Mathematical Modeling Handbook II Sample

Mathematical Modeling Handbook II:The Assessments

Prepared under the direction ofThe Program in Mathematics Education

Copyright © 2013 Comap, Inc.Printed in the USA

Publishedby

The Consortium for Mathematics and Its ApplicationsBedford, MA 01730

www.comap.com.

Front Cover Photo Credits. See respective Module

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TABLE OF CONTENTS

Mathematical Modeling Modules

COULD KING KONG EXIST? ..................................................................................1A MODEL SOLAR SYSTEM......................................................................................2FOR THE BIRDS ..........................................................................................................3ON SAFARI ....................................................................................................................4CHOOSING A COLLEGE............................................................................................5A TOUR OF JAFFA ......................................................................................................6GAUGING RAINFALL ................................................................................................7NARROW CORRIDOR................................................................................................8TALE OF THE TAPE ..................................................................................................9UNSTABLE TABLE ..................................................................................................10SUNKEN TREASURE ..............................................................................................11ESTIMATING TEMPERATURES ........................................................................12BENDING STEEL ......................................................................................................13A BIT OF INFORMATION......................................................................................14STATE APPORTIONMENT ..................................................................................15RATING SYSTEMS ..................................................................................................16THE WHE TO PLAY ................................................................................................17WATER DOWN THE DRAIN................................................................................18VIRAL MARKETING................................................................................................19SUNRISE, SUNSET ..................................................................................................20SURVEYING THE ANCIENT WORLD ..............................................................21PACKERS' PUZZLE ..................................................................................................22FLIPPING FOR A GRADE ......................................................................................23PRESCIENT GRADING ..........................................................................................24PICKING A PAINTING ............................................................................................25CHANGING IT UP ....................................................................................................26

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INTRODUCTION Assessing Mathematical Modeling

Prof. Dr. Pauline Vos University of Agder, Norway

Member of the ICTMA Executive Committee (International Commission on the Teaching of Mathematical Modeling and Applications)

Assessment of Professional Mathematical ModelersProfessionals in all kinds of fields—from furniture movers and business managers to airport con‐trollers and climate researchers—carry out mathematical modeling activities to predict, plan, se‐lect, understand, and solve real‐world problems. Assessment of their mathematical modelingactivities is part of the complete modeling process—the proof is in the pudding. If the outcome of themathematical modeling activities is wrong, furniture may end up stuck in the corner of a narrowcorridor, businesses may face bankruptcy, planes may be delayed, or CO2 emissions may be en‐hanced rather than reduced. The assessment of professional mathematical modeling takes place inthe natural habitat of the professional modeler. As a result, the consequences may be severe (maybechange to “the impact may be significant”). If the modeler fails to create an accurate and effectivemathematical model, he or she may face serious consequences, but also the future of others may beat stake. Conversely, if the modeling work is carried out well, the world may be a bit brighter, not onlyfor the modeler, but for other stakeholders as well.

Mathematical modeling has become an integral part of modern society. In business, engineering, andmany other fields, mathematical models play a vital role to let things run. With such a vital role, itmakes sense that mathematical modeling entered into education, starting at colleges and universi‐ties and, more recently, at high schools and below. Mathematical modeling increasingly has becomepart of regular mathematics education, because we want students to develop competencies in ap‐plying mathematics and building mathematical models. Through modeling, students learn how math‐ematics is a useful tool for solving real‐world problems in many areas of modern society. In the firstMathematical Modeling Handbook, the modules “Narrow Corridor,” “Unstable Table,” and “BendingSteel” give students a sense of how modeling is important in engineering. The modules “GaugingRainfall,” “Estimating Temperatures,” and “Water Down the Drain” demonstrate the use of mathe‐matical models to explore environmental issues. Also, the Handbook contains a number of modulesthat focus on optimization, which is a mathematical way of rationalizing a decision process – anddecision‐making processes are pivotal in life – demonstrating again the usefulness of mathematics.

Modeling also is a tool for mathematics learning, because it gives meaning to mathematics and servesto support mathematical understanding. For example, the module “Flipping for a Grade” explains tostudents the mathematical concepts of standard deviation and expected value by comparing twogames: in one game you flip a coin and either win 2 points or lose 1 point; in the other game you flipa coin and either win 100 points or lose 99 points. This module assists students in developing a bet‐ter understanding of probability concepts; this module could be used to introduce standard devia‐tion and expected value before these concepts are formalized. For a curriculum developer, thismodule demonstrates that modeling education serves to support mathematics education. Mathe‐matical modeling does not necessarily need to be an application after students learn the requiredmathematics. On the contrary, modeling may also be useful at the beginning of the learning processes

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in the mathematics classroom. In this way, learning mathematics and mathematical modeling areintertwined. In fact, a number of new mathematical discoveries were made by professional model‐ers in search of better mathematical models.

