2− P10 + t

,

which the professor has rewritten as

(10 + t)dPdt

+ P = 2(10 + t). (2.1)

By this time, the class had solved problems byseparation of variables, but had not yet seen in-tegrating factors.1 The goal of the discussionaround the problem was to recognize that theleft hand side of Equation 2.1 was equivalent toddt ((10 + t)P). The following exchange occurs:

Tony: I was thinking initially, like, tome it looked like a chain rulekind of thing.

Professor: Chain rule? . . .Tony: But I couldn’t get anywhere

with that, though.Ron: Yeah. . . .Dan: Differentiation by parts?

In any class, and especially in discussion-oriented classroom, a teacher must in real-timemake sense of what students say. Then, ateacher must decide whether or not to open aclass conversation around the student’s contri-bution. There are many judgement calls to bemade.

In the first episode, a professor must decidewhether to pursue the idea of chain rule, differ-entiation by parts, both, or neither. In this case,the student Dan who brought up differentiation

1 M(t) is an integrating factor for a differential equationdPdt + Q(t)P = N(t) if it satisfies MQ = dM

dt . This is usefulfor solving the equation, as multiplying both sides by Myields M dP

dt + MQP = ddt (MP) = MN. Then we can solve

for P by integrating both sides.

by parts was recorded by the his table micro-phone as explaining to his group, “like u dv andv du and that whole deal.” Indeed the point ofrewriting the equation is that the left hand sideof the equation at hand has a form that can bederived from the form u dv + v du. Ultimately,it turned out that the student who brought upchain rule had the product rule in mind, butmisremembered the name.

The knowing of differential equations andcalculus entailed in interpreting this situationinvolves the subject matter knowledge of relat-ing u dv + v du to the differential equation athand; and the pedagogical content knowledgethat students might conflate the chain rule andproduct rule as situations involving the differ-entiation of multiple functions.

In general, analytic scaffolding requires thatthe instructor:

• Recognize and make sense of students’ mathe-matical (correct and incorrect) reasoning.

• Recognize or figure out how students’ ideashave the potential to contribute to the mathe-matical goals of the discussion.

• Recognize or figure out how students’ ideas arerelevant to the development of students’ under-standing of mathematics; and finding a way forthe reasoning to propel their own understand-ing or their peers’ understanding.

• Prudently select which contributions to pursuefrom among all those available.

While discussing this class featuring theabove episode, the professor observed:

“I just don’t understand and haven’t thought enoughabout differential equations as a subject to be taught so thatI feel any flexibility at all. ”

Being able to recognize or figure out the po-tential utility of students’ contributions requirescourse-specific pedagogical content knowledgeand subject matter knowledge. Understandingsomething “as a subject to be taught” takes ef-fort and experience.


Summary and further readingThe professor’s experiences in Speer and Wag-ner’s study corroborate previous analyses of el-ementary and secondary content that commoncontent knowledge is not enough. Mathemat-ics instructors need to know their discipline “asa subject to be taught.” The message from thework on mathematical knowledge for teachingis that we need greater understanding about theknowledge needed to support various instruc-tional practices, and to design professional de-velopment opportunities to help teachers learnall the types of knowledge needed for teaching.At the same time, we need to refine our notionof what it means for students to comprehend amathematical idea.

We have laid out several strands of researchon undergraduate teaching and learning. Theselines of work draw inspiration from varioussources and methods of observation. TheAction-Process framework was influenced bydevelopmental psychology, especially the workof Piaget. The covariation framework was con-structed through careful analysis of cognitiveinterviews. Work on problem solving largelytraces back to Polya’s writings. Mathematicalknowledge for teaching has its predecessors inthe thinking of Shulman, among others.

Below we have listed references for furtherreading on these topics. The thinking in theseworks can guide us as we grapple with how tobalance the demands of teaching.

◦ } ◦Articles

• Knowing mathematics for teaching: Who knowsmathematics well enough to teach third grade, andhow can we decide? (Expository article by Deb-orah Ball, Heather Hill, and Hyman Bass onMKT).

• Conceptual analysis of mathematical ideas: Somespadework at the foundations of mathematics edu-cation. (Paper by Patrick Thompson analyzingseveral foundational ideas in the undergraduatecurriculum, including angle, trigonometry, andrate of change.)

• Beyond mathematical content knowledge: A mathe-maticians knowledge needed for teaching an inquiry-oriented differential equations course. (Paperby Joseph Wagner, Natasha Speer, and BerndRossa, Journal of Mathematical Behavior 26 (2007)247-266. See especially Section 3 (pp. 253-263),which describe Rossa’s personal struggles andgoals in teaching this differential equations cur-riculum.)

• Critical variables in mathematics education: Find-ings from a survey of the empirical literature. (Ar-ticle by E. G. Begle from 1979 on the statis-tical impact of various characteristics on stu-dent achievement. Published by the Mathemat-ical Association of American and the NationalCouncil of Teachers of Mathematics.)

• Applying the Science of Learning to the Univer-sity and Beyond: Teaching for Long-Term Reten-tion and Transfer. (By Diane Halpern and Mil-ton Hakel in Change 35(4): 36-41, Jul-Aug 2003.ERIC abstract: Discusses why experts fromdifferent areas of the learning sciences con-clude that higher education’s primary goals–enhancing long-term retention and the transferof knowledge–may result from applying knownprinciples of human learning.)

Books• How to Solve It: A New Aspect of Mathematical

Method. (A classic, by George Polya.)

• Making the Connection: Research and Teachingin Undergraduate Mathematics Education. (AnMAA publication, edited by Marilyn Carlsonand Chris Rasmussen, which brings togetherpractical pedagogical examples from precalcu-lus to group theory.)

Media• Let us teaching guessing. (DVD of materials for

problem solving, by George Polya.)

CHAPTER 3Portraits of teaching and mathematics

Suppose an instructor decides to change teach-ing style from primarily lecture to one drivenby student dialogue. During a lesson follow-ing this decision, this instructor may grapplewith determining when to tell students a pieceof mathematics– as well as how, what and why.It is often helpful to see concrete examples ofwhat others’ pedagogical techniques.

In this section, we sketch portraits of teach-ing: inquiry questions with “action”, “conse-quence”, and “reflection”; “advanced mathe-matics from an elementary standpoint” as usedby the Algebra Project; and leading discus-sions in an inquiry-oriented differential equa-tions course.

3.1 Teaching with inquiry, action,and consequence – Wade Ellis

3.1.1 OverviewWade Ellis began by presenting four tenets ofinstruction: (1) Students learn by doing; (2) Fo-cused time on task is important; (3) Studentsremember what they think about; (4) Contextand relevance help student learn. These threetenets fit well with the “Action-Consequence-Reflection Principle”: students should act onmathematical objects, transparently observe theconsequences of their actions, and then reflecton the mathematical meaning of those conse-

quences. By using technology or physical ma-nipulatives to examine mathematical objects,students are doing; the observations focus thetask; reflecting on the mathematical meaninghelps students remember; and context and rele-vance is created by the action and observation,when they are doing something with the math-ematics. As Ellis observed, “Students find rele-vant what they do. One of the things that comesto my mind is the handshake problem.1 Oncethey begin working on it, it becomes relevant tothem, because they are very interested in the an-swer, even though it is not directly influential totheir long-term life goals.”

