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**Mathematical models** are useful tools for engineers who study the effects of traffic on a bridge, telephone companies that want to know the best price to charge for long distance service, and social scientists who wish to predict trends in population and disease.

Mathematical modeling is the process of using various mathematical structures – graphs, equations, diagrams, scatterplots, tree diagrams, and so forth – to represent real world situations. The model provides an abstraction that reduces a problem to its essential characteristics.

Mathematical models are useful for a variety of reasons. Foremost, models represent the mathematical core of a situation without extraneous information. The equation a = b + 6, for instance, removes all of the unnecessary words from the original statement “Alison is 6 years older than Brenda.” In addition, people can use models to explore various scenarios cost-effectively. A jet engine manufacturer would prefer to design a mathematical model and conduct simulations on a computer, rather than incur the costs of building engines for testing purposes.

Other examples:

- The linear equation y = 0.5x models the transmission factor of two gears when the driver gear (x) is half the size of the follower gear (y).
- When a population triples each year, the function P(n) = I 3
^{n}represents the population P after n years, where I is the initial population. - The town of Koenigsberg (below, top) can be modeled with a diagram (below, bottom) where each line segment represents a bridge over the Pregel River and each dot represents a section of the city.

The town of Koenigsberg is famous because of the well known Koenigsberg Bridge Problem, which asks: Is it possible to cross each of the seven bridges in Koenigsberg exactly once and return to the original starting point? Alternatively, using the model of the town, the question becomes: Is it possible to draw the model, tracing each segment only once, without lifting your pencil from the paper? (The answer to both questions is no.)

A model is a description of a system, theory, or phenomenon that accounts for its known or inferred properties and may be used for further study of its characteristics.

By extension, a mathematical model is a mathematical structure that can be used to describe and study a real situation.

While most dictionaries and textbooks do not explicitly define “mathematical model,” Peter Tannenbaum and Robert Arnold provide an explanation in their book *Excursions in Modern Mathematics*:

When a mathematical structure such as a graph is used to describe and study a real world problem we call such a structure a mathematical model for the original problem.

(p. 170)

The National Council of Teachers of Mathematics (NCTM) suggests:

Modeling involves identifying and selecting relevant features of a real-world situation, representing those features symbolically, analyzing and reasoning about the model and the characteristics of the situation, and considering the accuracy and limitations of the model. In the program proposed [by NCTM], middle-grades students will have used linear functions to model a range of phenomena and explore some nonlinear phenomena. High school students should study modeling in greater depth, generating or using data and exploring which kinds of functions best fit or model those data.

(Principles and Standards for School Mathematics, p. 302)

Mathematical modeling is part of the mathematics curriculum at all grade levels. Even for preK-2, NCTM’s *Principles and Standards for School Mathematics (PSSM)* recommends that “students should learn to make models to represent and solve problems.” (*PSSM*, p. 94)

Read what David C. Webb has to say about mathematical models:

**Transcript from David C. Webb**

Modeling is used to promote student engagement. And by “modeling,” here we’re looking at a mathematical model. This is in contrast to a physical model, which might be a construction of a human skeletal system or a cube [made] out of clay and straws. The purpose of the mathematical models is to have students develop the mathematics from a problem context so they can make sense of the situation and make sense of the mathematics at the same time. They develop informal ideas and representations, and eventually lift those ideas into patterns and attempt to generalize the situation to describe what they are seeing. And by having students model, this gives students a sense of what it means to do mathematics.

NCTM explicitly mentions mathematical models in the Algebra Standard of *Principles and Standards for School Mathematics*. Students in middle school and high school are expected to learn to “use mathematical models to represent and understand quantitative relationships” and to develop the following skills:

- Model and solve contextualized problems using various representations, such as graphs, tables, and equations.
- Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships.
- Use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts.
- Draw reasonable conclusions about a situation being modeled.

(*PSSM*, pp. 222 and 297)

Algebra students should encounter examples of real world problem situations that they can model mathematically. These examples should appear in each section of the algebra curriculum, including linear and quadratic equations, functions, recursion, and exponential curves. The following examples are cited in *Principles and Standards for School Mathematics* as modeling situations:

- Students should have frequent experiences in modeling situations with equations of the form y = kx, such as relating the side lengths and the perimeters of similar shapes …Scatterplots and approximate lines of fit can model trends in data sets. Students also need opportunities to model relationships in everyday contexts, such as the “cellular telephone” problem. Students also should have experience in modeling situations and relationships with nonlinear functions, such as compound-interest problems, the relationship between the length of the radius of a circle and the area of the circle.

(p. 226) - Students could conduct an experiment to study the relationship between the time it takes a skateboard to roll down a ramp of fixed length and the height of the ramp … In this situation, as the height of the ramp is increased, less time is needed, suggesting that the function is decreasing. Students can discuss the suitability of linear, quadratic, exponential, and rational functions by arguing from their data or from the physics of the situation.

(p. 302) - In making choices about what kinds of situations students will model, teachers should include examples in which models can be expressed in iterative, or recursive, form. Consider the following example, adapted from National Research Council … of the elimination of a medicine from the circulatory system.