Linking Mathematical Modeling Competencies to Assessment AimsThe difference between professional modelers and student modelers is clear: a professional aims tosolve real‐world problems by using mathematical models, while students aim to learn how to model.This means that mathematical modeling education has many goals. First, students should learn tomake the distinction between a certain problem and the mathematics that can be used to solve thisproblem. Students need to select the relevant information from the problem situation, structure theproblem, identify and name variables, and translate the problem into a mathematical model (for‐mula, graphs, diagrams, etc.). Thereafter, they will need to do the mathematics and interpret the an‐swer in terms of outcomes that help solve the initial problem posed. If the results do not suffice,students need to improve or correct their work, for example, by refining the mathematical model.

By comparing the work of professionals we can see that there are many more competencies neededin modeling. Real‐life problems are often fuzzy and not well defined; therefore, making assumptionsabout a problem is another modeling competency. Justifying one’s choices, arguing why a certainassumption was made, and presenting a solution are part of a professional modeler’s work. There areeven more modeling competencies. During the modeling process, one may need to search for addi‐tional information, consult with others, and improve one’s mathematical knowledge and skills; thus,monitoring one’s own learning process is another modeling competency.

Of course, we cannot expect (and we do not want) students to be professionals. Having the goal thatstudents learn to model, we need environments in which the work of professional modelers is sim‐ulated. In this way, we offer students a training ground, allowing them to feel safe and get exposureto modeling experiences. The Mathematical Modeling Handbookoffers this environment. In this sim‐ulation environment there are many differences from the world of professionals. First, students arenot given full responsibility over the outcomes of modeling activities. In education, students are al‐lowed to make errors and learn from them. We assist students in making progress by scaffoldingtasks, dividing activities into smaller chunks, and offering additional hints.

In education, we want to facilitate students’ learning. And in education we need to find out whetherlearning was achieved. This also applies to mathematical modeling. After students have completedone of the modules, we may want to find out what students have learned. This is the type of assess‐ment that comes after a module and I will write more about this in the paragraphs to come. Findingout what students have learned, however, is only one reason for assessment. There also can be otheraims for assessment. We see this in this book in the Readiness Tests that come with each module. Thistype of test has the purpose of diagnosing, that is, to inform teachers on whether students have suf‐ficient basic knowledge about the mathematics required in the module. Simultaneously, this type oftest reveals to students the mathematics needed in the forthcoming two lessons. For example, thereadiness test for “Could King Kong Exist?” tells students that calculations relating length and area(measurement conversions between dimensions) will be the mathematical topics needed in theforthcoming lessons. Because of their nature, the readiness tests are quite short and focus primarilyon mathematical readiness. These tests contain some pure mathematical items. These tests also con‐tain some contextual items involving mathematical modeling, but these items are more like wordproblems in which mathematics is “dressed up” and the required answer is simply a number ratherthan a solution to a problem worth solving.

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Linking Mathematical Modeling Aims to Assessment PracticeIf we want to assess all possible modeling competencies, we need to create a complete simulation ofa professional modeler’s work. This would mean that we would need to expose students to complex,open‐ended problems that are not well delineated, require more mathematics than students initiallyhad at their disposal, and for which an incorrect solution may have big consequences. Of course, thisis out of the question.

After students complete a module, we want to know what they have gained in terms of modelingcompetencies. This is an important educational justification of assessment. However, the assess‐ment of mathematical modeling competencies is not an easy issue, especially because mathematicalmodeling entails a large variety of activities. We cannot say that mathematical modeling is made upof a linear chain of activities that we can parse and assess one activity after another. Mathematicalmodeling is far more complex, requiring competencies such as monitoring the modeling progress,justifying choices, and reasoning back‐and‐forth. These competencies are needed throughout themodeling process and cannot easily be separated from other activities.

There is little doubt that the standard assessment modes that are traditionally used in mathematicseducation are too limited to assess the wide variety of modeling competencies. And therefore, in as‐sessing modeling we need to rethink and renew assessment modes. To assess mathematical modeling we need to be innovative and practical, and therefore we need tomake a number of choices, for example, between the extent to which we break up modeling into sep‐arate steps or take modeling holistically. Another choice pertains to the question of how far we ex‐pose students to new contexts and ask them to model a different problem than they encountered inthe lessons.

In assessing modules from the Mathematical Modeling Handbook, the first choice is dealt with intwo ways. On the one hand, there is the assessment of separate modeling competencies. For exam‐ple, the Multiple Choice Items test for a variety of competencies, some of which are purely mathe‐matical, while others deal with different modeling competencies such as interpreting models or usinggiven models to reason about model properties. We see also that students are asked to reflect onthe validity of their models in light of the phenomena that are modeled. In the Error Analysis items,students are asked explicitly to reflect on the power of the mathematical model using mathematicaltools.

On the other hand, the modeling process is assessed holistically through the Cloze Paragraph. In thistest format, students must fill in the blanks of a given text. An outsider would not recognize theseitems as mathematical items, because the required answers are NOT numbers but words. This for‐mat is not easy to carry out because the required words are not stated at the side of the text. To fillin the blanks, deep mathematical thinking is required. One needs to remember definitions and rela‐tions within the mathematical model, and have insight into the limitations of the model.

