Project Mosaic: Motivation and Structure

Daniel Kaplan

June 30, 2010

Welcome to Project MOSAIC!

Kick-Off Workshopat the

Institute for Mathematics and Its Applications

Project PIs: Danny Kaplan, Nicholas HortonEric Marland, Saber Elaydi, Randall Pruim

IMA: Fadil Santosa

Assessment: Joan Garfield, Bob delMas, Andy Zieffler,Rebekah Isaak, Jiyoon Park

And You!

Our Love/Hate Relationship with Mathematics

We put huge amounts of resources into math education, allleading up to calculus and heavily oriented toward algebra. Wedon’t expect that this will be useful for anything.What people tell me when I say I teach mathematics: “You mustbe smart.” Not: “That must be exciting.” or “You must beinfluential.”What people tell me when I say I teach statistics: “I had to takestatistics in college. I hated that course.”

An old jokeOne old lady to another, telling about her stay at a hotel resort:

“Don’t go there. The food is terrible ... and such small portions!”

The Multiple Roles of Mathematics

As a society, we do not agree on what mathematics is for. Toillustrate, I’ll summarize three purpose-oriented critiques ofmathematics education:

1. “A Mathematician’s Lament”2. “What is Mathematics For?”3. Failure to Support the Uses of Mathematics

We don’t actually agree on the purpose of mathematicseducation. Often, educators don’t acknowledge that there aremultiple legitimate purposes. This is a large handicap inproviding an effective education.

Critique: Math is an Art Form

Paul Lockhart, A Mathematician’s Lament: HowSchool Cheats Us Out of Our Most Fascinatingand Imaginative Art Form

I Published on the MAA web site.I “... one of the best critiques of current K-12

mathematics education I have ever seen.”— Keith Devlin

“The first thing to understand is that mathematics is an art.”

“Math is not about following directions, it’s about making newdirections.”

“... the glory of it is its complete irrelevance to our lives.”

Mathematician’s Lament (cont.)

“Now hold on a minute, Paul. Are you telling me thatmathematics is nothing more than an exercise in mentalmasterbation? Making up imaginary patterns and structures forthe hell of it and then investigating them and trying to devisepretty explanations for their behavior, all for the sake of somesort of rarified intellectual aesthetic?”

“Yep. That’s what I’m saying. In particular, pure mathematics(by which I mean the fine art of mathematical proof) hasabsolutely no practical or economic value whatsoever. Yousee, practical things don’t require explanation. Either theywork or they don’t.” [5, p. 120]

Mathematics as Training in Reasoning

Underwood Dudley (2010) “What is Mathematics For?” Noticesof the AMS, 57(5):608-613

“ ... by ‘mathematics’ I mean algebra, trigonometry, calculus,linear algebra, and so on: all those subjects beyond arithmetic.There is no question about what arithmetic is for or why it issupported. Society cannot proceed without it.”

Dudley’s complaint: The study of mathematics is being justifiedby it’s claimed applications in the workplace, rather than as ameans to teach “the race to reason.”

“I keep looking for uses of algebra in jobs, but I keep beingdisappointed. To be more accurate, I used to keep looking untilI became convinced that there were essentially none.”

What is Mathematics For? (2)

“Many times people have told me that they liked mathematics(though they call it ‘math’) because it was so definite and it wassatisfying to get the right answer. Have you not heard the samething? They liked being able to reason correctly. They knewthat the practice was good for them. No one has ever said tome, ‘I liked math because it got me a good job.’ ”

“[W]hat mathematics education is for and what it has alwaysbeen for is to teach reasoning, usually through the medium ofsilly problems.”

“We no longer have the confidence in our subject that allows usto say that. We justify mathematics on its utility in the world ofgetting and spending.” [2]

What is Mathematics For? (3)

“Everyone supports the teaching of algebra. The public wantsmore mathematics taught, to more students. The requirementskeep going up, never down.”“The reason for this, I am convinced, is that the public knows, orsenses, that mathematics develops the power to reason. Itshows, better than any other subject, how reason can lead totruth. Of course, other sciences exhibit the power of reason, buttheres all that overhead — ferrous and ferric, dynes and ergs —that has to be dealt with. In mathematics, there is nothingstanding between the problem and the reasoning.”