To carry out the Action-Consequence-Reflection Principle, Ellis has used “inquiryquestions” that extend mathematical environ-ments so that students can understand under-lying mathematics through their own reasoningand reflection. The inquiry questions help tocreate a classroom setting where students areconfident in answering the questions and ul-timately posing their own. Before class, hebrainstorms questions, and then selects andsequences them. The questions ask students to

• compare and contrast phenomena;

• predict forward and backward: What is an ac-tion that can result in . . . / Given this action, . . . ;

1There are 20 people in a room. If everyone shakeshands with everyone else, how many handshakes will takeplace?

29

30 CHAPTER 3. PORTRAITS OF TEACHING AND MATHEMATICS

• analyze a relationship: This happens when . . . ;

• make a conjecture;

• provide mathematical reasoning, justify a con-jecture, or prove a conjecture.

3.1.2 SnapshotEllis has developed several pieces of calculatorsoftware. He demonstrated two interactive sys-tems, one for lines and another for angles.

The system for lines featured two movablepoints, p1 and p2, and the line through them. Inhis experience, students naturally begin mov-ing around the points, observing that the linemoves with the points. Inquiry questions for

Figure 3.1. Set-up for inquiry ques-tions on slope.

this software might include:• How do you move p2 to get a negative slope?

(prediction)This is a good question because it involvessomething that students can do; and false startscan generate interesting conversation.

• How do you move p2 to get “no slope”? (pre-diction)This question, which was phrased by a math-ematician at a previous workshop Ellis ran,is interesting because it piques students withprovocative language while simultaneouslyraising a need to improve vague but compellinglanguage. This question can provide an oppor-tunity for the students to prompt for precision,by asking what “no” slope means.

• When does p1 below p2 make the slope nega-tive? (relationship)

This question could be scaffolded, “Move p1 be-low p2 make the slope negative”, then askingif moving p1 below p2 always makes the slopenegative.

• What happens when you move p1 to p2? (pre-diction and compare/contrast)Though this is a prediction question, it couldlead to a host of compare and contrast questionsthat could be used to set up reasoning about in-finitesmal quantities.

• Given a point q [a point such that p1 p2q is aright triangle with legs parallel to the axes], tryto move p2 so that the length of p2q is 10. . . . sothat the length of p2q is 0.75. Is this possible?(prediction)If p1 and p2 are constrained to a grid, the possi-ble lengths of p2q that the students can createdepend on the grid’s specifications. Studentscan reason about the grid properties.

• Move below the x-axis in a rigid transformation.What happens to the coordinates? Why? (rela-tionship, make a conjecture, provide reasoning)This question could be a setting for discussingthe invariance of length under rigid transforma-tions, or the relationship between coordinatesand the axes.

• What is a slope? Why doesn’t it change with theaspect ratio of the screen? (provide reasoning)

Ellis’s second demonstration featured a cir-cle with a highlighted arc and the correspond-ing length of the arc on a number line. As heprompted the audience for questions, a silencefell across the room. Ellis quipped, “That’s theproblem. Boy, we think, it’s so much fun, wecan move it around! But if there’s no math, it’snot going to go anywhere.”

Inquiry questions for this demonstrationmight include:

• Does the angle measure change when youchange r? Explain. (compare and contrast)

• What are all the angles that have a sin of 1? (pre-diction)

• What are all the angles that have a sin of 2? (pre-diction)

• What angles have the same sin? cos? (conjec-ture)

• How many r’s are there in the arc? (relationship)

31 3.2. ADVANCED MATHEMATICS FROM AN ELEMENTARY STANDPOINT

Figure 3.2. Set-up for inquiry ques-tions about angle.

Ellis closed his presentation with a graph-ing calculator of “What’s my rule?” Typically,teachers use “What’s my rule?” to introducefunctions: students are given a list of pairs ofnumbers, where the first number is related tothe second by a function that the students areasked to figure out and describe. Ellis translatedthis to an activity on transformations: as a stu-dent moves around a point p, a second point,which is always related to p via a rigid trans-formation, moves correspondingly. Students at-tempt to figure out and describe the transforma-tion.

3.2 Advanced mathematics froman elementary standpoint –Bill Crombie and Bob Moses

3.2.1 OverviewBob Moses comes at math literacy from thepoint of view of the black civil rights move-ment from the 1960’s, and trying to work the“demand side” of the problem. The AlgebraProject, founded by Moses, is focused on thebottom quartile of the nation’s students, askingwhether or not these students can be acceleratedrather than remediated.

The Algebra Project began its work at LanierHigh School in Jackson, MS, in 2002. Bystate standards, Lanier was near the bottom ofschools in Mississippi; and by national stan-dards, Mississippi is at the bottom of the coun-

try. They asked students there to do math withus for 90 minutes a day, for all four years,and to set several expectations: to graduatehigh school, to conquer the state standards andthe ACT, and to enter college ready to do col-lege math. This cohort of students graduatedin 2006. During this time period, the Alge-bra Project began working with research mathe-maticians to spend time in the classroom as wellas on curricular materials.

Bob Moses observed, “You know, you can’tmake the kids do all this math. It’s a coali-tion of the willing. So if I look at the AlgebraProject and working the demand side, I thinkour biggest success has been developing youngpeople who take on this idea of math literacy.We have to understand that we have no choice.We are faced with this transition of technologyfrom industrial age to technology age. Thisbrings with it this new literacy: quantitative lit-eracy. My interest in all of this is that I wouldlike the standard for all this to be for these kidsto leave high school, and not be fodder for thecriminal justice system.”

Bill Crombie, an Algebra Project developer,has been looking at the transition from alge-bra to calculus. Casting his inspiration inthe philosopher Søren Kierkegaard, who wrote“Life is to be understood backwards, but it islived forwards”2, Crombie is interested in whathe calls “backwards curriculum design”.

The learning trajectory, for students prepar-ing for college in the US, must proceed fromAlgebra, Geometry, and Trigonometry to Cal-culus. Many times, designers consider the skillsnecessary from previous classes to be successfulin a capstone class, usually calculus. Yet algebrais not enough for the understanding of calculus.The Algebra Project interprets “backward cur-riculum design” as not just asking about skillsand related concepts, but also asking how muchof that actual content – from the destination ofcalculus – can be brought back in a coherent,true, adequate fashion?

The Algebra Project is interested in a learn-ing trajectory from high school to college, withthe goal that students – including those in the

2“Livet skal forstas baglaens, men leves forlaens.”


bottom quartile – are able to make their own ca-reer choices. There are many reasons for an in-dividual to pursue or choose another career, butmathematical under-preparation should not beone of those reasons.

3.2.2 SnapshotAs an example, Crombie presented the follow-ing problem, which he has used in teachinghigh school students in Geometry and in Alge-bra II.

Problem 3.1: Area Problem.Given a parabola y = x2 and a displacement on thex-axis, determine the rectangle with area equal to thearea between the parabola and the x-axis across thegiven displacement.

x

y

Figure 3.3. Diagram for the AreaProblem.

Behind this problem is a characterization ofa discipline – in this case, the study of calculus– via the questions that it asks, rather than bythe techniques it uses. Crombie characterizedthe problems intrinsic to the study of calculususing the diagram in Figure 3.4.

To tackle the Area Problem, a teacher asksstudents to find relationships between a se-quence of various regions derived from the pic-ture of the parabola, on their graphing calcula-tors (see Figure 3.5).

As the students talk through the similaritiesthey find between the regions, they might noticethat some areas, such as in pictures D and B, arecomplementary (relative to the area bounded bythe axes and lines parallel to the axes and goingthrough the marked point of the parabola).

tangent problem

distanceproblem

speedproblem

area problem

Figure 3.4. Defining problems of el-ementary calculus.

