The Mathematical Imagination, by Mika Munakata

As I sit in front of the television watching the Wimbledon men’s quarterfinal match (the one Roger Federer will eventually lose), I am reminded of a classic mathematics problem:

In a single-elimination tennis tournament featuring 128 players, how many matches will be played?

This seems to be a practically-minded problem. If you are the tournament director, you would need to know how many matches to schedule, in addition to when to schedule them, and on what courts.

To one interested in mathematics, however, the problem provides many opportunities for rich exploration. One could start by drawing a diagram, or by counting the number of rounds, or by considering a series that includes the number of matches played each round. Depending on the level of mathematical experience and on one’s learning tendencies, the notation used can vary from simple addition to the use of summation or binary decision trees. One can spend minutes, even hours, trying to find a suitable way of expressing the solution.

But, in every roomful of people, there is bound to be one person who will scarcely need a second to state the solution. In fact, this person would not be bothered if we were to change the parameters of the problem.

For example, what if 279 players entered the tournament? What if 8962 did? The same person would be able to use the same creative strategy to calculate the answers to these problems in a practical instance.

While it is tempting to disclose this clever person’s strategy at this point, I will resist doing so, in case there is someone out there who sees the challenge in finding this “shortcut” strategy. So…allow me to stall…

When Neil Baldwin asked me to write this article for the Cteative Research Center, I pondered the different ways in which creativity and mathematical problem solving intersect.

In a traditional mathematics class, students are rarely asked to “problem solve.” Sure, they solve many problems, but as Alan Schoenfeld — Elizabeth and Edward Conner Professor of Education and Affiliated Professor of Mathematics at the University of California at Berkeley — writes:

A problem is only a problem (as mathematicians use the word) if you don’t know how to go about solving it. A problem that has no ‘surprises’ in store, and can be solved comfortably by routine or familiar procedures (no matter how difficult) is an exercise.” (Schoenfeld, 1983, p. 41)

Consider the typical mathematics textbook that introduces the “what” (finding the area of a circle), gives examples of the “how” (multiply the appropriate numbers) and the “why” (to find the amount of carpet needed), then sets students free to answer questions #1-31 odd. This textbook is merely providing exercises. Creativity is not necessary in this example of mathematics teaching and learning.

In order to encourage creativity in mathematics, students have to be engaged in true problem solving and allowed to gain ownership of the learning that takes place. Creativity is involved when students are asked to “create” the mathematics, at least for themselves. That is, the problems that the students encounter must be new to them, and they must be encouraged to transfer their mathematics experiences and skills to this new problem. This requires some creativity. Of course, this creation doesn’t happen spontaneously, nor is it independent from practice and mastery.

The brand of creativity needed in this particular situation (of finding the area of a circle) may be different from the creativity necessary to paint a painting or compose music (two fields typically associated with the notion of creativity).

But, on the other hand, perhaps it isn’t so different. Painters and musicians don’t become so overnight. Master painters develop a sense for the use of dimensions and colors. Musicians have knowledge and experience of what notes sound good together. Similarly, the mathematics student can be guided to derive (newly for them) the formula for the area of a circle. Perhaps the teacher would ask, “how is it related to the area of a rectangle” or “how might you divide up a circle into shapes for which you know the area? How might that help you?”

The process of organizing and integrating previous knowledge is a creative one, and one that is crucial to the development of new mathematical ideas.

Now, let’s return to the original problem, “how many matches will be played?” Finding a shortcut requires one to resist the temptation to resort to formulas and algorithms (e.g., summation, powers of two, even addition). Rather, one is encouraged to take on a different perspective, to be flexible in one’s thinking, and to consider the context of the problem. Even imagining the events of the tournament, and the outcome of each match may help.

So, as a way to keep you thinking, I will end with one question: “How many winners will there be?

— Mika Munakata is associate professor of Mathematical Sciences at Montclair State University. Her research interests include Problem Solving, Program Assessment, and Teacher Development.

As she tells the CRC, “I began my career as a middle school and high school mathematics teacher. My work now at the collegiate level has direct connections to my experiences teaching at the secondary level. I typically teach undergraduate students training to be teachers; general education mathematics courses for students not necessarily training to be teachers; and content, methods, and research courses for master’s and doctoral students. I am currently co-director of an NSF-funded project that pairs graduate research students with middle school teachers. For that project, we are collecting quantitative and qualitative data to investigate the impact of the project on graduate students and middle school students and teachers.”

[Reference: Schoenfeld, A.H.: 1983, ‘The wild, wild, wild, wild, wild world of problem solving: A review of sorts’, For the Learning of Mathematics 3, 40-47.]

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