Applications of Mathematics Abound at ICME-10 (Daren Starnes, The Webb Schools, Claremont, CA).

From the opening session at the International Congress on Mathematical Education (ICME) in Copenhagen, it became abundantly clear that applications in the mathematics curriculum would permeate the lectures, discussion groups, poster sessions, national presentations, and the thematic afternoon. My intent in this brief report is to provide an overview of the variety of mathematical applications that were shared at ICME-10, along with some ideas for using such applications in mathematics classrooms. At the end of the paper, I will also relate some potential obstacles that might discourage teachers from incorporating applications in their classes on a regular basis.

ICME-10 was held at the Danish Technical University (DTU) in Lyngby, about 20 kilometers outside Copenhagen. During his welcoming remarks, the mayor of Lyngby surprised most of the 2300 mathematics educators in the audience by revealing that he had earned a Ph. D. in mathematics from DTU some years earlier. At one point in his presentation, the mayor displayed a graph showing the population of Lyngby over time.
The graph showed what appeared to be exponential growth in Lyngby’s population in the mid-to-late 1900’s, followed by a small decline (coinciding with the relocation of DTU from Copenhagen to Lyngby, which eliminated considerable land from being available
for housing development), and then small growth more recently. The question the mayor posed to the assembled group was, “What happens next?” Would exponential growth resume, or would some other mathematical function better describe the population of Lyngby over the short term? This preliminary application nicely represents the idea of using functions to model relationships between two variables.

Classroom ideas: Some related examples for the classroom would be asking students to construct a model of world population growth over time, or of the number of Starbucks stores that were open in each year since the company’s founding. It seems critical that students should see examples of both exponential (unconstrained) and logistic (constrained) growth.

Hundreds of posters addressing topics in mathematics education were presented at ICME-10. Several of them focused on applications and mathematical modeling. Two posters especially captured my interest. The first, which was designed by Thorir Sigurdsson from Iceland, examined the stunning decline of the herring population in the ocean between Iceland and Norway in the 1960’s. In his poster, Thorir useds available data from 1953-1963 together with a combination of trigonometric and sinusoidal functions ( S = Aebt [1 + C sin(Dt )] , where S is the stock size and t is time) to construct a very reasonable model for the herring stock from 1964 onward. He went on to assess the quality of his model in light of actual herring stock sizes.

The second poster, authored by Chris Haines and Rosalind Crouch from England, was titled “Real World – Mathematical Model Transitions.” Their primary modeling task was to have students examine whether opening an express line for customers with a small number of items to purchase would decrease waiting times at a supermarket with numerous checkout lines. This scenario differs in an essential way from the two earlier population modeling examples; namely, the goal is not to fit a functional model to existing data. Instead, students must make some simplifying assumptions about the number of checkouts, the distribution of customer arrivals at the checkout, and the number of items that each customer will purchase. Students can then use simulation techniques to compare the average waiting times for customers under each of the two plans: no express lane or one express lane. Although this is a somewhat “messier” modeling problem, it is quite rich in mathematical thinking and in applicability. I should mention that Haines and Crouch also suggested modeling the height of a sunflower while it is growing, a more data-based modeling problem.

An especially entertaining part of the ICME-10 evening program was the final round of the KappAbel Competition, in which students from the five Nordic countries made dynamic presentations on this year’s theme, mathematics and music. A charming group of five Danish 8th graders dazzled audience members with a brilliant blend of song and accompaniment on water glasses. (Their command of spoken English was equally impressive.) After their musical performance, the team members proceeded to analyze the mathematics behind the tones that they had produced on the water glasses. They
explained that the frequencies of consecutive notes on the scale are in a ratio of 12 2 . The students also performed simple experiments to investigate such things as the effect of the temperature of the water in the glass on frequency and of using a liquid that was not water on the frequency (some effect). This engaging activity is an excellent illustration of applications of mathematics in other disciplines.

There is perhaps no subject that is more riferipe with mathematical applications than physics. I attended two lectures that reaffirmed this belief: “Use of mathematics in other disciplines” and “Mathematics Learning and Experimenting with Physical Phenomena”. In the former, the speaker acknowledged that in many high schools, teachers tend to stay compartmentalized by discipline. As a result, mathematics and physics teachers do not always interact comfortably. Not surprisingly, students sometimes feel that there are missing connections between mathematical ideas and physics concepts.

Very few advanced students who were taking both advanced math and physics classes could solve this problem, in spite of the fact that they had been taught all of the necessary mathematics and physics to do so. The speaker hypothesized that many physics teachers try to minimize the mathematical level of the physics topics that they present and that the math teachers generally avoid using such complex physics examples.