But Mathematics is Useful!I am sympathetic to both Lockhart’s and Dudley’s critiques.Each is rooted in a notion of the purpose of teachingmathematics:

I Mathematics education is to teach abstract reasoning skillsand an appreciation of argument leading to truth.

I Mathematics education should foster creativity.These purposes are legitimate. But they are not the onlypurposes for mathematics, especially for university-level STEMstudents.

It’s important to remember that mathematics is tremendouslyuseful — “unreasonably effective” — in dealing with the realworld, particularly the world of science and technology whichextends beyond the intuition formed in everyday life.

The deductive reasoning that Dudley is properly concernedwith is not the only type that’s important to science. Inductivereasoning is critical.

Do you agree with these statements?

Science is ...I Studying more complex relationships than ever before.

(Example: Genome)I More quantitative than ever before.I More driven by data than ever before.I More computational than ever before.

The same could be said for technology (e.g., the smartelectricity grid), business and commerce (e.g., amazon, itunes,financial derivatives), news and publishing, government (e.g.,GIS in community planning) ...

“The avalanche of data generated bycontemporary high-throughput biology haschallenged traditional methods of drawingbiological inferences, primarily becauseinferences based on these data requiresimultaneous consideration of multiplevariables, multi-dimensional visualization, andoften also multi-disciplinary analysis. Twodisciplines that have extraordinary potential tohelp address these biological challenges aremathematics and computer science.Furthermore, mathematics will be necessaryto the development of succinct summaries ofthe causal mechanisms underlying suchcomplex phenomena.” — John Jungck [4, p.1]

Meanwhile ... back at the UniversityWhat in our curriculum has changed to support studying morecomplex, quantitative, data-driven, computational methods ofwork?

I Are STEM departments requiring more math to build onthe traditional foundation? No. (But see Bio2010.)

I Are they requiring courses in computation? No.I Are they requiring courses in statistics? Some, but not

much.

Why not?

I There’s no time. Science has itself become more complex.I There’s little confidence that the mathematics curriculum

will deliver the goods. The reader of Lockhart or Dudleywould not conclude that mathematics education is aboutsupporting science.

My Critique

It’s important to include as a substantial part of mathematicaleducation those aspects of mathematics that directly engagethe work our students will be undertaking.

Evidence we have NOT been doing this

I Ask faculty in a variety of disciplines — biology, geology,etc. — how much of their college math they use.

I Look in textbooks in these fields — how muchmathematics are they using?

The CUPM/CRAFTY Reports: 1999-2001

11 separate meetings of mathematicians and clients held in1999-2001 to provide a framework for reform of the first twoyears of the mathematics curriculum.

I Physics and CSI Math, Physics,

EngineeringI EngineeringI Health-Related Life

SciencesI Technical MathematicsI Statistics

I Business andManagement

I Teacher preparation andMathematics Education

I Biology and ChemistryI Math preparation for the

major.I All disciplines.

The CRAFTY Summary

“Students do not see the connections between mathematicsand their chosen disciplines; instead, they leave mathematicscourses with a set of skills that they are unable to apply innon-routine settings and whose importance to their futurecareers is not appreciated. Indeed, the mathematics manystudents are taught often is not the most relevant to theirchosen fields. For these reasons, faculty members outsidemathematics often perceive the mathematics community asuninterested in the needs of non-mathematics majors,especially those in introductory courses.”’

From Susan Ganter and William Barker (2003) “A CollectiveVision” in The Curriculum Foundations Project: Voices of thePartner Disciplines. Mathematical Association of America.

MAA CRAFTY Recommendations Across the Groups

What to do:I Modeling (in every report)I Multivariate topicsI Computational skills useful

in other fieldsI Statistics useful in other

fieldsI Units, scaling, dimensional

analysisI Only the “fundamental and

applicable results fromcalculus.”

What not to do:I Non-trivial algebraic

manipulations, e.g., chainrule beyond f (ax + b) orintegration by parts.

I ProofsI Use of graphing

calculators.

