Tuesday, August 16, 2011

A Declaration of Independence

I used to be roommates with a magician. He kept all of his materials locked up in a trunk in our hall closet. Although he had devoted himself to the study of human psychology, I still convinced him to crack open the trunk and show me a trick from time to time. This experience was one of the most frustrating yet intellectually stimulating experiences of my life. I was a mathematics undergrad immersed in a stressful environment of number theory, numerical analysis, and abstract algebra. I was being trained to reason effectively, and his antics refreshed my perspective on reality. Life often muddies mathematics; such is the unfortunate reality. 

I was again reminded of this fact this morning when I followed a twitter conversation between @davidwees and @mathfour. The conversation was based on the nature of irrational numbers. Measuring them in the plane yields a rational number, and this could place them in the same, unknowable category of imaginary numbers. In mathematics theory, irrational numbers are very knowable, but when they are transferred into a concrete, classroom environment, they are forged. 

Another intriguing example of this phenomenon is the classification of independent events. We claim that two actions (or events) are independent if one has no bearing on the other. The events could happen in succession or concurrently and it would remain immaterial. When we toss a coin 10 times, the events are independent, but when we ask a student to predict the outcome of tossing a coin 10 times in succession, their guesses may not behave in this way. The involvement of the human psyche tampers with independence.

Testing this influence is a great way to introduce elementary probability to students. I would begin with showing this intriguing video from Derren Brown:

A typical response is immediately available in the comments:

"If he answered the same thing every time, he would have won eventually" - Guitarhero0904

This is a common reaction to such ploys, as evidenced by the 19 "likes" on this individual's comment. (at time of writing) It only makes Brown's statement ring true:

"Our tendency to think that we're not predictable is probably one of our more predictable traits."

This is an excellent conversation to have with your students. If Steve Merchant were to choose randomly, what are the odds that he would get the word correct? How can you represent your answer? Why did you choose this answer? Can you broaden your definition? Before long you are conversing about the sample space, outcomes, and favourable outcomes to a particular experiment. This opens the door to the fundamental counting principle. What are the odds that Merchant would get all 4 wrong? 

You suggest to the students that there must be a way to explain the large difference. Extending this to more trials makes the multiplication of intersection second nature before moving into the next phase. Allow the students to pair up and attempt the game on one another. As the trials continue, take the data on which students are guessing wrong and which are guessing right. This data collection can be done in many different ways, and it is best to allow students to decide which stats to keep. The goal should be to determine which students are the most predictable. You essentially are using the deviance from the theoretical probability to measure predictability of students. 

The power of the experiment is its underpinning in theoretical probability. Students begin to understand probability as an estimate of the norm. If you are lucky, students may question if their results are significant enough. Maybe getting 3 out of 4 wrong isn't that unlikely? This builds a convenient bridge into the union operator. 

Another great way to introduce human impact on probability is through the familiar game of Rock-Paper-Scissors. (RPS). Take a couple minutes to calculate the probabilities of a win, loss, or draw if all trials are deemed independent. Become immersed in the expected value before introducing the human element. Ask the students if they have any strategies to win in a standard game. You will get several responses varying in complexity. A great place to introduce RPS strategy is the strategy guide from the World RPS Society entitled "How to beat anyone at Rock Paper Scissors". The link is found below:

The effectiveness of human influence can then be tested with two online resources. The first game is built on a random number generator. It essentially uses no human influence to calculate its moves. If the students play enough trials, they should notice an even amount of wins, losses and draws. The link for this version is found below:

The next version debuted in the New York Times science section online. The goal of the program is to develop an artificial intelligence based around the game. The computer tracks your trends and makes selections based on prior behaviour. You can choose a novice or expert setting. In the novice, only your decisions are used against you. Students may find that it is harder to win as the trials continue. The expert setting uses thousands of iterations by developers to devise global patterns. It essentially is trying to play like a human--this includes human influence on the probability. The link for this version if found below:

Students may begin to develop strategies to beat the artificial intelligence. The teacher should encourage them to verbalize. Most often, they involve using randomness to trick the computer. The goal of these exercises is to understand what it means to be independent. Students work on the theoretical calculation of probabilities along the way and learn terms like sample space, event, and may even dive into union and intersection. These learning outcomes fall out of the investigation. Probability is mathematical fortune telling, and some of us are more predictable than others.


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