A. B. C.

D. E. F.

G. H. I.

J. K.

.Figure 3.5. Sequence of regions usedto solve the Area Problem.

How is one picture related to the next? Howare the shaded regions and its area related to thenext? Some of the pictures, such as D and B, arecomplementary. The area in picture 6 is equalto the area in picture 5; sometimes students ar-gue this by saying that pencil-thin slices of E can“fall down” to the x-axis to create region F. Not-ing the equivalence of the areas in 5 and 6 is thekeystone of this activity. As the students move

33 3.3. INQUIRY-ORIENTED DIFFERENTIAL EQUATIONS

through the rest of the pictures, they eventuallypiece together that the area under the parabolais one third the cube of the displacement.

Formally, an argument could be written: Let dbe the displacement, and let A be the area of thefirst region, B be the area of the second region,etc. Note that A = B = C, so D = d3 − C =d3 − A and E = d3 − 2A. From the equation forthe boundary of the picture in 6, namely y =

−2(x− d)2, we may deduce that 12 D = d3− 2A.

The sequence of pictures G through J show thatd3 − E = A, whence A = d3

3 .The goal is not for the students to commu-

nicate the argument in this formal way; theabove argument was provided simply as a ref-erence for mathematicians. The point is to bringan area problem into a context where studentscould use Algebra II skills to reason about thenotion of areas under a curve, and therefore toset up foundations for calculus.

3.3 Inquiry-Oriented DifferentialEquations – Chris Rasmussen

3.3.1 OverviewMuch of the literature on undergraduate math-ematics education has focused on the construc-tion of reasoning or proofs rather than theo-rem usage. The inquiry-oriented differentialequations curriculum developed and studiedby Chris Rasmussen provides a context for stu-dents to use classical theorems, such as the ex-istence and uniqueness of solutions, in a mean-ingful way.

3.3.2 SnapshotRasmussen and Ruan [32] describe the interac-tions between class members as they grapplewith the solutions of the equations

dPdt

= 0.3P(1− P12.5

) (3.1)

which had arisen as a model for a deer popula-tion, and the equations

dhdt

= −h (3.2)

and

dhdt

= −h13 , (3.3)

which had arisen as models for airplane de-scent.

One goal of this unit was for students to un-derstand how and why the following existenceand uniqueness theorem was significant:

If f (t, y) is continuous on t ∈ (a, b), y ∈ (c, d), andthere exists a constant L such that | f (t, y) − f (t, z)| ≤L|y− z| for all y, z ∈ (c, d), then the initial value problemdydt = f (t, y) with y(t0) = y0 where t0 ∈ (a, b) andy0 ∈ (c, d) has at most one solution for all t ∈ (a, b) andy(t) ∈ (c, d).

To motivate this theorem, the students neededto have some intellectual need for finding so-lution curves, and to debate different proposalsfor solution curves. To generalize their ideas,students needed to have a sense of how theirreasoning could apply to other situations. Toharness physical intuition from the model con-text, students needed to be comfortable with thedistinction between a model and precise predic-tion of reality. Finally, students needed to ex-amine situations both with and without uniquesolutions.

A key tool for exploring these equations wasa computer system that plotted slope fields, butnot solution curves. As Rasmussen emphasizes,this feature of the program was critical to thesuccess of the pedagogy. The students coulduse the slope fields to bolster or argue againstclaims made about limiting behavior, but hadto rely upon the equation to argue about theuniqueness of solution curves around a givenpoint. The reader may note that of the threeequations above, only Equation 3.3 does not sat-isfy the conditions for existence and unique-ness – its solutions have the form h(t) = 0 orh(t) = 2

√2

3√

3(C − t)

32 , so for any point on the

line h = 0, there are two solution curves goingthrough it. All three equations have an equilib-rium solution at 0, though it is unstable in Equa-tion 3.1 and stable in Equation 3.2.

In Equation 3.1, the following featuresprompted discussion from students:


• the slope field around equilibria: whether thesolution curves around the P = 12.5 seemed totend toward 12.5, and in what manner.

• the equilibrium at P = 12.5, a non-integralvalue.

Some students initially thought that the deerpopulation would oscillate around P = 12.5,because “you can’t have half a deer runningaround”. This opened up a conversation inwhich students eventually reasoned about therelationship between the solution curves of amodel for population and the actual popula-tion behavior. As the conversations progressed,students turned to the mathematical relation-ship in Equation 3.1, reasoning directly aboutthe rate of change to find that P = 12.5 was astable equilibrium, and that no other solutioncurve would intersect it. The deer populationmodel was the first of many discussions thathoned students’ ability to reason using a combi-nation of empirical arguments (using the slopefield or the real world situation) and mathemat-ical reasons.

Later in the semester, the students discussedthe Equations 3.2 and 3.3. A feature thatprompted discussion from students included:

• differences in how the solution curves movedtoward 0, according to exploring the slope fieldsoftware, especially how suddenly the vectors“snapped” to 0 in the slope field for Equation3.3.

This observation motivated students to solvethe equation analytically and discuss the “rateof change of the rate of change.” They talkedabout the “snapping” of the curves in termsof changes in the rate of change of the heightover time. Thus the comparison between Equa-tions 3.2 and 3.3 opened conceptual conversa-tion about the quantity d

dt f (t, y) in the condi-tions of the uniqueness and existence theorem,giving a concrete context for whether or not thetheorem holds.

A conceptual understanding of the existenceand uniqueness theorem motivated these dis-cussions. Two pieces of evidence suggest suc-cess in this goal: a post-class survey of thestudents’ work found instances where studentscited the theorem even when the problem did

not specifically ask for the theorem; and a quan-titative study by Rasmussen, Kwon, Allen, Mar-rongelle, and Burtch [31] found that studentswho were taught differential equations usingthe inquiry-oriented curriculum were signifi-cantly more likely to use the theorem. Thisstudy, which looked at approximately 130 stu-dents in three sites (the midwest and the north-west of the U.S., and Korea), where each sitehosted a traditional ODE class and a inquiry-oriented ODE class, found no significant dif-ferences in the students’ ability to solve rou-tine problems and an improvement in the IO-DE students’ ability to reason conceptually.

35 3.3. INQUIRY-ORIENTED DIFFERENTIAL EQUATIONS

Summary and further readingIt is sometimes easier to implement ideas whenwe hear about how others have implementedtheirs. We have sketched out three ideas thatwere implemented in a variety of contexts. Eachcase depends on the teacher’s preparation onhow to use mathematical features and studentideas, and what responses were likely to arisefrom students. By thinking carefully about howto listen to students and stoke students’ mathe-matical reasoning, we can continue to refine ourpersonal teaching practices.

◦ } ◦

Books• Radical Equations: Civil Rights from Mississippi

to the Algebra Project. (By Robert Moses andCharles E. Cobb)

• Making the Connection: Research and Teachingin Undergraduate Mathematics Education. (AnMAA publication, edited by Marilyn Carlsonand Chris Rasmussen, which brings togetherpractical pedagogical examples from precalcu-lus to group theory.)

