Identifying Modeling ConceptsProject Mosaic Kick-Off Workshop

Daniel Kaplan

June 30, 2010

Why Teach Modeling? I

CRAFTY emphasis on modeling and “problem solving.”

“The importance ascribed to mathematical modeling byevery disciplinary group in every workshop was quite striking.After the first two years, students should be able to create, solve,and interpret basic mathematical models from informal problemstatements; to provide logical arguments (at an appropriate level)that the models constructed are valid; and to use the models tosolve problems.” — A Collective Vision, [7, p. 4]“The need to increase the emphasis on mathematical modeling inthe first two years of the undergraduate program is a strongmessage from the Curriculum Foundations Project.” — ibid

Bio2010 in the first paragraph!

“Biological concepts and models are becoming more quantitative,and biological research has become critically dependent onconcepts and methods drawn from other scientific disciplines.” [3]

Why Teach Modeling? II

From books ...“Models are one of the critical instruments in modern science.”Morgan and Morrison, “Models as Mediators: Perspectives onNatural and Social Science” [13, p.11]

“Modeling is a method used in disciplines as diverse asmicrobiology and macroeconomics. In fact modeling is anintegral part of problem solving in any discipline.” — How toModel It: Problem Solving for the Computer Age [16, p. 1]

Purpose of this SessionStart a conversation that will lead to the development of a list ofimportant and widely applicable modeling concepts.

I We do not need to start from scratch. There are lots ofexcellent materials and curricular resources available that wecan draw on.But ... we have a particular context:

I Integration of modeling into introductory Calculus, Statistics,and Computation. If we’re going to introduce modeling in thiscontext, we need to have a road-map to guide us — Whatconcepts are being covered? Which ones have been left out?How can we include them?

I Provide one form of index into the M-CASTs and othermaterials so that we can see which ones relate to whichmodeling concepts.

I Development of a modeling concept inventory for lowerdivision undergraduates from many fields.

Mathematical Modeling in Today’s Curriculum I

I Introductory level courses, e.g., COMAP For All PracticalPurposes, often seen as alternatives to introductory calculus.

I Integration of models into calculus textbooks. Large and smallscale coverage. (Some examples, later.)

I Statistics coverage is spotty. Dominant paradigms inintroductory statistics are “tests” and “statistical graphics.”

I Junior/Senior-level course in mathematical modeling.

Modeling is even in the mainstream ... sort of.

Modeling in Statistics Courses I

I Mainstream Introductory Statistics

“Models” in fact refers to a small set of functional forms:straight-line, groupwise constant.

Modeling in Statistics Courses II

I Wide view that a 2nd statistics course should be aboutgeneral linear models and generalized linear models. (Howmany people in the audience are aware of the difference?)

I My statistical modeling approach to introductory statistics.Market percentage is zero to two decimal places.

Modeling in Computer Science Courses

Computer scientists have their own notions of “models,” distinctfrom the sorts of models used in science.

In terms of science modeling, there seems to be very little, despiteconsiderable potential, e.g.:

I Stochastic simulations.

I Sensitivity analysis.

Modeling in Calculus Courses

I Small-scale models.

I Large-scale models.

I Bullshit models.

Small-scale models.

Modeling is not the main issue here ...

I Steps aside the units issue.

I Functional form isn’t justified.Why not this? “It reaches amaximum then gradually trails offin effectiveness.”

I Ignores parameterization offunction.

Main point is to take derivatives andfind maximum analytically.

Large-scale models. I

Calculus in Context: Chapter 1. [2, p.1]

Large-scale models. II

Adler: Biocalculus text

Adler then assumes a simple form for the absorption function A(t)and finds the rate of oxygen uptake. He then considers severalother forms for the function and shows which ones lead topanting.[1]What’s very nice here is how he decomposes the problem: The keyunknowns or parameters are the fraction of air exchanged eachbreath, absorption as a function of time. These are placed in adynamical model.

Bullshit models.