CRAFTY emphasis on Modeling

“The importance ascribed to mathematical modeling byevery disciplinary group in every workshop was quitestriking. After the first two years, students should be able tocreate, solve, and interpret basic mathematical models frominformal problem statements; to provide logical arguments (atan appropriate level) that the models constructed are valid; andto use the models to solve problems.” — A Collective Vision, [3,p. 4]

“The need to increase the emphasis on mathematical modelingin the first two years of the undergraduate program is a strongmessage from the Curriculum Foundations Project.” — ibid

Bio2010RECOMMENDATION #1.5Quantitative analysis, modeling, andprediction play increasingly significantday-to-day roles in today’s biomedicalresearch. To prepare for this seachange in activities, biology majorsheaded for research careers need tobe educated in a more quantitativemanner, and such quantitativeeducation may require thedevelopment of new types of courses.The committee recommends that allbiology majors master the conceptslisted below. — Bio2010, pp. 41-46

Bio 2010: Transforming Undergraduate Education for FutureResearch Biologists, National Research Council. Committee onUndergraduate Biology Education to Prepare ResearchScientists for the 21st Century, National Academies Press,2003

The Bio2010 Topics

I CalculusI Linear AlgebraI Dynamical SystemsI Probability and Statistics

I Information andComputation

I Data Structures

“In addition, the committee recommends that life sciencemajors become sufficiently familiar with the elements ofprogramming to carry out simulations of physiological,ecological, and evolutionary processes. They should be adeptat using computers to acquire and process data, carry outstatistical characterization of the data and perform statisticaltests, and graphically display data in a variety ofrepresentations.” — Bio2010, p. 63

Are We Ready for 2010?

Bio2010 RecommendationsI am unaware of any school that has come anywhere close tothe recommendations using the existing curriculum. A fewattempts to change introductory courses, e.g., providingbiocalculus courses. (e.g., Univ. of Tennessee, AppalachianState, ... ).To fulfill Bio2010 with the conventional curriculum would meanthat all biology students would take almost as muchmathematics as most math majors, as well as statistics andcomputing.

CRAFTYAnecdotally, any progress toward the CRAFTYrecommendations has been spotty at best.Do your introductory courses include multivariate topics,modeling, computation, units?

Even Earlier: “Calculus in Context” from the 1990s

“We believe that calculus can be for students what it was forEuler and the Bernoullis: a language and a tool for exploringthe whole fabric of science. We also believe that much of themathematical depth and vitality of calculus lies in connectionsto other sciences. The mathematical questions that arise arecompelling in part because the answers matter to otherdisciplines. We began our work with a “clean slate,” note byasking what parts of the traditional course to include ordiscard.” — Callahan, Cox, Hoffman, O’Shea [1, p. xvi]Now available on line: http://www.math.smith.edu/Local/cicintro/cicintro.html

Calculus in Context (cont) I

Starting Points

I Calculus is fundamentally a way of dealing with functionalrelationships that occur in scientific and mathematicalcontexts. The techniques of calculus must be subordinateto an overall view of the questions that give rise to theserelationships. [DTK — modeling]

I Technology radically enlarges the range of questions wecan explore and the ways we can answer them.Computers and graphing calculators are much more thantools for teaching the traditional calculus.

I The concept of a dynamical system is central to science.Therefore, differential equations belong at the center ofcalculus, and technology makes this possible at theintroductory level.

Calculus in Context (cont) II

Impact of Technology

I Differential equations can now be solved numerically, sothey can take their rightful place in the introductorycalculus course.

I The ability to handle data and perform many computationsmakes exploring messy, real-world problems possible.

I Since we can now deal with credible models, the role ofmodeling becomes much more central to the subject.

From [1, pp. xvi-xvii]

“Context” in the Mainstream

Graphing Calculators Still DominateSummary 25

Mainstream Calculus I Mainstream Calculus II

0

10

20

30

40

50

60P

erc

enta

ge o

f sections

Graphing calculator

Writing assignments

Computer assignments

On-line resource systems

Group projects

FIGURE S.11.1 Percentage of sections of Mainstream Calculus I and Mainstream

Calculus II taught using various reform methods in mathematics departments at four-year

colleges and universities in fall 2005.

Dec 6;Sept25(former SFY.18.1);Sept 18; Sept 8, 2006; formerly SFY.20.1

Mainstream Calculus I Mainstream Calculus II

0

10

20

30

40

50

60

70

80

90

Perc

enta

ge o

f sections taught usin

g

Graphing calculator

Writing assignments

Computerassignments

On-line resource systems

Group projects

FIGURE S.11.2 Percentage of sections in Mainstream Calculus I and Mainstream

Calculus II taught using various reform methods in mathematics programs at public two-

year colleges in fall 2005.