Articles• Research Sampler. (An occasional online MAA

column on research on undergraduate math-ematics education. Includes summaries of re-search and pedagogical interventions on proof,ordinary differential equations, and problemsolving.)http://www.maa.org/t_and_l/sampler/research_sampler.html

• Capitalizing on advances in mathematics and k-12mathematics education in undergraduate mathe-matics: An inquiry-oriented approach to differen-tial equations. (Comparison study of inquiry-oriented differential equations course with tra-ditional differential equations course, by ChrisRasmussen, Oh Nam Kwon, Karen Allen, KarenMarrongelle and Mark Burtch, in the Asia Pa-cific Educational Review 7(1): 85-93. (2006))http://www.springerlink.com/content/k444461382116983/

Media• Let us teaching guessing. (DVD of materials for

problem solving, by George Polya.)

http://www.maa.org/t_and_l/sampler/research_sampler.html

http://www.maa.org/t_and_l/sampler/research_sampler.html

http://www.springerlink.com/content/k444461382116983/

http://www.springerlink.com/content/k444461382116983/

CHAPTER 4Assessments

Assessment is a perennial responsibility: howcan we pinpoint our students’ backgroundsand dispositions, whether or what our studentshave learned, or how they compare with otherstudents?

Broadly speaking, the two most common fla-vors of assessment are “formative” and “sum-mative”. Formative assessment is diagnostic,tends to take place during the term, and givesinformation for planning subsequent activitiesto help form the students as intellectuals, prob-lem solvers, or towards a instructors’ learninggoals. Formative assessment is often contrastedwith summative assessment, which takes placeat the end of a unit, and gives information forsummarizing what students have learned. In thissense, a midterm examination could be seen asboth formative and summative: it may help aninstructor recalibrate goals for the second halfof the term as well as lay out what studentslearned in the first half of the term. On theother hand, questions asked with clickers dur-ing a particular lesson might be more formativethan summative – the students’ collective an-swers serve more to structure the ensuing dis-cussion than as a final report of the students’mastery of the subject. The point is that whetherassessment is formative or summative dependson how an instructor uses the information – andthat no matter what form the assessment takes,it should match the instructor’s ultimate goals.

Whenever using an assessment instrument,a primary concern is whether or not the re-sults accurately represent what the instrumentset out to evaluate. An unintentionally mis-leading phrase may throw students off for rea-sons other than their mathematical skills. In thiscase, the students’ collective answers would notbe very informative as either formative or sum-mative assessment. In tension to the concern ofvalidity is the pragmatic issue of scalability: re-alistically, an assessment instrument should beeasy to use, with minimized time investmenton the part of the instructor. The quickest as-sessments use multiple choice items; but whenwe only know whether students answered (a)or (b) or (c), we must be sure that these an-swers accurately represent the misconceptionswe think they capture. Using written examsameliorates this to some extent – reading stu-dent explanations gives more insight into howstudents were thinking. However, most instruc-tors do not have time to read through hundredsof written answers. The usual way that thisis handled by the developers of assessments isthrough pilot testing. In these trial runs, stu-dents are either interviewed or asked to elab-orate upon why they chose or eliminated par-ticular answers. Based on the results of thisin-depth information, developers revise assess-ment items. Each project discussed in this chap-ter went through such a vetting process.

37

38 CHAPTER 4. ASSESSMENTS

Below, we highlight the assessment endeav-ors of Maria Terrell, who heads the Good Ques-tions Project, and of Jerome Epstein, who isknown for his work on the Basic Skills Diag-nostic Test and the Calculus Concept InventoryExam. Before describing their work, we be-gin outside of mathematics, in physics. Thework associated to the Force Concept Inventoryhas had wide influence on the physics reformmovements, and inspired both Terrell and Ep-stein.

In each section, we give the personal or intel-lectual context in which the projects were con-ceived, example assessment questions wherepossible, and selected findings.

4.1 The Force Concept Inventoryand related diagnostic tests

4.1.1 Initial developmentIn the mid 1980’s, Ibrahim Halloun and DavidHestenes were interested to see how students’common sense theories of physics influencedtheir ability to learn physics. These observa-tions prompted them to create the Mechan-ics Diagnostic Test, an multiple-choice instru-ment to assess students’ common sense con-cepts about motion. Halloun and Hesteneswrote their instrument to reflect two generalcategories: principles of motion, correspondingto Newton’s Laws of Motion; and influenceson motion, corresponding to specific laws offorce in Newtonian mechanics. In the 1990’s,Hestenes worked with Wells [23] to create theMechanics Baseline Test, and with Wells andSwackhamer [24] to extend the Mechanics Diag-nostic Test to the now well-known Force Con-cept Inventory (FCI). They focused on the no-tion of force as a foundational concept for learn-ing physics for which students come in withmany erroneous ways of thinking. A report ofthe concepts in the inventory can be found in[24]. Here, we focus on the findings by Hal-loun and Hestenes, as they set in motion muchof physics education reform at the college leveltoday.

Halloun and Hestenes [22][21] found that stu-dents may believe that the trajectory of a rocketfiring its engine is similar to the parabolic trajec-tory of a ball, not recognizing that the signaturetrajectory of a ball is due to constant force. Stu-dents may also believe that an object subjectedto constant force will move at constant speed, orthat the time interval needed to travel a partic-ular distance under a particular constant forceis exactly inversely proportional to the magni-tude of the force. Each of the above examplespoints to concepts that are covered in almostany standard physics course in mechanics – soone might optimistically predict that after sucha course, most students would apply the ideaslearned from class to analyze these situationsrather than relying on incorrect common sensetheories.

Here is an example item from Halloun andHestenes’ work (emphasis as in the original):

Problem 4.1: Mechanics Diagnostic Problem.The accompanying figure shows a ball thrown ver-tically upwards from point A. The ball reaches apoint higher than C. B is a point halfway betweenA and C (i.e., AB = BC). Ignoring air resistance:

What is the speed of the ball as it passes point Ccompared to its speed as it passes point B?

(a) Half its speed at point B(b) Smaller than that speed, but not necessarily

half of it(c) Equal its speed at point B(d) Twice its speed at point B(e) Greater than that speed, but not twice as great.

A

B

C

(Reprinted with permission from Halloun, I.A. & Hestenes, D.American Journal of Physics 53(11), pp. 1043-1055 (November1985). c©1985, American Association of Physics Teachers.)

39 4.1. FORCE CONCEPT INVENTORY AND RELATED DIAGNOSTIC TESTS

Halloun and Hestenes [22] found that, in-deed, “(1) Common sense beliefs about motionare generally incompatible with Newtonian the-ory. Consequently, there is a tendency for stu-dents to systematically misinterpret material inintroductory physics courses.” However, con-trary to the hope that an introductory coursemight correct students’ misconceptions aboutthe physical world, “(2) Common sense beliefsare very stable, and conventional physics in-struction does little to change them.” Thesefindings corroborate the conclusions of priorwork done by physics education researchers;the work of Halloun and Hestenes is distin-guished by its large sample size and use of amassively-scalable instrument.

Halloun and Hestenes collected pre- andpost-test performance from nearly 1500 stu-dents in introductory level college physicsclasses at Arizona State University, and 80students beginning physics at a nearby highschool. They additionally collected informa-tion on a mathematics pretest performance. Inanalysing correlations between their data, Hal-loun and Hestenes found that “pretest scoresare consistent across different student popula-tions”, “mechanics and mathematics pretest as-sess independent components of a student’s ini-tial knowledge state”, and “the two pretestshave higher predictive validity for studentcourse performance than all other documentedvariables” including gender, age, academic ma-jor, and background courses in science andmathematics. Students come in with strongmisconceptions about physics that are not eas-ily dislodged.

4.1.2 Interactive Engagement methodsWith scalable tests and compelling results byHestenes and his colleagues, a natural nextresearch question was: is there an instruc-tional method that might help students over-come erroneous common sense theories aboutphysics? In the 1990’s, Richard Hake used theMechanics Baseline Test, the Mechanics Diag-nostic Test and the Force Concept Inventory tostudy the effectiveness of “interactive engage-ment” methods.