Philosopher Harry G. Frankfurt, in On Bull-shit (2005), writes about the bullshitter’scomplete disregard for whether what he’ssaying corresponds to facts in the physicalworld. He “does not reject the authorityof the truth, as the liar does, and opposehimself to it. He pays no attention to it atall. By virtue of this, bullshit is a greaterenemy of the truth than lies are.” — [6]

Calculus Optimization as BS

The number of people, P,visiting a certain beach on aparticular day in January dependson the number of hours, x thatthe temperature is below 30◦Caccording to the rule

P = x3 − 12x2 + 21x + 105

where x ≥ 0.Find the value of x for themaximum and minimum numberof people who visit the beach.From MathsQuest Maths B Year 12

for Queensland, [15, p.57]

0 2 4 6 8 100

50100

150

200

Hours below 30C

Num

ber o

f Vis

itors

to B

each

What the problem is really about

An algorithm for finding extrema symbolically.

I Set dP/dx = 0 and solve.3x2∗ − 24x∗ + 21 = 0 implies

x∗ = 24±√576−2526 = 24±

√324

6 = 4±√

9 = 4± 3.

I Check d2P/dx2 at each extremum.d2P/dx2 = 6x − 24. Negative at x∗ = 1, therefore maximum.Positive at x∗ = 7, therefore minimum.

The function is posed as a cubic because the differentiation steptransforms it to a quadratic and students have learned how tosolve quadratic.

This problem teaches the wrong things!

I Should be: “Sketch out a graph that describes what youimagine the relationship is between temperature and numberof visitors.”

I The problem gives no hint that precision of the model is anissue. As a rule, calculus books don’t touch on this eventhough it is an important application of derivatives:Conditioning of calculations.

I What’s important about df /dx = 0 is not that it locates theextremum, but that near the extremum the value of theextremum is not sensitive to x . (Unless there are activeconstraints, but this is not covered in calculus books.)

I What’s important about d2f /dx2 is not that it tells whetheran extremum is a maximum or a minimum, but that itindicates how important precision in x is.

Some Examples of Models

I like to have a set of concrete examples in mind before trying togeneralize and abstract. These are drawn from my own experienceand aren’t necessarily representative.

I The sleep-wake cycle.

I Synchronization of the circadian rhythm.

I Planetary motion from Copernicus to Kepler.

I Cars at a traffic light.

I The Macroeconomy.

The Sleep-Wake Cycle: The PhenomenonData on sleep length versus previous wake length in free-runningsubjects (from Strogatz, “The Mathematical Structure of theHuman Sleep-Wake Cycle” [17]

Sleep-Wake Cycle (cont.)Organizing this by temperature oscillator phase

Sleep-Wake Model

A simple model of the ski jump pattern:

From Winfree [19, p.44]

Circadian Rhythm Phase Resetting

The Poincare Oscillator

From Leon Glass (2001) “Synchronization and rhythmic processesin physiology” Nature, [8]

Planetary Motion: Copernicus

The Sun is in the center!

Copernican model (cont.)

Two ways the new model changes the interpretation ofobservations:

I “What appear to us as motions of the sun arise not from itsmotion but from the motion of the earth and our sphere, withwhich we revolve about the sun like any other planet. Theearth has, then, more than one motion.”

I “The apparent retrograde and direct motion of the planetsarises not from their motion but from the earth’s. The motionof the earth alone, therefore, suffices to explain so manyapparent inequalities in the heavens.”

Quoted from Rosen (2004) translation. [18]

Planetary Motion: KeplerIs the circular orbit really an ellipse?Kepler computed a heliocentric angle and radius for eachobservation:

●●●

●●

●●

●●

●●

●●●● ● ●

●●●

●

●●

●

●

●●●

−2 −1 0 1

−1.

5−

0.5

0.5

1.5

x

y ●

158215821582

15821584

15841584

15841586

15861586

15861588158815881588 1590

159015901590

1593

15931593

1593

1595159515951595

Data source and motivation: [9]

Planetary Motion: Estimating Parameters

Parameterization of an Ellipse: radius = A1+B cos θ

●

●●●●●●●●

●●

●●

●●●

●●●●

●●●●●

●●

●

0 1 2 3 4 5 6

1.40

1.55

Angle

Radius

With modern statistics: How many years’ data would Kepler haveneeded to estimate the parameters convincingly? What precision isrequired for the measurements?

Cars at a Traffic Light

Problem: What’s the maximum capacity of a traffic light?

Basic Structure: Each car k is stopped a distance dk from the redlight. The light turns green. After a delay of Tk , the caraccelerates to the speed limit.Find the largest k for which the distance travelled by the car bringsit through the light before the light turns yellow.