Dec 6; Sept25(formerSFY.18.4); Sept 18; Sept 8, 2008; formerly SFY20.4

From Statistical Abstract of Undergraduate Programs in theMathematical Sciences in the United States: Fall 2005 CBMSSurvey

University Calculus Enrollments

2005 Fall Enrollments inCalc I - 251K; Calc II - 104K; Calc III/IV - 85KFrom http://www.maa.org/columns/launchings/

launchings_06_08.html.

Dramatic Progress in Another DirectionWhatever small changes have been made in the direction of theCRAFTY recommendations, the dominant change in the earlycurriculum over the last decade has been toward teachinguniversity math in high school.

AP Calculus/Statistics Enrollments

0

50,000

100,000

150,000

200,000

250,000

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

ABBCTotal CalculusStatistics

Source: http://apcentral.collegeboard.com/apc/public/courses/teachers_corner/2151.html

BC Calculus is growing at 9-10% per year, Statistics at 10-15%.

Progression of Students Through the Sequence

Enrollments data is generally cross-sectional, not longitudinal.

Longitudinal Data from Macalester ...For 2005 Graduates (entering in Fall 2001)

Entering in ... TotalCalc I Calc II Calc III Enrollment

Calc I 112 112Calc II 46 77 123Calc III 24 28 26 78Linear algebra 12 22 13 47Diff. Eq. 5 11 10 26

Individual student retention from each course to the next is <50%. This is hidden since total enrollment from course tocourse falls much more slowly.Attrition: First course 215 → second course 87 → third course56

A Strategy for ChangeThe CRAFTY Reports“Changes in the mathematics curriculum that are advocated inthis volume will not occur easily or naturally. They will require agreat deal of effort, insight, and hard work, and will not happenuntil the mathematics community believes in the need for thesechanges.”CRAFTY calls for “establishing mechanisms by whichcolleagues from partner disciplines can collaborate withmathematics faculty” and “establish[ing] mechanisms for thedevelopment, review, and dissemination of effectiveinstructional materials and techniques.”

Bio2010“Faculty from biology, mathematics, and physical sciences mustwork collaboratively to find ways of integrating mathematics andphysical sciences into life science courses as well as providingavenues for incorporating life science examples . . . into coursestaught in mathematics and physical science.”

Developing Collaborations and Partnerships

At many universities and colleges, it is hard to use theCRAFTY/BIO2010 strategy. The partners are not ready.

I The “partner disciplines” are often not prepared to increasetheir use of mathematics in their own teaching.

I There is a disconnection between the partners and themathematics faculty:

I How to teach statistics, computation, and modeling in a waythat is useful to the partners. Example: multivariatestatistical modeling.

I What are the essential foundations even if they are notdirectly useful. Example: Partners teaching computerpackages rather than concepts.

Suggestions that are uneconomical.Bio2010 has ambitious, but utterly unrealistic,recommendations for the quantitative curriculum.

I To follow the Bio2010 recommendations with the currentmath curriculum, every biology major would also need tobe a math major and take additional computation andstatistics courses.

We need to be thoughtful about how to deliver themathematics/statistics/computational skills efficiently, withoutrequiring unrealistic numbers of additional courses.

From “What is Mathematics For?”:“People seem to think that because something involvesmathematics it is necessary to know mathematics to use it.Radio does indeed involve sines and cosines, but the personadjusting the dials needs no trigonometry. Geologists searchingfor oil do not have to solve differential equations, thoughdifferential equations may have been involved in the creation ofthe tools that geologists use.” [2, p. 611]

Insufficient Opportunity for Training

I Lack of opportunity for existing faculty to receive training.I Professional disparagement of “training.”I The training, such as it is, goes on with graduate students

who are using the books selected by an elder generationand whose perspective is strongly disciplinary.

I Standard textbook approach: Have a vision, write a book,engaging in marketing. There’s little opportunity for facultydevelopment — faculty either accept a book as a whole ornot.

The MOSAIC Strategy I

Our PremisesI There are already partners who are ready: the other

quantitative disciplines. Statistics, Computer Science,Applied Math, ...

