Rationale

I chose this lesson plan because it has several salient features that fit in well with the methods suggested in the readings. This lesson features a problem that is accessible to high school students; moreover, the problem has a history that is interesting – it was featured on a game show and was moderately famous as a result of a Parade magazine feature. The problem also has an unexpected outcome. All of these things combine to form a lesson that can create a memorable discrepant event.

Fisher & Frey mention that discrepant events should be memorable and command attention in order to facilitate recall and learning. I think that this lesson has those features. The teacher must work to stage a memorable production of the Monty Hall problem (for example, dressing as a game show host, encouraging students to participate in the game show environment, etc), but once this is accomplished it creates a highly memorable setting that, hopefully, will help the students to learn about conditional probabilities, probability trees, and monte carlo simulation.

Fisher & Frey also emphasize the importance of discrepant events in facilitating recall of prior knowledge in subsequent lessons. I believe that this lesson has the potential to be a salient memory feature to function as such; in future lessons, as students are working with dependent probabilities, they can always “remember Monte Hall!”. That is, when examining probabilistic events, they can use this problem as a touchstone to recall the process of examining potential outcomes. If students can remember this lesson and how they analyzed the outcomes of this problem, then they will be able to apply this knowledge to future problems and situations.

In addition, Chiappetta indicates that discrepant events can lead to and encourage inquiry based learning. This lesson features inquiry-based activities involving discovering optimal strategies for the Monty Hall problem. He states that

A discrepant event puzzles students, causing them to wonder why the event occurred as it did. Puzzlement can stimulate students to engage in reasoning and the desire to find out (Piaget, 1971). Discrepant events can be used to promote inquiry. (Chiappetta, page 25)

Chiappetta’s primary area of focus is science education, but the principles should transfer to mathematics education. The inquiry component of the lesson should compound the discrepant event of the presentation once the optimal strategy is found, because the optimal strategy turns out to be quite unexpected (to many people). This compounding effect should work to make a memorable lesson that emphasizes to the students the importance of the methods that they have learned; using these methods is not only correct, but can lead them to results that are highly unexpected.

The lesson also requires a reasonable degree of math-related reading. Students are required to interpret graphs, directions, and results and translate them into the context of a real-life scenario (or, at least, a real-life game). These types of critical reading skills are frequently not found in math lessons, but are crucially important skills in the context of mathematics. Students need to be able to move between “real life” and “math” fluently.

Reference

Chiappetta, E. (1997). Inquiry-Based Science. The Science Teacher, October 1997, pp. 22-26.

STICK OR SWITCH: The Monty Hall Problem

Adapted from an Illuminations Lesson

http://illuminations.nctm.org/LessonDetail.aspx?id=L377

and from http://www.montanamath.org/lessons/montyslp.doc

Monty’s Dilemma is a lesson designed to develop the theoretical probability, using a simulation, for the chance of winning what’s behind door number 1, 2, or 3 on the game show “Let’s Make a Deal”. The students simulate the game show using an applet from Shodor Education Foundation’s website, Shodor Interactive (http://www.shodor.org/interactivate/index.html).

The students use experimental probability and the Law of Large Numbers to make and test conjectures about the theoretical probabilities of winning if they stay with their original door choice or switch to a new door after seeing what is behind one door.

This is a fun way to learn about Monte Carlo simulations, experimental and theoretical probabilities, and the Law of Large Numbers. In addition, the history of this problem gives the teacher an opportunity to create a memorable teaching experience by replicating a game show environment.

The lesson is intentionally written in stages: a discrepant event introduction (the gameshow!) and the simulation stage. The introduction is designed to introduce the students to the rules of the game in a way that is not heavy on written words and easily accessible. This is intended to scaffold the students’ understanding of the rules prior to introducing the rules in a written, formal sense. The simulation stage takes the problem to its final formalized mathematical setting.

Monty’s Dilemma: Should You Stick or Switch?

Experimental Probability

I. Objective: The learner will use a probability tree and Monte Carlo simulation to determine theoretical probabilities by examining experimental probabilities and using the Law of Large Numbers. The learner will use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations. The learner will compute probabilities for simple compound events, using such methods as tree diagrams and area models. The learner will apply and adapt a variety of appropriate strategies to solve problems.

II. Materials: Computer, projector, access to internet, spinners and copies of a spinner divided into three equal areas, coins.