1. What’s an appropriate dk?

2. What’s an appropriate Tk?

3. How does distance go with time after the car starts?

Based on Example 2.2.5 in [4].

The Macroe-conomy: The“Moniac”Described in “Like

Water for Money”,Steven Strogatz,NYTimes June 2,2009, Link toarticle. Video link

What is a Model?I “We can come very close to a definition of a model by

thinking of it as a purposeful representation. The word“purposeful” is an essential part of the definition. We cannotbuild a model if we do not know why we are building it, andwe cannot criticize or discuss a model except in terms of itspurpose.” — How to Model It: Problem Solving for theComputer Age [16, p. 1]

I “A model is a purposeful representation of reality. ...Mathematical models ... are models built using the tools andsubstance of mathematics (including computers and computersoftware).” — Mooney and Swift, A Course in MathematicalModeling, [12]

I “It is also important to realize at the outset that mathematicalmodelling is carried out in order to solve problems. The ideais not to produce a model which mimics a real system just forthe sake of it. Any model must have a definite purpose whichis clearly stated at the start.” — Edwards and Hamson, Guideto Mathematical Modeling [4, p. 3]

I “A mathematical model is an attempt to capture, in abstractform, the essential characteristics of an observed phenomenon.The success of the attempt depends as much (if not more) onthe modeller’s empirical knowledge of that phenomenon as onher or his mathematical ability.” —Mesterton-Gibbons, AConcrete Approach to Mathematical Modeling [11, p. xx]

I “Mathematical modeling is the link between mathematics andthe rest of the world. You ask a question. You think a bit,and then you refine the question, phrasing it in precisemathematical terms. Once the question becomes amathematics question, you use mathematics to find ananswer. Then finally (and this is the part that too manypeople forget), you have to reverse the process, translating themathematical solution back into a comprehensible,no-nonsense answer to the original question. Some people arefluent in English, and some people are fluent in calculus — wehave plenty of each. We need more people who are fluent inboth languages and are willing and able to translate. Theseare the people who will be influential in solving the problemsof the future.” — Meerschaert, Mathematical Modeling [10,p. xv]

Teaching Modeling

I “Mathematical modeling is a subject that is hard to teach. Itis what applied mathematics (or, to be precise, physicalapplied mathematics) is all about, and yet there are few textsthat approach the subject in a serious way. Partly this isbecause one learns it by practice: There are no set rules, andan understanding of the ‘right’ way to model can only bereached by familiarity with a wealth of examples.” — Fowler,Mathematical Models in the Applied Sciences [5]

I “Modeling is more like a craft than a science, and the processof learning should therefore be more like an apprenticeshipthan a course of study.” — Starfield, Smith, & Bleloch Howto Model It: Problem Solving for the Computer Age [16, p. 1]

I “It is important to explain not only what modelling is, butalso why it is worth doing. It is not merely a means of makingthe usual first-year curriculum in mathematics and statisticsmore lively and applicable. To accept that is to miss thepoint. The objective is to provide an approach to formulatingand tackling problems in terms of mathematics and statistics.Eventually, when entering employment where real problemshave to be dealt with, mathematicians will require additionalskills to those fostered by study of conventional topics on thecurriculum. The study of modeling promotes the developmentof these extra skills.” — Edwards and Hamson, [4, p. iv]

Phases of Modeling and their Purposes

Particular models have their intended purposes, but modeling as ageneric activity has it’s own purposes. There are two distinctphases: building and using models.

Building models

I Integrating component parts. Connecting things together.

I Providing a framework for the organization and analysis ofobservations and data.

I Calibration: Estimating parameters by matching observations,either quantitative or qualitative (e.g., extinction, stability,oscillation)

Phases of Modeling and their Purposes

Using models

I Theory confirmation. Can a proposed mechanism reproducean observed phenomenon?

I Sensitivity analysis. How do assumptions affect results?

I Design. It’s cheaper/safer/quicker to evaluate a design with amodel than by building the real thing.

I Prediction. What will happen? (Closely linked to design.)

Which to Teach?

Few people will build models formally, but many use them. So it’simportant to focus on how they are used and give skills for usingthem. An understanding of how they are built can, however, giveinsight into using models, e.g., providing a level of skepticism, andunderstanding of what gets omitted, an idea of how the use ofmodels might inform revision of the model.