I There are already good ideas about how to improve theearly curriculum to support the growing need for strongerquantitative skills early.

I Curricular change is intrinsically coupled to facultydevelopment.

The MOSAIC Strategy II

Our Project

I Work to integrate the various topics emphasized byCRAFTY:M odeling, S tatistics, C omputation, and C alculus

I Bring together local curricular innovators to share theirinnovations in a an environment that emphasizes theconnections among calculus, statistics, computation, andmodeling.

I Provide faculty development opportunities.

Two Phases to the MOSAIC Project

(1) Diffusion of ideasMany, small-scale meetings through webinars: M-CASTS.

I Lowers the overhead of reporting innovations.I Opportunity for comments and feedback at an early stage.I Provides ready access to collaborators and adopters.

(2) Consolidation.Forming a coherent curriculum from the local ideas.

I Rather than starting with (another) textbook, bring leaderstogether to develop summer workshops for facultydevelopment. This provides both training for young faculty,exposure to new ideas, and important feedback for thedevelopers.

I Assessment and concept inventories.

The People and Institutions Involved

PIs and co-PIs on the NSF grant.I Danny Kaplan — Macalester CollegeI Saber Elaydi — Trinity UniversityI Nicholas Horton — Smith CollegeI Eric Marland — Appalachian State UniversityI Randall Pruim — Calvin College

IMA Fadil Santosa — key role in recruiting interestedfaculty and providing professional credibility.

Program Assessment — CATALST group at UMNLogistics — Project CAUSE (Coalition for the Advancement

of Undergraduate Statistics Education), Ohio StateUniversity

Assessment Staff: PhD Candidates in Stat. Ed. (UMN)

Rebekah IsaakBA Math. (Drexel)Use of technology in teachingstatistics and assessment ofstatistical literacy, reasoning, andthinking. Connections betweenstudents’ math background andlearning statistics. (w/JoanGarfield)

Jiyoon ParkBachelors Math. Educ. (EwhaWoman’s University)MA in Math. Educ. ( UT Austin)Development of the StatisticalTeaching Inventory (w/JoanGarfield) and use of computersimulation in teaching statistics(w/Bob delMas).

Goals for this Workshop

1. Set things up to Keep People Engaged.2. Start in developing a set of essential modeling concepts —

and a set of “concept inventory” questions — to give aspecific, operational definition of what it means to teachmodeling.

3. Create an initial schedule for M-CASTS.I Sign up every participant for one M-CAST this year and two

to three over the next two years.I Define the structure of an M-CAST to make this feasible.I Identify additional people, not at this workshop, who would

give good M-CASTS.

Keeping People Engaged

1. Syllabus Construction Crews. Contribute to modelsyllabi for

I an introductory MOSAIC Calculus Course.I an introductory computational course?

2. Implementer Reports. Sign up to try out an idea that youhear about on an M-CAST. Let your experiences guideimprovement.

3. An editorial board? A book of M-CASTS? Sign-ups forimplementation of ideas presented in M-CASTS?

4. Review goals for the project and make sure that we have arealistic way to achieve them. Take assessment seriously.

The Conference Schedule & Announcements

I Wiki accounts: IMA1 (password “one”), IMA2 (password“two”), and so on, up to 7.

I Sign up for this evening’s outing: Saint Paul Saints.I Thursday outing to the Walker.I Contact me about presentations you would like to make

here, perhaps motivated by something you hear here.

James Callahan, David Cox, Kenneth Hoffman, DonalO’Shea, Harriet Pollatsek, and Lester Senechal.Calculus in Context.Five Colleges, Inc., 1994.

Underwood Dudley.What is mathematics for?Notices of the American Mathematical Society,57(5):608–613, 2010.

Susan Ganter and William Barker.A collective vision.In The Curriculum Foundations Project: Voices of thePartner Disciplines. Mathematical Association of America,2003.

John R Jungck.Challenges, connection, complexities: Educating forcollaboration.In Lynn Arthur Steen, editor, Math & Bio 2010: LinkingUndergraduate Disciplines. Mathematical Associatio ofAmerica, 2005.

Paul Lockhart.A Mathematician’s Lament: How School Cheats Us Out ofOur Most Fascinating and Imaginative Art Form.Bellevue Literary Press, 2009.