Hake was interested in methods “designedat least in part to promote conceptual under-standing through interactive engagement ofstudents in heads-on (always) and hands-on(usually) activities which yield immediate feed-back through discussion and peers and/or in-structors, all as judged by their literature de-scriptions”, in contrast to those “relying primar-ily on passive-student lectures, recipe labs, andalgorithmic-problem exams”, which he terms“traditional”. As examples of interactive en-gagement, Hake considers courses at The OhioState University, Harvard University, and In-diana University. These universities used acombination of Overview Case Studies, Ac-tive Learning Problem Sets, ConcepTests, SDIlabs, cooperative group problem solving, andMinute papers (see Hake [20] for references tothese materials).

To measure the effectiveness of interactive en-gagement methods against traditional methods,Hake defines the average normalized gain g for acourse as the ratio of average percentage gainG to the maximum possible average percentagegain Gmax:

g =G

Gmax=

S f − Si

100− Si(4.1)

where S f and Si refer to the percentage scoredon the final and initial assessment.

Hake analyzed the average normalized gaing for 62 schools who replied to his requestsfor pre- and post-FCI test data. He classifiedcourses as using interactive engagement meth-ods or traditional methods via survey responseson activities of students and teaching methods.The resulting graphs of gains versus pretestscores strongly suggested that interactive en-gagement methods influence students’ reason-ing in physics more so than traditional methodsdo. This picture is corroborated by the collec-tive average gains. Of the 62 schools reportingdata to Hake, 14 were classified as traditionaland 48 as using interactive-engagement meth-ods. The 14 traditional courses exhibited an av-erage gain of 0.23 ± 0.04, in sharp contrast tothe 48 interactive engagement courses’ averagegain of 0.48± 0.14 – a nearly two-standard de-viation difference.


Table 4.1. Average diagnostic test results by course and professor. Maximumscores: 36 (physics), 33 (mathematics), including five calculus items which wereomitted in College Physics.

(Reprinted with permission from Halloun, I.A. & Hestenes, D. American Journal of Physics 53(11),pp. 1043-1055 (November 1985). c©1985, American Association of Physics Teachers.)

Table 4.2. Force Concept Inventory pre- and post-test data. (a)%〈Gain〉 vs.%〈Pre-test〉 score on the conceptual Mechanics Diagnostic (MD) or Force ConceptInventory (FCI) tests for 14 high-school courses enrolling a total of N = 1113students. (b) %〈Gain〉 vs. %〈Pre-test〉 score on the conceptual MD or FCI testsfor 16 college courses enrolling a total of N = 597 students. (c) %〈Gain〉 vs.%〈Pre-test〉 score on the conceptual MD or FCI tests for 32 university coursesenrolling a total N = 4832 students.

(a) (b) (c)

(Reprinted with permission from Hake, R.R. American Journal of Physics 66(1), pp. 66-74(January 1998). c©1998, American Association of Physics Teachers.)

41 4.2. BASIC SKILLS DIAGNOSTIC TEST AND CALCULUS CONCEPT INVENTORY

4.1.3 Peer instruction methodsIn recent years, Eric Mazur’s work has becomerecognized as an exemplar in the scholarship ofteaching and learning. His model of “Peer In-struction” is well-specified, his students showgains in qualitative and quantitative reason-ing about physical phenomena, and his instruc-tional methods evaluate positively.

In his speech “Confessions of a ConvertedLecturer”, Mazur relates the chain of events thatan article on the Force Concept Inventory setin motion. The article he read had concludedno matter who the teacher was – even if theywere teaching award winners – there was essen-tially no gain. The students were not overcom-ing their initial misconceptions. He reflects,

“At that time, we were doing rotational dynamics, thestudents had to do triple integrals to calculate dynamics.There was no comparison between what we were doing andthe Force Concept Inventory, they were way beyond it. . . .

“I was worried that my students would be offendedby the simplicity of this test once they would start on it.Oh boy, were my worries quickly dispelled. Hardly hadthe first group of students taken their seats in the class-room when one student raised her hand, she said, ‘Profes-sor Mazur, how should we answer these questions? Ac-cording to the way you have taught us, or how we usuallythink?’ I had no idea how to answer that question.”

After a damning performance by those studentson the Force Concept Inventory, Mazur began todevelop a method he now calls Peer Instruction,which partitions a class into a series of shortpresentations each centered around a physicalidea and followed by:

• a conceptual question (“ConcepTest” question)related to the idea

• a one-minute or two-minute period in whichstudents prepare individual answers to sharewith the instructor (via clickers, flashcards, orother method which is more visible to the in-structor than to fellow students)

• a two to four minute period in which studentsshare their answers and reasoning with peers

• a poll of students’ final answers to the question,followed by an explanation by the instructor ofthe answer.

As support for this basic structure, Crouchand Mazur [13] describe gradual refinements to

their pedagogy that they developed in the firstdecade of using Peer Instruction.

Here is an example ConcepTest:

Problem 4.2: ConcepTest (Blood Platelets).A blood platelet drifts along with the flow of bloodthrough an artery that is partially blocked by de-posits.

As the platelet moves from the narrow region to thewider region, its speed: (a) increases (b) remains thesame (c) decreases.(Reprinted with permission from Crouch, C.H. & Mazur, E.,American Journal of Physics 69(9), pp. 970-977 (September2001). c©2001, American Association of Physics Teachers.)

(The answer is that the speed decreases.)Mazur found that the average normalized

gain g for the Force Concept Inventory, definedas in Hake’s study (Equation 4.1), doubled from1990 to 1991, the transition between traditionalinstruction and the Peer Instruction method.

The Force Concept Inventory represents con-ceptual mastery. Mazur also recorded gains inhis students’ quantitative problem solving abil-ity, represented by performance on the Mechan-ics Baseline Test. Mazur’s results are taken fromsections taught by different instructors, and areconsistent with results from Hake’s study [20],suggesting that the gain can be attributed to theinstructor’s choice of method rather than the in-structor or the location.

4.2 The Basic Skills DiagnosticTest and the Calculus ConceptInventory

4.2.1 Basic Skills Diagnostic TestJerome Epstein developed the Basic Skills Di-agnostic Test (BSDT) out of a laboratory pro-gram in 1980 with an NSF grant for workingwith a group of entering college students “whotested as having the mathematical and concep-tual level of a 10 year old”. At the time, he


Table 4.3. Peer Instruction results using the Force Concept Inventory and Me-chanics Baseline Test. The FCI pretest was administered on the first day of class;in 1990 no pre-test was given, so the average of the 1991-1994 pre-test is listed.In 1995 the 30-question revised version was introduced. In 1999 no pretest wasgiven so the average data of the 1998 and 2000 pre-test is listed. The FCI post-test was administered after two months of instruction, except in 1998 and 1999,when it was administered the first week of the following semester to all studentsenrolled in the second-semester course (electricity and magnetism). The MBT wasadministered during the last week of the semester after all mechanics instructionhad been completed. For years other than 1990 and 1999, scores are reported formatched samples for FCI pre- and post-test and MBT. No data are available for1992 (the second author was on sabbatical) and no MBT data are available for1999.

(Reprinted with permission from Crouch, C.H. & Mazur, E., American Journal of Physics 69(9),pp. 970-977 (September 2001). c©2001, American Association of Physics Teachers.)

believed that he was working with an outliergroup. The BSDT, which examines pre-algebraand algebra concepts, has now been adminis-tered to thousands of students in high schooland college. What Epstein has concluded overtime, as more institutions generate data on theBSDT, is that there are many students of compa-rable level who are hidden from our view – andthat the students he worked with in 1980 maynot be as much of an exception as we might liketo think.

