**Report: November, 2013
**Committee on Mathematical Modeling Courses:

Members:

Charles Collins (University of Kentucky)

Dawn Lott (Delaware State University)

Judith Miller (Georgetown University)

Joseph Malkevitch (York College (CUNY))-Chair

Phenomena to model might include the trajectory of a firework (to ensure safety at a July 4th celebration), the scheduling of the operating rooms at a hospital (to serve more patients and cut down on waiting time by doctors and patients), the blending of different gasolines (to cut carbon emissions), the search for molecules that might make safe and effective drugs, the task of ranking pages based on a string entered into a Web browser, or the restoration of electricity after a hurricane, such as Hurricane Sandy that devastated Long Island, New Jersey and the New York City area.

Many applied mathematics courses do not treat modeling as a topic but concentrate entirely on mathematical concepts and techniques, especially differential equations, integral equations, transforms and other topics in analysis. These particular tools involve continuous mathematics but many of these tools have discrete analogs. New continuous and discrete mathematical tools are constantly being developed. Modeling may draw on a wide range of mathematical tools but, crucially, embeds the use of these tools in a larger framework, non-mathematical insights. Key aspects of the modeling process include:

* looking for new problems to analyze;

* recasting verbal or graphic descriptions of problems in mathematical terms;

* making simplifying assumptions which allows a problem with many complex aspects to be replaced by a more focused simpler problem amenable to mathematical analysis;

* developing new mathematical methods or modifying or combining old methods for attacking these problems;

* interpreting mathematical results in the context in which the problem arose.

* and, mathematical modeling goes beyond problem solving in that simplifying assumptions must be made, sometimes data collected, and after a solution is found, an attempt must be made to see if the results are insightful for the original situation.

The goal of this process is to achieve insights into the "real world" and, perhaps also, to further develop the "pure mathematics" tools that are fundamental to solving the problem.

While an area of mathematics like number theory might draw on analysis, combinatorics, geometry, and many other parts of mathematics to provide gain understanding into old and emerging problems of interest to number theorists, mathematical modeling not only draws on other areas within mathematics but also builds direct ties between mathematics and other disciplines. For example, research on DNA strings has motivated a variety of problems in the developing area of the combinatorics of words and pattern matching, but constructing mathematical models to find genes in the DNA of a recently sequenced species raises many complex questions. To solve them, one might have to know a significant amount of biology and study the biology of genes as well as the combinatorics of words.

* Make students aware of the vast range of environments in which mathematics can play a vital role. In particular, students should see that mathematics can be applied in the behavioral and social sciences and perhaps even in the humanities, as well as in the physical sciences and engineering.

* Develop students' ability to formulate mathematical problems and to choose appropriate techniques for their solution. We note that these skills are crucial in pure as well as in applied mathematics.

* Develop students' ability to communicate professionally to a variety of audiences, including the “end users” of a modeling problem, who may be utterly unfamiliar with the mathematics used.

Traditionally, mathematics majors have taken a core curriculum (Calculus, linear algebra, abstract algebra, etc.) with room for electives. This core has not necessarily included a course treating modeling or even substantial applications. However, in the last 10 years many mathematics departments have moved to degree programs in which students choose one of a variety of tracks. There may be a common core across tracks, but fundamentally each track has its own core courses and electives. When an applied mathematics track is an option, a modeling course may be offered. However, the non-applied tracks may not explicitly treat applications and modeling in a way that best serves students.

We suggest that a suitably designed modeling course cannot only strengthen the background of students already committed to mathematics, but also help recruit students to the mathematics major. Conversely, experiences in modeling can lead to deep questions about the mathematical tools being used and thereby help students to appreciate the need for rigor.

We also note that broad contrasts can be drawn between types of mathematical models, which may be continuous or discrete and deterministic or probabilistic. Some students may have a particular aptitude for working with one type of model above all. Touching on several different types of models might therefore give more students a chance to shine during the course of a semester or quarter.

