Applications of Mathematics Abound at ICME10 (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 ICME10, 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.
ICME10 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 midtolate 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
ICME10. 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 19531963 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 databased modeling problem.
An especially entertaining part of the ICME10 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 nonuniform 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 statisticsrelated sessions
that I attended at ICME10. 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
technologysupported statistics education. They chose to use the software Fathom
in their work with students aged 1719 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 computergenerated 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 databased modeling problems and application problems takes time. Many
teachers feel pressured by contentladen 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 databased 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 databased 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 ICME10 certainly reinforced my belief that engaging
students with authentic and varied mathematical applications will enhance their
understanding of the mathematics involved. Both databased and situationbased
modeling problems can help students see the wide utility of mathematics in
solving real problems.