Themes among the BSDT data suggest thatareas of particular difficulty include placevalue, proportional reasoning, and the equiva-lent fractions.

We present example questions, discuss po-tential interpretations of the students’ errors,and tabulate performance data on these items.For reasons of protecting the validity of test re-

sults, we have obscured some features of thequestions. Interested readers can obtain the ac-tual Basic Skills Diagnostic Test by contactingJerome Epstein at [email protected] and agreeingto test security conditions.

From the perspective of understanding learn-ers, the data on these questions is interesting forthe information it encodes about the sources ofstudent ways of thinking. From the perspectiveof assessment item writ in, the questions are in-teresting in how they are crafted to detect error.

Problem 4.3: Place Value.Put the following numbers in order from smallest tolargest. n1 n2 · · · nk

(In an actual test setting, n1, n2, . . . , nk are re-placed with a finite set of numbers.)

43 4.2. BASIC SKILLS DIAGNOSTIC TEST AND CALCULUS CONCEPT INVENTORY

The numbers include an assortment of care-fully chosen fractions, improper fractions, anddecimals designed around common misconcep-tions and rote strategies.

For example, a common error is concludingthat a/b is less than c/d when as a < c andb < d. There are two misconceptions that maycontribute to such an error. Students may notunderstand the role of the numerator and de-nominator of a fraction, and students may notrealize that a fraction can represent a numbergreater than 1.

Problem 4.4: Proportional Reasoning (Piaget’sShadow Problem).(Piaget) Yesterday an m-foot tall man was walkinghome. His shadow on the ground was sm feet long.At the same time, a tree next to him cast a shadow stfeet long. How tall was the tree?

(In an actual test setting, m, sm, and st arereplaced with numbers.) The most commonwrong answer is st + (m − sm) feet. As Pi-aget noted, students begin by “thinking addi-tively” – that is, because the difference betweenthe man’s height and his shadow was m − smfeet, students erroneously apply this differenceto the relationship between the tree’s heightand its shadow. This problem, which assesseswhether or not a student is able to think propor-tionally, has an extraordinarily high correlationwith success in a college algebra course. Whengraphing the distributions of students who an-swer incorrectly and who answer correctly overtheir level of success in a college algebra class,two Gaussians result with very little overlap.As Epstein remarked, “It is as though the stu-dents who answer this question correctly are in

a different mathematical world than those whodo not.”

Another common misconception arises withlinear equations with fractional coefficients anda constant term (cf. the discussion about stu-dents’ difficulties with fractions and linear sys-tems in Section 2.3).

The above questions concern pre-algebraicand algebraic skills where faulty yet commonmodes of thinking can cause error. The skillsrepresented are foundational for success inmathematics. The results of the Basic Skills Di-agnostic Test are sobering, and a call to developpedagogies that will help students overcome er-roneous ways of operating with numbers andalgebra, as well as the tendency to fall backupon rote procedures pursued without under-standing.

4.2.2 Calculus Concept InventoryThe Calculus Concept Inventory (CCI) is di-rectly inspired by the Force Concept Inventory.In Epstein’s words, “There is a basic level ofconceptual understanding that [students] musthave, or anything that they have learned forthe final exam will disappear.” The CCI seeksto assess this understanding. Epstein receivedNSF funding in 2004-2007 for initial develop-ment and evaluation, and has collected data onthe CCI ever since. The CCI has proved popu-lar, with roughly one request per week from thebeginning of the project.

Themes among the data for performance sug-gest that students have difficulty with reason-ing exponentially as opposed to linearly. Aswell, the data from the CCI corroborate theBSDT’s data that students have difficulty think-ing proportionally as opposed to additively. Wepresent examples of each.

Problem 4.5: Exponential Reasoning.We are growing a population of bacteria in a jar. At11:00 a.m., there is one bacterium in the jar. Thebacteria divide once every minute so that the popu-lation doubles every minute. At 12:00 noon, the jaris full. At what time was the jar half full? (a) 11:01(b) 11:15 (c) 11:30 (d) 11:45 (e) 11:59.


Table 4.4. Percentages denote the percentage of students in the course obtainingthe correct answer to the questions concerning place value, proportional reasoning,and linear equations.

School Course Place Val. Proport. Eqns.Hofstra CS-005: 63% 51% 26%

CS-015: 84% 69% 50%CS-132: 85% 85% 77%

Wellesley WRT-125: 65% 72% 87%NYU GenPhys: 80% 79% 69%

Astron: 80% 71% 54%.

CS-005: Overview of Computer ScienceCS-015: Fundamentals of Computer Science I:

Problem Solving and Program DesignCS-132: Computational ModelingWRT-125: Introductory writing courseGenPhys: General Physics I (for non-physics majors)Astron: Origins of Astronomy.

The most common wrong answer is (c),which suggests that instead of reasoning aboutan exponential situation, students are fallingback upon familiar linear patterns: 11:30 ishalfway between the start and end times of thescenario, and the question asks when the popu-lation is halfway to the end population.

Problem 4.6: Proportional Reasoning (Num-bers Close to Zero).(Terrell) A number close to 0 is divided by a numberclose to, but not equal to 0. The result (a) is a numberclose to 0, (b) is a number close to 1, (c) is a verybig number, (d) could be any number, (e) is not anumber.

In the performance data on this question,one wrong answer stands out: (b). It is asthough students reason that the numbers areclose to identical because they are “running outof room” close to 0. This is symptomatic of Pi-aget’s description of students who think addi-tively, but who cannot yet think proportionally.Questions that ask students to think proportion-ally, and where students exhibit additive think-ing, show remarkably little gain from a semesterof traditional instruction.

Consistent with the hypothesis tested byHake and Mazur, interactive-engagement sec-tions show dramatically more gain on this ques-tion than traditional lecture instruction – ap-

proximately two standard deviations. More-over, as we will see in the next section, thisquestion was enormously productive as a PeerInstruction question in the Good QuestionsProject data.

Thus additive reasoning in calculus seems toplay a similar role to erroneous common sensetheories in physics (see Section 4.1.1). This anal-ogy holds in two ways: first, in both cases, thereis a tendency for students to fall back on linearpatterns instead of reasoning about the situa-tion; and second, in both cases, the only knownway for a course to dislodge the misconceptionis through substantive discussion amongst stu-dents. The discussion must happen after engag-ing with a question that gives a common intel-lectual experience.

In Hake’s and Mazur’s studies, interactiveengagement results in a two standard devi-ation difference above traditional lecture sec-tions. The most dramatic result in mathemat-ics so far has taken place at the University ofMichigan, with roughly 1500 students across55 sections of Calculus, and with normalizedaverage gains between 0.27 and 0.40. In con-trast, the typical gain in CCI data for lecture-based classes is between 0.05 and 0.25. Thedepartment at the University of Michigan useslesson plans and classroom seating that neces-sitate at least minimal interactive engagementmethods. Though this gain size is promising,it is still less than the gains seen in recent stud-

45 4.3. THE GOOD QUESTIONS PROJECT

ies of physics courses; on the other hand, thephysics movement toward interactive engage-ment precedes the mathematics movement bynearly two decades. The results so far indicatethat substantive interaction amongst students iscritical for growth in their powers of reasoning.