I Model building has the potential to develop general reasoningskills and can be very engaging.

I Teach building, but make sure that model use is also taughtin a serious way.

Question: To what extent do we actually teach about usingmodels? Too often, the model is an excuse to solve, rather thancriticize or evaluate.

CRAFTY and ModelingCRAFTY does not define modeling, or describe a set of modelingskills or concepts. To quote:

“Emphasize mathematical modeling.

I “Expect students to create, solve, and interpret mathematicalmodels.

I “Provide opportunities for students to describe their results inseveral ways: analytically, graphically, numerically, and verbally.

I “Use models from the partner disciplines: students need to seemathematics in context.

“Emphasize modeling through student projects that are engaging,meaningful, and relevant to student learning and interests. Modeling is apowerful problem solving process that helps students use their skills,knowledge, and creativity to produce results and products that canbenefit society. Therefore, modeling can build student confidence,introduce them to useful and powerful elements of mathematics, andprovide a mechanism for communication, expression, and reasoning thatis cross-cultural and cross-disciplinary.” [7, p. 4.]

CASE STUDY: Functional Approximation in Modeling

I’ll give a broad list of modeling concepts later, but for now I wantto focus on a small aspect of one modeling concept: functionalapproximation.

Calc I classes routinely cover Taylor Series:

From http://en.wikipedia.org/wiki/Taylor_series.

Taylor Series/Polynomials in Calculus

Why is this so standard in calculus?

I Convergence of a series.

I Matching limits of two functions.

I Use of derivatives.

I Approximation ... but is it really? It’s an approximation thatserves the goal of finding analytic solutions, but not onewidely used in modeling.

Approximation is a broader idea, and relates to simplification andparsimony, the selection of influences, etc.

Approximation in Modeling

I Approximate a relationship over a range, not just at a point.

I The function being approximated is often known fromobservations, not from analytic derivatives.

I The criterion used is not to match derivatives, but to comeclose to the function.

We should cover least squares in calculus:

I Optimization: a calculus topic.

I Applied to polynomial model functions, but also to otherfunctional forms.

I Can be used with relationships for which the derivatives arenot known.

I Can be contrasted with Taylor polynomials, highlighting theidea of how one quantifies the quality of an approximation.

My Starter List of Modeling Concepts and Skills I

1. The selection of modeling goals.

2. The identification of an appropriate level of accuracy andprecision. “If the purpose of a model requires only a low-resolution

answer, then a low-resolution model is completely acceptable.” —

Mooney and Swift, A Course in Mathematical Modeling [12]

3. Approximation

3.1 Measuring difference and residual.3.2 Absolute and relative difference.3.3 Criteria for matching, e.g., least squares.

4. Selection of phenomena

4.1 Growth/Decay/Saturation4.2 Balance and equilibrium. (Example: ecological models without

density dependence can still capture balance, if not theequilibrium size.)

My Starter List of Modeling Concepts and Skills II

5. Decomposition into simpler subproblems.

5.1 Combination of “forces,” each of which is simply modeled.(Example: traffic light. damped harmonic oscillator forces.)

5.2 Interaction

6. Representation techniques6.1 General purpose

6.1.1 Linear functions6.1.2 Quadratics

6.2 Growth and decay

6.2.1 Exponentials6.2.2 Oscillations6.2.3 Sigmoidal functions

6.3 Data-driven models

6.3.1 Fitting/regression6.3.2 Splines/Smoothers

6.4 Randomness

6.4.1 Probability models: uniform, gaussian, poisson6.4.2 Partitioning into deterministic + random residual

My Starter List of Modeling Concepts and Skills III

7. Dimensions and Scaling

7.1 Units and dimensions7.2 Returns to scale, e.g., chemists’ law of mass action,

economists’ Cobb Douglas

8. Objective vs contraint

9. Describing differences (e.g., difference or differentialequations)

10. Detail vs parsimony

11. Extrapolation vs interpolation.

12. Vocabulary: variables, parameters, sensitivity, trade-offs,resolution, lumping, heuristics, algorithms, ....

Randy Pruim’s List of Key Features of Models

Focus Since no model can represent everything, a goodmodel must focus on those aspects of the situationthat matter most for the purposes at hand.

Simplicity The simpler the model, the easier it is to manipulate,to understand, and to communicate.