In the second lecture connecting mathematics and physics, one of the speakers (Apolinario Barros) presented the results of a mathematical experiment involving circular motion that his students had performed. Using a motion detector connected to a computer data collection program and a wooden wheel with a speed control, students were challenged to match a position versus time graph of a large orange ball that was “velcroed” to the wheel. It was fascinating to listen to students’ reactions to an initial attempt to match the graph, as well as to the suggestions that led them through subsequent repetitions of the experiment. Apolinario offered what he described as a “paradigm for describing mathematical experiments”:
• Action/response: One takes an action and measures a response to that action.
• Collective interpretation: Teacher and students offer interpretation in the process of trying to solve a problem.

This relates directly to the process of mathematical modeling, in which one : defines a research question/problem to solve, collects appropriate data, formulates a mathematical model, the tests the model, and refines the model. An additional problem that was posed in the same session that illustrates connections between calculus and physics.

Problem: Consider a bar of charge with charge density of (5 + 7 x)µC / m , where x is measured in meters and x = 0 at the left end of the bar. If the bar is half a meter long, what is the total charge on the bar?

Classroom idea: Construct a similar non-uniform density problem with a baseball bat.

Because my own interests have pulled me deeper into statistics teaching these past few years, I was particularly interested in the statistics-related sessions that I attended at ICME-10. My favorite was a talk given by Rolf Biehler from the University of Kassel in Germany. He spoke in some detail about results from research that he and his colleagues have conducted in the area of technology-supported statistics education. They chose to use the software Fathom in their work with students aged 17-19 and with student teachers at the university level.

As a preliminary example of using simulation for studying variation in probability, Biehler Rolf suggested the following activity: have students write down a sequence simulating a sequence of random births of boys and girls. He compared students’ typical sequences to computer-generated sequences from Fathom. Students’ sequences tended to switch back and forth from boys to girls more frequently, and included fewer long “runs” of births of the same gender than did the Fathom sequences.

Rolf’s presentation also illustrated the kinds of questions students can ask and then produce data to answer in high school mathematics classes:
• How do males and females differ in TV watching time?
• Do those who watch more TV tend to read less?
Students can easily collect data related to their questions of interest, then use mathematical tools to analyze the data they have collected. Questions like the first one above allow for graphical and numerical comparisons of univariate data sets – boxplots, stemplots, histograms, as well as measures of shape, center, and spread. The second question is ideal for examining correlation, and perhaps fitting a regression model to the data.

What obstacles might prevent mathematics teachers from incorporating applications from other disciplines or to use modeling problems in their classrooms on a regular basis?

1. Lack of subject matter knowledge.
Some math teachers’ ready recall of physics content may be quite limited. As a result, they would be hesitant to utilize rich physics applications in their classes. Suggestion: Arrange a conversation with a colleague who teaches physics. Forming such a collaborative partnership can be mutually beneficial! It will allow you to use
interesting physics applications and it will also help the physics teacher feel more comfortable incorporating additional mathematics content in the context of their physics teaching. Your students may begin to transfer their understanding from one class to the other more readily.
2. Time.
Solving data-based modeling problems and application problems takes time. Many teachers feel pressured by content-laden syllabi, which discourages them from allocating class time to modeling and applications.
Suggestion: Examine the feasibility of adding one engaging application/modeling problem per unit. Students could be given the problem description and some instructions for thinking about the problem before class to reduce the amount of class time required.
3. Discomfort with problems that have multiple solutions.
Most standard textbook questions (and even classroom examples) have a single correct answer. A mathematical modeling problem can have a host of reasonable solutions, depending on the assumptions that are made and the approach to solving the problem that is taken.
Suggestion: Encourage students to employ alternative methods in the course of solving a data-based modeling problem. For instance, have students compare linear, power, and exponential models for a data set showing a relationship between variables. Ask students to select the one they consider the “best” model, and to be prepared to justify their choice.
4. Difficult to assess/evaluate.
Unlike standard computational problems, modeling and data-based application problems cannot be easily evaluated using an analytic grading system. These more complex problems can also require more time to evaluate.
Suggestion: Consider using a rubric that focuses on the critical elements of the problem’s solution and on the quality of a student’s communication of his or /her mathematical thinking. Remember that you are assessing the process leading to the development of the model, as well as the quality of the model that the student has selected.

My own experiences at ICME-10 certainly reinforced my belief that engaging students with authentic and varied mathematical applications will enhance their understanding of the mathematics involved. Both data-based and situation-based modeling problems can help students see the wide utility of mathematics in solving real problems.

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