4.3 The Good Questions ProjectThe Good Questions Project, led by MariaTerrell, adapts the ideas behind Eric Mazur’sConcepTests and Peer Instruction for freshmanphysics, to classes in the freshman calculus forliberal arts majors. About seeing Mazur’s ques-tions for the first time, Terrell [36, p. 5] writes,

I was excited. I wondered if it would be possibleto craft such questions in calculus; questions that werenon-computational, that were related to students experi-ences, questions that were memorable, and surprising, thathelped build on students partial understanding. I won-dered if instructors would take time out from class to usegood questions if they were attractive, if they led to ac-tive lively discussions, and if they helped students connecttheir intuitive understanding of the world to the conceptsof calculus.

A group at Cornell resolved to write such ques-tions, and the Good Questions Project was born.

4.3.1 Fall 2003 StudyA Good Question is intended to:

• stimulate students’ interest and curiosity inmathematics

• help students monitor their understanding

• offer students frequent opportunities to makeconjectures and argue about their validity

• draw on students’ prior knowledge, under-standing, and/or misunderstanding

• provides instructors a tool for frequent for-mative assessments of what their students arelearning

• support instructors’ efforts to foster an activelearning environment.

Miller, Santana-Vega, and Terrell [27] exam-ined the effect of peer instruction using a setof “Good Questions” in Fall 2003, after a pilot

phase the previous spring. In Fall 2003, Ter-rell invited instructors of a large, multi-sectioncalculus course to use these questions. Therewere 17 sections of 25-30 students each, and14 instructors. They informed the instructorsthrough a short series of workshops of Mazur’ssuccess and how the Good Questions were in-tended as an analogue of the ConcepTests. AsTerrell remarks of her instructors, “We knewthat we couldn’t tell people how to teach. Ev-eryone has their own ideas on how studentslearn, and give examples, but when they closethe door they will do what they want to do.”

Terrell’s team gathered data on the instruc-tors use and non-use of the questions. On sur-vey questions throughout the semester, theyasked which questions the instructors used. Theuse of the Good Questions were facilitated byloading the questions onto a laptop that the in-structors took to class, and the use of clickers, torecord which questions were used as well as thedistribution of answers from the students. Ad-ditionally, volunteer students were interviewedon their reaction to the Peer Instruction methodand incorporation of the Good Questions.

Instructors’ usage of Good Questions fell intofour profiles:

• Deep: Good Questions used 1-4 days per weekwith peer discussion, and many Deep/Probingquestions. (These questions are distinguishedby their ability in the pilot trial to elicit discus-sions about the conditions of theorems, or theimportance of precise language in mathemat-ics.)

• Heavy plus Peer: Good Questions used 3-4 daysper week with regular use of peer discussion.

• Heavy plus Low Peer: Good Questions used 3-4days per week, with minimal or no use of peerdiscussion.

• Light to Nil and Low Peer: Good Questionsrarely or not at all used, with no use of peer dis-cussion.

The profiles were determined by a combina-tion of student surveys, instructor surveys, andrecorded histogram data from the laptops. Thesurveys asked for information about the lengthand nature of discussions, and to what extentMazur’s model for peer instruction was fol-lowed. Where there was a disparity between


student reports and instructor reports, Terrell’steam used the student reports.

To relate the instructor’s usage to perfor-mance, Terrell’s team gathered data on the stu-dents’ common preliminary and final exams,which included both conceptual and procedu-ral questions.

The data suggest that different profiles af-fected student learning, with the students of in-structors who fit the Deep profile attaining amedian 10 points higher than the students of in-structors who fit the Light to Nil profile. Inter-estingly, the median was lowest from studentsof instructors who fit the Heavy plus Low Peerprofile. This suggests that peer discussion is acritical piece of the intervention, and that wheninstructors who want to engage in peer discus-sion of good questions are given the resourcesto do so, their students benefit.

A common concern about interventions fo-cusing on conceptual problems is low ability inrelatively procedural problems. However, Ter-rell’s data did not indicate any such risk. In fact,students in lower SAT math bands gained moreon their performance on procedural problemsthan on conceptual problems. An interviewedstudent suggested, “If you understand the con-cepts, it can help you with the numbers.”

Overall, the interviewed students reactedpositively to Peer Instruction. In the words ofone student, “I liked it. First of all I thoughtit made the learning more enjoyable than justwatching a lecture and then pulling out a penmaybe and finding the answer. I thought it wasmore valuable to look at it from a more analyt-ical approach, and to go to take a question, andthink about it for a while, and then respond toit. It was fun to talk about it, with other people,it helps you work out your own logic. It helpsyou figure out if there were any flaws in yourlogic, help you fill in a whole.” Another studentcommented, “This method forces you to en-gage, forces you to think, and forces you to getinto what you are doing. Because even if youput any answer, you don’t have to think, wellmaybe no one else has my answer . . . there’ssafety in your answer.” These students felt they

benefited from the safety, the anonymity, andthe expectation to articulate their conclusions.

4.3.2 Sample Good Questions

Problem 4.7: Height Problem.True or False: You were once exactly π feet tall.

In Terrell’s Fall 2003 data for sections with PeerInstruction, a typical distribution of votes was40% of students selecting True for their firstvote, and then 85% of students selecting Truein the re-vote.

This question can incite students to defendtheir thinking by elaborating their assumptions.Some students will argue that because atomsare discrete, a person may skip over the heightof π feet tall. Others may argue that humangrowth should be thought of as a continuousprocess. This discussion sets a context for in-troducing the Intermediate Value Theorem andwhether or not the hypotheses of the theoremare satisfied in a particular situation.

Interestingly, when Terrell’s instructors askedthe same question, with π replaced by 3, thedata showed very different vote distributions.Students voted 80%, then 90% for True.

As mentioned previously, another example ofa Good Question is Problem 4.6 from Section4.2.2, about division of numbers close to zero.Students’ votes were initially spread across alloptions. At 31%, option (c) held the mostvotes. Only 20% opted for the correct answerof (d). Upon revoting, 98% students chose (d)and 2% choose (c), with 0% across the other op-tions. The initial spread advantages Peer In-struction: the idea that quotient “could be anynumber” comes out through group discussion,when students share different examples. Dis-cussion around this question can be used to in-troduce limits with indeterminate forms.

4.3.3 Sample concepts unaffected bygroup discussion

While the Good Questions helped dislodge stu-dent misconceptions, Terrell’s team also found

47 4.3. THE GOOD QUESTIONS PROJECT

Table 4.5. Results of the Good Questions Project.

examples of questions whose votes for incorrectanswers seemed largely unaffected by peer dis-cussion, such as the following.

Problem 4.8: Repeating 9’s.True or False: 0.999 · · · = 1.

Here, 50% of students, then 45%, voted forthe correct answer of True. Some problems suchas this one were eliminated. However, a fewproblems were kept to help instructors see howdeeply students held some partial understand-ings of decimals and limits, for example:

Problem 4.9: Adding Irrationals.If p = .39393939 . . . and q = 0.676677666777 . . . ,then p + q is (a) not defined because the sum of a ra-tional and irrational number is not defined, (b) not anumber because not all infinite decimals are number,(c) is defined using successively better approxima-tions, (d) is not a number because the pattern maynot be predictable indefinitely.

