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December 2012
 
 

Live Probability Experiments to Engage Learners
From Richard C. Larson

Dick LarsonMost STEM Pals readers have probably heard of the “5 E” pedagogical model: Engagement, Exploration, Explanation, Elaboration, and Evaluation. Probability and statistics provide a wonderful playground in which to design lessons with the 5-E approach. Without Engagement as a starter, too much of the teaching of probability and statistics involves rote memorization of arcane formulas from textbooks. Students can become bored, disinterested and may tune out entirely from learning such amazing material, concepts that arise every day in so many aspects of our lives. The future is uncertain, and knowledge of probability and statistics provides a way to think about that, to cope with and plan for uncertainty.

So, how to do this in a class? One way is to conduct live probability experiments. One simple example is to give a “Heads-Tails” fair coin to each student in the class and then have each student flip the coin under a given scenario. For instance, you can start the class by asking the students about flipping the coins until the first Heads occurs, and then they stop. We are interested in the number of flips N until we see that first Heads. Have an active discussion about this, which hopefully eventually includes the idea of a histogram showing an empirical distribution of the number of flips N until the first Heads. That is, Johnny may get Heads on his first flip. Sallie may take 5 flips to see her first Heads. Then have them guess the overall shape of that histogram, that is, the ‘theoretical’ probability distribution. As you may know, that is a geometric distribution with each ‘probability bar’ one half the height of the bar to its immediate left. The students should be able to intuit this property.

Step 2 of the 5-E method is Exploration. Here, objects and phenomena are explored as hands-on activities, with guidance. This means that each student flips his or her coin until observing the first Heads, and he/she records that number. Then you as teacher collect the experimental results. You can do this by a raising of hands, such as, “How many of you got Heads on the first flip?” That should be about half the class. “On the 2nd flip?” That should be about one quarter of the class. Etc. You could also obtain the results by the students using ‘clickers’ or, if that technology is unavailable, by writing them on pieces of paper and handing them up to you. You draw the results on the blackboard, perhaps over the theoretical geometric distribution. Lo and behold, for any class size over about 25 or so, the chances are excellent that the results of this live probability experiment will resemble closely the theoretical distribution. The students become believers as they are engaged and have participated in obtaining the results.

Once you have the results shown very visibly on the blackboard (or on a screen via projector), you move to Explanation. You ask the students to engage in a discussion explaining their understanding of the concepts that led to the observed results. This includes their explanation of why the empirical distribution is not exactly the same as the theoretical distribution. They may want to do the experiment again, one or more times, to obtain other histograms to compare with the theoretical geometrical distribution.

Now you go move to Elaboration, activities allowing students to apply concepts in new contexts, and building on and extending understanding and skill. One way to do that is to undertake a different live probability experiment with the coins. A classic experiment would be to flip each coin ten times and count the number of Heads that results. Have the students discuss this in the same sequence as you designed for the first experiment. The theoretical distribution here is the Binomial distribution. And one can continue with one or two additional live experiments, each becoming a bit more complex (e.g., the number of coin flips until you see the 2nd occurrence of Heads).

At the end of these experiments, one can move to Evaluation, where students discuss and assess their knowledge, skills and abilities. They discuss the experiments, the probability principles involved, the reasons why the empirical histograms don’t exactly match the theoretical distribution, why successive repeated experiments give different results, etc.
I have done these live probability experiments in front of small classes (35) and large classes (200). Especially with clickers, as the students’ results come into the system and the histogram begins to build on a large screen, the wonder of the students is amazing to see as they watch their own coin flips collectively yield results very close to the theoretical. They are engaged, and in the end, they understand in ways that no textbook reading and memorization could convey.

We have also done a variety of other live probability experiments, two of which are available as BLOSSOMS videos. With the Broken Stick Problem http://bit.ly/PEFlLX, one obtains two random numbers uniformly and independently distributed between 0 and 100. Then with a piece of chalk, you mark those values on a wooden meter stick (marked in centimeters), then you saw the stick into pieces at those marked locations and then you ask the students, “For what fraction of these experiments will we be able to create a triangle with the resulting three stick pieces?” Before you do this experiment, have each student guess the answer and place each answer on the blackboard with the student’s name next to the guess. They become fully engaged quickly! The second is Flu Math Games http://bit.ly/WvKmgE, in which students ‘infect’ each other with the flu by pulling random numbers from a hat, the probabilities adjusted each time by the extent to which the students engage in good hygienic behaviors to avoid getting and passing on the flu virus. This lesson also shows the relationship between mathematics and biology, that the process on infection spread over a population can be accurately modeled with mathematics linked to human behavior.

Richard Larson is the Mitsui Professor of Engineering Systems at MIT. He is also the Director of MIT LINC and the Principal Investigator of MIT BLOSSOMS.

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