Another format to assess the modeling process holistically is the Model‐Building Activities. This for‐mat does not ask students to reproduce what was done in the lessons as in the Cloze Paragraph, butrather asks students to produce something new. This format offers students a different situationthan the one from the module, with a different problem, but nevertheless related. Students cannotfully rely on their routines and procedures but need to transfer from the previous lessons to a newsituation. This format assesses many important modeling competencies, in particular, the compe‐tencies of creativity and being able to apply prior knowledge to new situations.

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Coda for MathematiciansOn first sight it may look as if the assessment of mathematical modeling in this volume contains toolittle mathematics and too much text. In particular, the Cloze Paragraph has no calculations or proofs;it is narration only. But a closer look reveals that this assessment format asks students to use a richmathematical vocabulary. They have to make sentences with words such as function, factor, dimen‐sion, and mathematical model. These tasks ask students to reason about properties of their mathe‐matical models and about what is being modeled. Students have to verbalize this reasoning. Isn’tthis also part of doing mathematics?

This type of work is different from the mathematics that I learned when I was a university studentin the 1980’s. I took the course Mathematical Models, which consisted of two parts: continuous mod‐els and discrete models. The first part started with a symbolic notation, dp/dt =f(p), and thereafterwe learned about initial conditions, separating variables, homogeneous parts, characteristic, ordinaryand first‐ and second‐degree equations, and we learned about procedures to transform one type ofequation into another. All the work was symbolic, and there were no graphs and no applications.We did many proofs, which consisted of demonstrating that certain classes of formulae fitted certaindifferential equations. I passed the test because it consisted of given equations, and I was good atmemorizing the tricks to solve them. I do not remember that my classmates and I were told that theequations could be descriptions of phenomena outside mathematics. Nor were we told that the tech‐niques we were taught worked only for a limited class of equations, and that many other equationswere unsolvable. We learned symbolic language only. We did not learn where “the givens” camefrom, or what could be done with the answers we found. It was the time when slope fields were notyet available to demonstrate differential equations graphically, and terms such as chaos and attrac‐tors were not yet part of the mathematical jargon.

In three decades much has changed. Courses on mathematical models nowadays look different, andso does assessment. To train students to become able mathematical modelers, we may consider as‐sessment formats that do not just ask for numerical answers or solutions to difference and differ‐ential equations. This volume offers innovation in assessment. Making students verbalizemathematical reasoning through the Cloze Paragraph is an alternative way of assessing mathemat‐ical competencies. The required mathematical knowledge is by no means simple. Try them!

Coda for Mathematics EducatorsWhen we want students to learn about mathematical modeling and to engage in mathematical mod‐eling activities, these activities will show variety. This variety is inherent in mathematical modeling.Since creativity is part of mathematical modeling, collaboration is part of mathematical modeling, andeven searching the internet for additional information is part of mathematical modeling.

For the assessment of mathematical modeling, we face a number of problems: how to evaluate cre‐ativity, collaboration, or the search for additional information? How do we deal with the fact thatmodeling tasks always will lead to a variety of students’ answers? From the viewpoint of mathe‐matical modeling, we perceive this variety in possible answers from students as inherent to model‐ing activities. From the viewpoint of assessment, we perceive this variety as problematic: how toevaluate students’ answers? What do we do if one student comes up with a very sophisticated modelbut makes a careless calculation error, while another student oversimplifies the problem, creates amodel with little understanding of the phenomena described but carries out the calculations per‐fectly? How do we compare the performances of these students?

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My answer is that this assessment problem should not stop us from incorporating mathematicalmodeling into education. Should we only teach what can be assessed? That would severely hindereducation. Instead of adapting education to the needs of assessment,, we can develop new assess‐ment strategies that match the needs of education. This handbook offers innovative assessment for‐mats for mathematical modeling. Criteria for their evaluation are also given. Maybe not all students’answers are covered. In that case, we can discuss better assessment criteria with our colleagues atschool or in other forums. If there are alternative answers that weren’t covered by the criteria, weare blessed that our students haven’t turned into robots yet.

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Assessing“OnSafari”

AvailableJuly2013

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5 7.2 17.18

6 6 16.97

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9 4 18.03

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y+ 5

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c = x +3( )2+ 3x + 2( )

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Assessing “Sunken Treasure”


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Assessing “Sunken Treasure” Rebecca Johnson

Teachers College Columbia University

Brandie Waid

Teachers College Columbia University

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Assessing“BendingSteel”

AvailableJuly2013

Assessing “A Bit of Information”

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Assessing“Sunrise,Sunset

AvailableJuly2013

Post-Test Description

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Assessing“PrescientGrading”

AvailableJuly2013

Assessing“PickingAPainting”

AvailableJuly2013

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c.

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Mathematical Modeling Handbook II Sample

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