Comprehensiveness In opposition to our desire for simplicity is theneed for the model to include all the importantaspects of the situation. In words attributed toEinstein, we want our models to be “as simple aspossible, but no simpler.”

Falsifiability All models are wrong, but we should be able to tellwhen our model is “good enough” and when it is sowrong that it is no longer useful. Typically we do thisby comparing the model’s predictions with actualoutcomes.

From [14, p. 170 ]

Tasks

1. To define what aspects of modeling are particularly useful toSTEM students.

2. To identify a small set of modeling concepts and skills that awell-educated STEM student should be expected to know,and translate these into learning goals that can be assessed.

3. To associate each modeling concept with a level (introductory,intermediate, advanced) and the associated mathematical,statistical and computational background.

4. To illustrate the links between mathematical, statistical, andcomputational concepts and techniques and modelingconcepts and techniques to support teaching modeling in away that illuminates mathematics, statistics, andcomputation, and vice versa.

Topics in Calculus Relating to Modeling

1. smoothness

2. change

3. accumulation

4. optimization: fitting, parameter selection

5. approximation: linearization, taylor polynomials

6. algebra: exponentials, power laws, trigonometric forms

7. dynamics: growth, decay, oscillation

Topics in Statistics Relating to Modeling

1. randomness

2. assessment of fit (sum of square residuals)

3. model selection (is an explanatory variable significant?)

Topics in Computation to Modeling

1. Implementation of models.

2. Decomposition of problems and abstraction.

3. Iteration and accumulation

A Starter List of Student Learning Goals

I Students should recognize when SATURATION is an issueand know an appropriate technique for recognizing it.Example: Calibration curves in biosciences. Speed versusdistance in running.

I Students should recognized when EXTREMA are an issue.

I Recognize when the estimation of a STEADY RATE based ona collection of factors e.g., separating birth, death, predation,etc. each of which might be modeled simply. Example: woodproduction in a forest involves submodels for germination,growth, death, harvesting. (Mahogony model.)

I Students should recognize the TYPE OF MODEL, e.g.Dynamical (difference or differential), probabilistic, ...

I Students should recognize when each of the following isappropriate: describe typical behavior, describe uncertainty orrange of behavior.

References I

Frederick R Adler.Modeling the Dynamics of Life: Calculus and Probability forLife Scientists.Brooks/Cole, 1998.

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

National Research Council, editor.BIO2010: Transforming Undergraduate Education for FutureResesearch Biologists.National Academies Press, 2003.

Dilwyn Edwards and Mike Hamson.Guide to Mathematical Modeling.CRC Mathematical Guides, 1990.

References II

Andrew C. Fowler.Mathematical Models in the Applied Sciences.Cambridge University Press, 1997.

Harry G Frankfurt.On Bullshit.Princeton Univ. Press, 2005.

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

Leon Glass.Synchronization and rhythmic processes in physiology.Nature, 410:277–284, March 2001.

References III

Michael P McLaughlin.A tutorial on mathematical modeling.internet, 1999.

Mark M Meerschaert.Mathematical Modeling.Academic Press, 2nd edition, 1999.

Michael Mesterton-Gibbons.A Concrete Approach to Mathematical Modelling.Wiley Interscience, 1995.

Douglas Mooney and Randall Swift.A Course in Mathematical Modeling.Mathematical Association of America, 1999.

References IV

Mary S Morgan and Margaret Morrison.Models as Mediators: Perspectives on Natural and SocialScience.Cambridge Univ. Press, 1999.

Randall Pruim.Foundations and Applications of Statistics: An IntroductionUsing R.American Mathematical Society, 2010.

Nick Simpson and Rob Rowland.MathsQuest Maths B Year 12 for Queensland.Wiley, 2003.

Anthony M. Starfield, Karl A. Smith, and Andrew L. Blelock.How to model it.McGraw-Hill, 1990.

References V

Steven H. Strogatz.The Mathematical Structure of the Human Sleep-Wake Cycle.Springer, 1986.

Edward Rosen (trans.).Three Copernican Treatises:The Commentariolus ofCopernicus; The Letter against Werner; The Narratio Prima ofRheticus.Dover, 2nd revised edition, 2004.

Arthur T. Winfree.The Timing of Biological Clocks.W.H. Freeman, 1987.