In the Appendix we describe four different types of modeling courses. The diversity of both modeling problems and student audiences has led us to de-emphasize specific lists of topics for the courses, as might be appropriate for, say, linear algebra. Instead, we have tried to highlight the different niches that the four courses mentioned might fill at schools with different characteristics. What is appropriate for a school with 30,000 students and a full range of graduate programs might not work at a college with 600 students and no graduate courses. Likewise, picking the right approach for a particular institution involves understanding the career paths of its mathematics majors. Are they likely to go into teaching? Business and finance? Engineering, or even law? The answers must shape the content of the course.

Students in a modeling course should learn by example and experience that the formulation and analysis of a variety of mathematical models can be carried out entirely “by hand” without the use of mathematical software. However, students should also learn how to judge when computers (including calculators) can be helpful or essential for modeling and how to choose appropriate software in this case. If mathematics majors are required to learn to use a specific software package, such as Matlab, Maple or Mathematica, in a class separate from the modeling course, then the instructor can assume a basic competence and build on students' knowledge by showing how the software can best be deployed to solve specific problems. If no such experience can be assumed, however, instructors should not underestimate the time and effort required for students to learn to use a sophisticated software package.

Traditionally, mathematical modeling courses have not emphasized the role of using statistics for modeling purposes. Given the range of problems and tools of a non-statistical variety that can be used to get mathematical insight into a real world situation, it may be desirable not to greatly emphasize statistics in a modeling course. On the other hand, many mathematics majors are not required to learn very much statistics. These students may graduate with less knowledge of statistics than would be the case if they took a liberal arts course where statistics was a unit or took a service course offered by the mathematics department (which cannot usually be used for credit towards a mathematics major) with a title such as "Introduction to Statistics." Since new standards for K-12 mathematics instruction include probability and statistics, it is especially desirable to include a statistical unit in the modeling course if the audience includes future teachers who otherwise might not be exposed to statistical reasoning.

More standard assignments and assessments:

* Drill problems to give students practice in applying new mathematical tools

* Problems in which students (individually or in groups) develop new models by modifying or extending models developed in class or in a textbook

* Problems in which students (individually or in groups) analyze parametrized models and report the results. For example, students might find the fraction of a population that must be vaccinated to prevent an epidemic (the critical vaccination threshold), given specific values of model parameters.

* Assignments in which students explore how predictions derived from a model depend on the model's parameters. For example, students might describe how the critical vaccination threshold depends on the mean number of secondary infections derived from one infected individual (the effective reproductive number) and the duration of immunity provided by the vaccine.

* In-class exams to assess mastery of the skills developed through the assignments described above.

Assignments and assessments of a type less familiar to many mathematics instructors:

* Instructors should assign essay questions throughout the term and provide guidelines for style and content of written work that correspond to a grading rubric. Mathematics instructors sometimes feel at sea when it comes to assessing technical prose, but guidance is available.

* An extended project culminating in an essay and/or presentation is desirable for two reasons. First, it requires students to integrate all the skills they are expected to acquire in the course. Second, it requires students to explain their work to an audience not intimately familiar with the problem. Indeed, it can be useful for students to write an “executive summary” of their report intended for an audience that is completely unfamiliar with the relevant mathematics. The nature of the project, and in particular the degree to which students are on their own in identifying a problem, creating a suitable model and choosing appropriate methods for its analysis, will depend on the students' background and maturity.

Reference:

Pollak, H.O., What is Mathematical Modeling?, in Mathematical Modeling Handbook, ed. H. Gould, D. Murray, A. Sanfratello, COMAP, Bedford, MA., 2011, pp. vii-xi.

Appendix I

Recommendations

Recommendations related to applied mathematics and

mathematical modeling appear below. They are given in no particular order.

1. Calculus courses should specifically include modeling tasks which show how what one learns in Calculus lays the foundation for what was to become "classical applied mathematics."

2. Provisions should be made that all students who specifically plan to do K-

12 teaching and major in mathematics be exposed to mathematical modeling,

ideally in a course with this title.

3. Mathematical modeling and applications of mathematics should be infused in all courses, no matter how theoretically their material is viewed.

