Issues in Education
Probability, Stats & 'Playing Games'
By James J. Cochran
I vividly remember the first time I was exposed to the central limit theorem. Every student in this undergraduate introductory statistics course (or so it seemed at the time) was having difficulty with the professor's explanation of the normal approximation to the binomial distribution. At that point the professor passed a Galton board among the students and encouraged us to discuss it.
Galton Board
As the ball bearings bounced through the maze of pegs, the notion that each peg represented a Bernoulli trial with equal probabilities for success and failure (the ball bearing going left or right at a particular juncture) became evident. Furthermore, it was obvious that a ball bearing would have to bounce one way an unusual number of times to land in an extreme bin.
Through the use of this deceptively simple visual aid, what had been difficult for the professor to articulate had become astonishingly clear. While this demonstration helped me understand the central limit theorem, the real lesson I learned was the potential power of active learning.
This event later motivated me to implement my first active learning exercise into my introductory statistics courses. In this exercise students form pairs - one student in each pair flips a coin a given number of times while the other student records the results. As a class we construct a frequency distribution and histogram of the results. Naturally, the histograms become more normal in appearance as we increase the number of times each student pair flip their coin. This exercise is a live version of the Galton Board and effectively demonstrates the normal approximation to the binomial distribution when the probability of success is 0.50.
Of course, I eventually was confronted by a quick student who wanted to know what would happen if the probability of success differed from 0.50. Coins do not easily facilitate such experiments, so I substituted 20-sided die. By altering the definition of success relative to the outcome of the 20-sided die we can conduct the experiment for success probabilities in increments of 0.05 (fine enough to satisfy the curiosity of almost any student). If you desire even greater flexibility, you can build a Java applet (or use one of the many that can be found on the Internet). However, the 20-sided die allows you to exchange 10 minutes of class time for memorable hands-on insight into the central limit theorem.
The Three-Door Problem also lends itself to the development of a constructive classroom game. In this problem a contestant chooses from three closed doors, two of which hide undesirable "prizes," while the desirable prize is hidden behind the remaining door. After the contestant selects a door, the game administrator (who knows the locations of the prizes) opens one of the two remaining doors to reveal an undesirable prize. The contestant is then offered an opportunity to swap the door he or she initially selected (which has yet to be opened) for the other unopened door. Of course, most people are surprised to learn that a contestant who swaps doors is twice as likely to win the desirable prize.
Students form groups of three for the classroom execution. I distribute sets of three playing cards, two of which are of the same value, to the groups. Turned face down, the cards represent the three doors, with the odd card representing the desirable prize. Group members assume the roles of game administrator, contestant and recorder. The administrator places the three cards face down and oversees the game. Each group plays 15 rounds (rotating their assigned roles) and discusses their results. You can also build a Java applet for this experiment or find one the Internet, but it is hard to duplicate the understanding that students develop by playing the hands-on version (which can be administered in less than 10 minutes).
More recently, I created "Who Wants To Be A Millionaire: The Classroom Edition." WWTBAM-TCE is Microsoft PowerPoint-based, so it is highly portable and can be used for a brief in-class Q&A session. The format also makes WWTBAM-TCE appropriate for any subject.
WWTBAM-TCE
My experience suggests that WWTBAM-TCE maintains course momentum, keeps students engaged, and provides exposure to, and reinforcement of, course topics while providing a short respite from the lecture. WWTBAM-TCE seems to provide a tremendous pedagogical punch while increasing classroom interaction, all at a relatively small cost. See INFORMS Transactions on Education (http://ite.informs.org/) Vol. 1, No. 3 (May 2001) for an article on WWTBAM. This article provides a detailed description of WWTBAM-TCE as well as instructions on how to use and edit the downloadable, fully functional copy of this game.
My experiences using these and other physical and computer-based games in the classroom have been consistently positive. A well-planned and executed game can quickly turn something perceived by students to be abstract and conceptual into a concrete and understandable concept.
James J. Cochran (jcochran@cab.latech.edu) is an assistant professor with Louisiana Tech University's College of Administration and Business, vice president-Projects for INFORM-ED and a member of INFORMS' Education Committee.
OR/MS Today copyright © 2001 by the Institute for Operations Research and the Management Sciences. All rights reserved.
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