Using Role-Playing to Teach Mathematics:
A Hare-Brained Idea?

Philip J. Owens
Austin Community College
Austin, TX
powens@austincc.edu
http://www.austin.cc.tx.us/powens/
6/24/97
 Abstract The Question Month 1 Introduction The Method Month 2 The Fibonacci Sequence Conclusion Month 3 Population Growth Historical Note Month 4 The Fibonacci Bunny Month 5

Abstract

At Austin Community College we teach a general education mathematics course whose content is nicely described by its title: Mathematics: Its Spirit and Use.

Getting and keeping student attention in any math class can be a challenge. But if we as teachers can engage our students actively and amusingly in the process of math, education suddenly becomes fun! Here's a lesson that allows students to "give their bodies to science" in order to learn math.

Our lesson is based on "The Fibonacci Bunny", as introduced by the excellent book Excursions in Modern Mathematics, by Peter Tannenbaum and Robert Arnold (Prentice-Hall, 1995), chapter 10, p. 330. The nomenclature "Fibonacci Bunny" is theirs, for which I thank them. The method of teaching the Fibonacci Bunny is mine, having come to me about 30 minutes before the class in which I was to present it. It worked very nicely - hence this note!

Introduction

One usually encounters "role-playing" as a device to be used in courses having to do with social psychology, law enforcement, clinical psychology, and so on. It is used with great seriousness and to good effect in those disciplines. But I believe it can be used, in perhaps a more playful way, to teach certain aspects of mathematics. Here is how I have used it to teach a lesson for the course Mathematics: Its Spirit and Use.

Let me start by establishing the mathematical setting of this lesson, which involves putting together the ideas of population growth and the Fibonacci sequence.

The Fibonacci Sequence

The students will already have been introduced to the Fibonacci sequence of numbers:

1, 1, 2, 3, 5, 8, 13, 21, … and so on forever

Population Growth

One way to study population growth is to think of it in terms of transition rules. For example, if we put two fish in a pond, and on the first of each month, we count the number of fish, we might come up with the following tabulation:

Month 1: 2
Month 2: 4
Month 3: 8
Month 4: 16

We would conclude that the fish population is doubling every month. That would be the transition rule for this population.

The Fibonacci Bunny

We now imagine a population of bunnies (why "Fibonacci" will become apparent as we proceed). We describe the transition rules:

• We start with one pair of bunnies (meet Mr. and Mrs. Fibonacci Bunny):
• At the end of month one, they produce a baby male-female pair.
• It takes a month for a newborn pair to achieve adulthood, and another month for them to produce another pair, after which they produce a pair every month.
• In the meantime, Mr. and Mrs. F are producing a pair each month.
• Etc. etc.

The Question

How many pairs of bunnies will there be at the beginning of each month, according to the above population transition rules?

The Method

One obvious method would be to explain one's way laboriously through the process one month at a time, keeping track of numbers at the board while the students sleep in their seats. It occurred to me that I might engage the students better if I had them play exploding bunny population, while I wrote the numbers on the board. Here's how it goes:

Two students volunteer to be the original Mr. and Mrs. F. They come to the front and stand in the "adult population bunny cage". Here is our population during month1:

Month 1
 Adult Population Cage Baby Population Cage

At the end of month 1, Mr. and Mrs. F are told to "reproduce" (two more volunteers come to stand in the "Baby Cage"). During month 2:

Month 2
 Adult Population Cage Baby Population Cage

The end of month 2 arrives. The baby pair is instructed to "grow up" (move to the adult cage), and the adult pair is instructed to "reproduce" again (two more volunteers):

End of Month 2 Transition
 Adult Population Cage Baby Population Cage

During month 3, our bunny population looks like this:

Month 3
 Adult Population Cage Baby Population Cage

At the end of month 3, the two adult pairs reproduce, and the baby pair grows up and moves into the adult cage:

End of Month 3 Transition
 Adult Population Cage Baby Population Cage

During Month 4:

Month 4
 Adult Population Cage Baby Population Cage

Baby pairs grow up, adult pairs reproduce:

End of Month 4 Transition
 Adult Population Cage Baby Population Cage

Month 5:

Month 5
 Adult Population Cage Baby Population Cage

Conclusion

While directing traffic into and out of the cages, I am also writing the numbers for the number of adult pairs during months 1-5 on the board:

1, 1, 2, 3, 5,

I think you see what's coming! After a couple more rounds, I run out of bunnies. They take their seats and I ask them to conjecture what the next number will be, and what this sequence of numbers appears to be (we have previously had a lesson on the Fibonacci sequence).

They, of course, respond in unison: "Fibonacci sequence!"

The whole demonstration has taken about ten minutes. We are now ready to settle down. I show them how the conjecture they have formulated can be convincingly argued, and then we all go home, exhausted by our (re-)productive labors, but happy little bunnies, nonetheless!

Historical Note

Fibonacci, also known as Leonardo of Pisa (c. 1170-c.1250), was the first of the great European mathematicians. He made many of the notable Greek and Arabic mathematical works available in Latin, including works by al-Khowarizmi, who gave algebra its name. His most famous work was entitled "Incipit liber Abaci Compositus", known later as Liber Abaci (Book of Calculation), in which he covered arithmetic and elementary algebra.

The Fibonacci sequence seems to pop up everywhere. We see one example here in the study of population growth. It also appears in the study of the "Golden Ratio" ((1/2)(sqrt(5)-1)), for which it is seen that the ratio Fn/Fn+1 of successive elements of the Fibonacci sequence converges to exactly that Golden Ratio. It is also seen in gnomonic (self-similar) growth patterns occurring in nature, tied as well to the Golden Ratio.