Comment: One way to accomplish this infusion is by using writing projects

which require students to learn about applications related to the theory that

they are covering in a course.

4. Students should be encouraged to take discrete mathematics early in their

mathematical careers to broaden their exposure to modeling situations and new mathematical tools. Mathematics departments could consider offering parallel tracks to enter the mathematics major (e.g. both as requirements) via Calculus and discrete mathematics. Calculus and discrete mathematics courses offer many opportunities to sample mathematical modeling techniques.

5. Majors should be provided with career information and graduate program

information that builds on using the applied mathematics and mathematical

modeling skills they have acquired. Students with undergraduate degrees in

mathematics often select to use their mathematical skills in the business

world, so attention should be paid to relating how American businesses (large

and small) rely on mathematics beyond arithmetic, algebra and generalized

reasoning skills.

Learning objectives for all of these courses:

a. Expose students to a broad collection of areas to which mathematics can be applied.

b. Develop mathematical modeling and problem solving skills.

c. Learn new mathematical tools to solve problems that can arise outside or within mathematics.

Audience: Highly prepared students - meaning those who have completed a three-semester calculus sequence and linear algebra as well as advanced or elective courses such as differential equations, discrete mathematics, or probability.

This course would be appropriate for schools with sufficiently large programs that having a course with many prerequisites would not make it hard for the class to run. It could serve as a capstone course option for the mathematics major or applied mathematics tracks.

Audience: Mathematics majors who have not necessarily yet taken a large number of mathematics courses.

This course could serve as an introduction to applied mathematics for students who may plan further coursework in this area.

Audience: Mathematics majors, with a focus on the needs of those majors who hope to pursue a career in pre-college mathematics teaching.

This course is aimed at mathematics majors who plan to teach in middle school and high school. The purpose of the course is to make students familiar with models which relate to and are of value to the curriculum of the CCSS-M (Common Core State Standards - Mathematics).

Audience: Prospective elementary school teachers, and liberal arts students in general.

This course would have no prerequisites beyond high school algebra. The course might be recommended or required by the college for students as part of a program to train elementary school teachers. Given the importance of statistical thinking in modern society, at least one unit in this course should focus on statistics.

Generic course description (Courses I, II, III)

Ways of using mathematics in subjects outside of mathematics will be developed. Real world situations will be examined and mathematical tools (continuous, discrete) applied to them in order to get insight. Situations will be drawn from different areas of knowledge and new mathematical tools will be developed as needed to analyze the situations examined.

Units for Course IV (Modeling for elementary school teachers and liberal arts students)

a. Statistics and Probability

b. Daily life models (urban operations research, financial mathematics)

c. Elections and voting

d. Fairness models

e. Models in the physical sciences and social sciences

Need for background in trigonometry and algebra should be minimized and reviewed as necessary for Course IV.

What should actually be taught in each of the courses above? For courses with titles such as Modern Algebra, Linear Algebra, Number Theory, etc. there is a core of topics that typically are taught, perhaps augmented with special topics that reflect the interests of the instructor. By contrast, mathematical modeling courses as they exist today are remarkably diverse in both mathematical content and areas of application. Topics for modeling should in part be chosen with an eye to students' interests and career plans, as we have noted above. They can also be chosen to reflect current events: in the early 2010s, for example, “hot topics” might include climate change, networks (or electricity suppliers, social networks or inter-bank transactions), ad auctions, influenza epidemics, and many others. Finally, it is legitimate for each instructor's taste to inform the choice of topics. Indeed, faculty members unaccustomed to modeling may be more willing to teach the course if they have latitude in selecting areas of application. With this in mind, some different ways to come at modeling are offered, and, of course, instructors are free to use a mixture of approaches.