A student strained her knee in an intramural volleyball game, and her doctor prescribed an anti-inflammatory drug to reduce the swelling. She is to take two 220-milligram tablets every 8 hours for 10 days. If her kidneys filtered 60% of this drug from her body every 8 hours, how much of the drug was in her system after 10 days? How much of the drug would have been in her system if she had continued to take the drug for a year? (p. 303)

- Students should encounter a wide variety of situations that can be modeled recursively, such as interest-rate problems or situations involving the logistic equation for growth. (p. 305)

Modeling allows students to experience the real life application of mathematics. In Sarah Wallick’s class, students learned concurrently about gears and crankshafts on a tangible level, and about ratios on a mathematical level. Likewise, students in Orlando Pajon’s classroom explored real world situations of population growth while implicitly learning about linear and exponential functions.

Read what Sarah Wallick has to say about using mathematical models in the classroom:

**Transcript from Sarah Wallick**

My hope, in the work that I do at this level, is to help them develop an appreciation for the power of mathematics – that mathematics isn’t something that just sits on a shelf, separate from the rest of the world; that mathematics is something that can be used to help explain the world that they are living in and the things they see in other disciplines, whether it be science, social sciences, arts, all of those types of disciplines. What I’m hoping is that they will connect mathematics to other parts of their life, and they will continue to see the value of studying mathematics. But my secret kernel of hope is that they will start to realize the poetry as a part of mathematics that I see: the beauty of mathematics as an art form in itself.

Read what Carol Malloy has to say about Sarah Wallick’s use of modeling:

**Transcript from Carol Malloy**

Students experiment in the classroom, and they can see the rubber band go around and actually count [the rotations of each gear] so that they have a very application-based tool to figure out their answers. [Then] they move to thinking about what this means mathematically, and they are allowed to abstract and think about it and justify their answers based on the work that they had done in their small groups. And after they’ve abstracted and written the different relationships down in equations, she then says, “Well, let’s go back now and look at what we’ve done.” So she brings them back to the crankshaft at the end of the lesson.

Read what Orlando Pajon has to say about using modeling in the math classroom:

**Transcript from Orlando Pajon **

[Mathematical modeling] helps them to learn better because, first of all, it’s meaningful to them – it’s not just symbols, it’s not just numbers, it’s not just algorithms that have no sense or meaning for them. Here, every activity they go through has a meaning. And they have to explore, they have to discuss what they found during the activity, and they do it in a group setting, modeling what happens in the real world. When you go [to work], you have to work in teams and you have to come up with solutions, and you have to explain your points, you have to justify things, you have to look for patterns, you have to make predictions. And all of that ties into the way our curriculum works.

Because mathematical modeling provides an opportunity to explore math in context, it plays a vital role in the algebra classroom. Without modeling, the algebra curriculum is a collection of skills, concepts, and processes that is disconnected from the outside world. But when students model real world situations with equations, graphs, and diagrams, algebra comes alive. Students experience mathematical power for themselves when they use algebra to explain and understand the intricacies of physical phenomena.

The links below are to pages within stable sites and are current as of the date of publication of this workshop. Due to the ever-changing nature of the Web, it is possible that some links may change. Should you reach a non-working link, we recommend entering a couple words from its description into the site’s search function, or into a Web-based search engine.

**Lesson Study:**

Fernandez, Clea and Sonal Chokshi. “A Practical Guide to Translating Lesson Study for a U.S. Setting.” *Phi Delta Kappan*, October 2000; vol 84, no. 2: pp. 128-134.

Hiebert, James and James W. Stigler. “A Proposal for Improving Classroom Teaching: Lessons From the TIMSS Video Study.” *Elementary School Journal*, September 2000; vol. 101, no. 1: pp. 3-20.

Hiebert, James, Ronald Gallimore, and James W. Stigler. “A Knowledge Base for the Teaching Profession: What Would It Look Like and How Can We Get One?” *Educational Researcher*, June/July2002; vol. 31, no. 5: pp. 3-15. *Educational Researcher* is available online at http://www.aera.net/pubs.

Lewis, Catherine, et al. “Lesson Study: Crafting Learning Together.” *Northwest Teacher*, Spring 2003; vol. 4, no. 3. A PDF version of the newsletter is available for free download at http://www.nwrel.org/msec/.

Stigler, James W. and James Hiebert. *The Teaching Gap: Best Ideas From the World’s Teachers for Improving Education in the Classroom.* New York: Free Press, 1999.

Watanabe, Tad. “Anticipating Children’s Thinking: A Japanese Approach to Instruction.” *Mathematics Education Dialogues*, November 2001 (NCTM). Available online at http://math.kennesaw.edu/~twatanab/Anticipation%20NCTM%202001.pdf.

**Mathematical Modeling:**

Bender, Edward A. *An Introduction to Mathematical Modeling*. New York: Dover Publications, 2000.

COMAP and Lynn Steen. *For All Practical Purposes: Introduction to Contemporary Mathematics (3rd Edition)*. New York: W. H. Freeman & Co., 1994.

Gershenfeld, Neil. *The Nature of Mathematical Modeling*. Cambridge: Cambridge University Press, 1999.

Tannenbaum, Peter, and Robert Arnold.* Excursions in Modern Mathematics *(2nd Edition). Englewood Cliffs, NJ: Prentice Hall, 1995.

“Using Graphs, Equations, and Tables to Investigate the Elimination of Medicine from the Body: Modeling the Situation.” *Principles and Standards for School Mathematics*. Reston, VA: NCTM, 2000. Available as e-example 7.2 at http://standards.nctm.org/.

**Listening To Students:**

Reinhart, Steven C. “Never Say Anything a Kid Can Say.” *Mathematics Teaching in the Middle School,* April 2000; vol. 5, no. 8: pp. 478-83.