Initial distributions resembled re-vote distri-butions: each option received 8%-41% of ini-tial votes, with approximately 20% selecting thecorrect answer of (c). The data from decimalproblems suggest that many students have anunderdeveloped sense of “equals” for limits,despite demonstrating some intuition. When

prompted, “Suppose I was your boss and I saidyou have to have something in 15 minutes,”students said that they would “go out reallyfar” and calculate an approximation. They re-marked that they could “always get it as closeas you want it, as long as you tell me how closeyou want it” – thus hinting at the ε-δ notion ofcontinuity. However, until students understandthat when it comes to real numbers, equals is aprocess, it will be difficult for them to connectthe definition of a limit with intuition for partic-ular limits (cf. Section 2.1.1).

The pilot data on questions addressing num-ber and decimal concepts suggest that studentshave conceptual gaps in their understandingthe real numbers, and that K-20 students wouldbenefit from a better articulation of conceptsfrom the arithmetic of real numbers prerequisiteto the study of calculus.


Summary and further readingThe Force Concept Inventory, MechanicsBaseline Test, and Mechanics Diagnostic Testsparked a wave of physics education researchbecause their questions were created with care,the initial results provocative, and the imple-mentation scalable. Mazur’s work, which usedthe Force Concept Inventory and MechanicsBaseline Test as measures, is valuable for itsclear pedagogical methods. Although we donot yet have published, large-scale results onthe analogues of these physics education find-ings in mathematics, the work by Terrell andEpstein is well worth further exploration andimplementation. Below are articles for furtherreading in the physics and mathematics edu-cation literature relevant to the material in thischapter.

◦ } ◦

Articles related to the Force Concept Inventory• The initial knowledge state of college physics stu-

dents. Article by Ibrahim Halloun and DavidHestenes introducing the Mechanics DiagnosticTest. In American Journal of Physics 53(11), 1043-1048 (1985).

• Peer Instruction: Ten years of experience and re-sults. Article by Catherine Crouch and EricMazur on pedagogical support for implement-ing Peer Instruction. In American Journal ofPhysics 69(9):970-977 (2001)

• Professors as physics students: What can they teachus? Article by Sheila Tobias and Eric Mazur,reporting on a study of 10 non-physical sci-ence professors who studied Newton’s lawsalongside approximately 300 students in a non-calculus-based introductory physics course. InAmerican Journal of Physics 56(9): 786-794 (1988).

• Resource letter on physics education research. Re-view article by Lillian McDermott and EdwardRedish on learning and teaching of physics.Contains extensive list of bibliographies on sci-ence education research regarding instructionalstrategies targeting conceptual understanding,analysis of assessment instruments. In AmericanJournal of Physics 67(9): 755-767 (1999).

Articles on the Calculus Concept Inventory• The calculus concept inventory. Article by Jerome

Epstein in Proceedings of STEM Assessment,Washington, D.C. (2006), edited by DonaldDeed and Bruce Callen.http://www.openwatermedia.com/downloads/STEM(for-posting).pdf#page=64

Articles on the Good Questions Project• Asking good questions in the mathematics classroom.

Prepared by Maria Terrell for the AMS-MERWorkshop “Excellence in Undergraduate Math-ematics: Mathematics for Teachers and Mathe-matics for Teaching,” Ithaca College, New York,March 13-16, 2003.

• Can good questions and peer discussion improve cal-culus instruction? Article by Robyn Miller, Ev-erilis Santana-Vega, and Maria Terrell report-ing on the Good Questions Project, in PRIMUSXVI(3): 193-200 (2006).

http://www.openwatermedia.com/downloads/STEM(for-posting).pdf#page=64

http://www.openwatermedia.com/downloads/STEM(for-posting).pdf#page=64

CHAPTER 5End Notes

Two themes that emerge from the precedingchapters are the work of seeing and listening. Torefine undergraduate mathematics education,we must, like a person peering through a mi-croscope, dial into the details within mathemat-ical concepts and pedagogical methods. To sup-port the work of seeing, we must be able to hearstudents when they reveal pieces of their waysof thinking or previous experiences. What stu-dents say gives us a context for observing thedetails of our own instruction and the coherenceof our institutions’ mathematical instruction.

This workshop launched with a comparisonof collegiate and secondary course-taking pat-terns, presenting a stark contrast between thedramatic trend of mathematical advancementamong high school students and the relativelyflatlined enrollments in collegiate mathematicsclasses. More disturbingly, there are anecdotesof students who excel in their high school’s APCalculus class, yet place into precalculus uponentering college, or who leave with a visceraldislike of mathematics. David Bressoud pushedthe mathematical community to gather moreand better information about the high schoolstudents who take calculus, what happens intheir classes, and the benefits and dangers to fu-ture mathematical success of taking calculus inhigh school. Moreover, Bressoud advocates are-examination of first-year college mathemat-ics, to build upon the skills and knowledge that

students carry with them to college. Such a callrequires that we identify these skills and knowl-edge, which in turn entails careful observationof and listening to our students.

“Voices of the Partner Disciplines”, a Cur-riculum Foundations project sponsored bythe MAA, synthesizes conversations betweenmathematicians and colleagues from STEMfields on what they would like to see in math-ematics courses their students take during thefirst two years of college. To see the math-ematical concepts integral to other disciplineswithin our curricula, the Curriculum Founda-tions project encourages mathematics faculty tomeet with faculty of fellow STEM fields at theirinstitution. To initiate the reforms supported bythe Curriculum Foundations project on a largescale, mathematical biologist John Jungck ar-gues that we need to “cross the chasm.” Wemust be able to work with, convince, listenand talk to departments outside our own andschools outside our own.

The Curriculum Foundations workshopswere unanimous in their emphasis on concep-tual understanding, in addition to computa-tional skills. Jerome Epstein, whose work onthe Basic Skills Diagnostic Test and the CalculusConcept Inventory has revealed an impover-ished understanding of foundational notionssuch as place value and fractions on the partof some college students, echoed this point.

49

50 CHAPTER 5. END NOTES

He urged us to continue investigations intopedagogical methods that help our studentsreorient their learned ways of thinking aboutmathematics from memorized routines to rea-soned consideration of mathematical concepts.If we take a cue from the research in physicseducation, then such pedagogies might involveinstructors setting up opportunities for studentsto listen to each others’ thoughts on mathemat-ical concepts. As work of Maria Terrell, theAlgebra Project, and Chris Rasmussen suggest,such opportunities can be extracted from close-up views of central mathematical ideas such asvariable, area, existence and uniqueness, inde-terminate limits, proportional reasoning, andexponential reasoning – combined with atten-tiveness to students’ existing ways of thinkingabout these ideas.

The Action-Consequence documents, theInquiry-Oriented Differential Equations cur-riculum, and the Algebra Project curriculumeach rely intrinsically on listening to students.The analysis by Natasha Speer and Joe Wag-ner, along with other work on mathematicalknowledge for teaching, argues that leadingmathematical discussion entails recognizingand making sense of students’ mathematicalreasoning, how these ideas have the poten-tial to contribute to the mathematical goalsof the discussion, and how students’ ideasare relevant to the development of other stu-dents’ understanding of mathematics. To makefiner-grained observations and assessments ofstudents’ thinking, Marilyn Carlson advocatedusing frameworks from the research literatureas a guide. These frameworks, which are pred-icated on close observations of students, mayhelp us see the mathematics from the perspec-tive of student learning.

With this document, we have tried to put to-gether a resource for mathematics instructorswho are interested in the ideas of other educa-tors, within and outside of mathematics. Manyof us pay attention to our teaching; as we know,there are many things to pay attention to, andsometimes these things are not precisely articu-lated in our minds even if we have an intuitionfor what they are. We have tried to articulate

these demands here. We hope that this guidewill help sharpen the ways in which we listento and observe our students and mathematics.

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