Actuarial Science

Annuities

Pricing futures and options

Compartment models

Ecological models

Classical genetics (Hardy-Weinberg equation)

Mechanism design

Scheduling

Flexible manufacturing

Resource allocation

Diffusion

Knotted molecules

Image analysis and manipulation

Random number generators

Codes and their complexity

Predator-prey

Animal and plant harvesting

Sustainability

Market design

Nash equilibria

Game theory

Price of stability and price of anarchy

Signal processing

Acoustics

Control theory

Numerical methods

Gene identification

Pattern matching

Phylogeny

Medical imaging

Medical databases

School bus routing, meals on wheels, truck deliveries

Facility location

Planetary orbits and satellite orbits

Special relativity

Quantum mechanics

Paired comparisons

Decision making and risk perception

a. Differential equations

* Newton's Law of Cooling

* Projective motion (falling bodies)

* Radioactive decay

* Population growth

exponential growth

logistic growth

* Animal harvesting

* Mixing problems

* Predator-prey

* Natural selection

* Diffusion between compartments

b. Derivatives

* Maximization

* Approximation of functions

* Stability analysis

c. Integrals

* Solving pure-time differential equations

* Solving autonomous equations with separation of variables

d. Partial differential equations

* Heat flow

* Diffusion through porous media

e. Fourier series

* Study of periodic phenomena

* Electrical engineering

* Quantum mechanics

f. Special functions and orthogonal polynomials

Discrete tools

a. Discrete-time dynamical systems (recursion equations)

* Population growth

* Animal harvesting Credit card and mortgage payments

* Annuities

* Newton's method

* Natural selection

* Diffusion in discrete time and space (mixing)

b. Graph theory models

* Garbage collection, snow removal, pot hole inspection (Chinese postman problem)

* School bus routing, meals on wheels, manufacture of integrated circuits (TSP)

* Critical path scheduling

* Six degrees of separation and connectivity of networks

c. Mathematical programming

* Resource allocation

* Scheduling

* Discrete optimization

Stochastic models

* Diffusion

* Birth/death processes

* Queuing theory

Queues at bridges, tunnels, supermarkets, banks, computer services

1. Optimization

* Maxima and minima problems (Calculus, calculus of variations, isoperimetry)

* Linear and integer programming (linear algebra)

* Urban operations research (graph theory, mathematical programming, queues)

2. Growth and Change

* Population growth for human, animals, bacterial and viral populations (differential equations and difference equations, exponential growth)

* Animal and plant harvesting

* Changes in temperature, rain patterns, availability of water, CO2 in the atmosphere

3. Information

* Codes to hide information (security for ATM machines, phone calls of government officials, email security)

* Codes to compress information (cell phone, HDTV, DVD, CD compression systems)

* Codes to correct information (cell phone, HDTV, DVD, CD, error correction systems)

* Codes to synchronize information

* Bar codes and codes to track information

4. Fairness and Equity

* Elections

* Apportionment

* Bankruptcy

* Games

* Weighted voting

* Fair division

* School choice

* Kidney exchange

5. Risk

* Decision making with uncertainty

* Analysis of lotteries

6. Shape and Space

* The nature of physical space

* Relativity

* Image processing

7. Pattern and Symmetry

*Mathematics for design

*Mathematics in art

8. Order and Disorder

*Randomness

*Ramsey Theory

9. Reconstruction (from partial information)

*Samples to reconstruct information about a population

* Data mining

10. Conflict and Cooperation

* War and escalation of crises

* Congestion games

* Mechanisms for regulation of businesses

11. Similarity and Dissimilarity

* Antiterrorism

* Workplace security

* Image processing

* Face, fingerprint, and iris identification

12. Close Together and Far Apart

* Development of new drugs

* Distance between species, bird songs, viruses, etc. (Hamming and Levenschtein distance)

13. Unintuitive Behavior

* Prisoner's dilemma

* Braess's paradox

* Vaccination and HIV intervention programs

* Paradoxes in scheduling

* Voting paradoxes

Model with Mathematics

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Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

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Even for mathematics majors who are not planning to teach at the time they are students, making sure that modeling course students are aware of the importance of modeling and applications in the CCSS-M would be of value to our society in general.

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Mathematical modeling competitions

Undergraduate competition developed by the Consortium for Mathematics and Its Applications