III. Lesson: Monty’s Dilemma: Should You Stick or Switch?

a. State the objectives of the lesson.

b. Introduce the game by having students play “Monty’s Dilemma”. The teacher should dress and act as a game show host, encouraging students to take part in the game. This serves as an introduction to the game.

c. Describe the game show “Let’s Make a Deal” hosted by Monty Hall. Read the situation from “Monty Hall, Three Doors Exploration Questions”. Introduce the rules to the students in a formal fashion.

d. Give students an opportunity to play the game, where you act as Monty and the students act as the contestant (or audience, shouting suggestions).

e. After several rounds, ask students to predict what is better: to switch doors or to stay with the original choice. Write down your prediction and the reason for your prediction. By a show of hands, how many would stay? How many would switch? Share some of the reasons with the class.

f. Go to the interactive applet to simulate individual trials of this game. http://www.shodor.org/interactivate/activities/monty3/index.html. Play the game with the students. Ask a student to choose a door and click on it. Monty will show another door with a pig behind it. Ask the student if they want to “stick” with their original choice of doors or “switch” to the other door. Click on their second door choice, if they see the boat and prizes they win a piece of candy. If they see the other pig, they win nothing.

g. After playing the applet simulation, ask students if they want to change their predictions. Write down their new predictions and reason for the change. Or write down no change and why they decided not to change.

h. Hand out the spinners and the copies of the spinner divided into three equal areas.

i. First the class will simulate the choice of “sticking” to the original door. The prize will always be behind door A. Students spin the spinner. The area it lands on is the students “door” choice. Monty shows you the other door without a prize behind it. However, you don’t switch from your original door choice. Ask students to tally mark the number of games where they won the prize and the number of total games played. Find the experimental probability after playing 25 times.

j. Collect the class data and obtain a class experimental probability. Compare your probability to the class probability. Which do you think is more accurate? Why? This introduces the “Law of Large Numbers”. The more trials performed, the closer the experimental probability will come to the theoretical probability.

k. What is the experimental probability of winning if you stick to your original door? What is the experimental probability of losing if you stick to your original door? How did you obtain your answer?

l. Use the applet for a large number of trials to demonstrate the Law of Large Numbers and how the experimental probability approaches a constant amount. http://www.shodor.org/interactivate/activities/monty/index.html

m. Now the class will simulate the choice of “switching” to the other door after Monty has shown the first door with a pig. The prize will always be behind door A. Students spin the spinner. The area it lands on is the student’s first “door” choice. Monty shows you the other door without the prize behind it. This time, you switch to the remaining door. Tally mark the number of games won and the total number of games played. Find the experimental probability after playing 25 times.

n. Collect the class data and obtain a class experimental probability. Compare your probability to the class probability.

o. What is the experimental probability of winning if you switch to the other door? What is the experimental probability of losing? How did you obtain your answer?

p. Use the applet for a large number of trials to demonstrate the Law of Large Numbers and how the experimental probability approaches a constant amount. http://www.shodor.org/interactivate/activities/monty/index.html

q. Does sticking or switching give a better chance of winning the prize? Why? How does this compare to your prediction? Write down your answers.

r. Determine the theoretical probability of winning by using the areas on the spinner and walking through the simulations of each strategy. Compare the theoretical probabilities to the experimental probabilities. Write down your observations.

IV. Summary: Misconceptions of chance are common. Since this particular problem involves the process of revising our probability estimates in the light of new information about the doors, it involves conditional probabilities. Strong evidence points to the fact that people are especially prone to misconceptions of chance that involve conditional probabilities. A good way to solve probability problems is to proceed from predictions to experiments to simulations to theoretical models. These steps are very helpful in understanding exactly what occurs within each of the strategies.

V. Assessment: Students turn in their write-ups about their predictions, changing their predictions, the results of their experimental probabilities and the class probabilities, and the theoretical probabilities.

a. Extension: Suppose you enter a sweepstakes and subsequently receive an announcement that after a random drawing out of a million entries, the winning ticket number is one of six listed numbers and your entry is among the six. What is the probability that you hold the winning ticket? Suppose you are given the opportunity to change your entry to any of those six listed. What should you do and what are your chances of winning?

i. Make a prediction, design and perform and experiment, find the experimental probability for a large number of trials, find the theoretical probability. Write up your results.

Spinner for Monty's Simulation

1

2

3

Monty’s Dilemma Assignment

1. Write a few sentences explaining whether you would “stay” with your original door choice or “switch” to a new door after seeing one of the choices. Explain why you would “stay” or “switch”.

2. Keep a tallied list of your experiment for “staying”. Copy the class list of experimental outcomes for “staying”. Write the experimental probability of “staying”. Explain how we determined this probability.

3. Write the experimental probability of “switching”. Explain how we determined the experimental probability.

4. Write a few sentences comparing your original prediction in #1 to how you feel now. Would you “stay” or “switch”? Explain why. Use the experimental and theoretical probabilities we obtained during this lesson in your response.