Randomness+Professional+Development

Elizabeth H Pawelka EMS 519

May 8, 2011

toc

= = = Randomness and Probability = = Professional Development =

Goals and Objectives
People need to be able to make risk analysis decisions, inferences, and predictions based on probabilistic information in workforce and life, e.g. medical test results and risks (Lee & Lee, 2009; Shaughnessy, 2003). And the [|GAISE report] says, "Every high-school graduate should be able to use sound statistical reasoning to intelligently cope with the requirements of citizenship, employment, and family and to be prepared for a healthy, happy, and productive life" (Franklin, Kader, Mewborn, Moreno, Peck, Perry, and Scheaffer, 2007, p.1). However, research has shown that teachers and pre-service teachers have misconceptions about randomness and probability and that teachers are passing these misconceptions on to their students (Batanero, Godina & Roa, 2004; Batanero, Arteaga, Ruiz & Roa, 2010). Therefore, we need to educate teachers so they understand randomness and probability and can properly teach statistics to their students.

The GAISE report is very clear about the importance of variability in statistics and, in fact, states that variability in data is what sets statistics apart from mathematics. "Statistical thinking, in large part, must deal with this omnipresence of variability; statistical problem solving and decision making depend on understanding, explaining, and quantifying the variability in the data" (Franklin et al., 2007, p. 6). Probability is discussed as an essential tool in statistics that can be used to find solutions and randomness creates chance variability which can then be described with probability models. As students progress through the development levels (A-C), variability and randomness play an increasingly important role. (Franklin et al., 2007).

This professional development class will provide teachers a deeper understanding of variability and randomness as well as the role that variability and randomness play in probability and the role they play in understanding statistics. In additional, this class will provide tools and activities that teachers can use in their own classrooms to help their students develop an understanding of randomness and statistics.

The audience for this class is high school math teachers. It is designed to be delivered in two hour sessions over three days and addresses the following standards:


 * [|NCTM (2000)]**:
 * use simulations to construct empirical probability distributions;
 * compute and interpret the expected value of random variables in simple cases;
 * understand the concepts of conditional probability and independent events;
 * understand how to compute the probability of a compound event.

Calculate expected values and use them to solve problems Use probability to evaluate outcomes of decisions
 * [|Common Core Standards (2010):]**
 * S-MD.1. Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space.
 * S-MD.2. Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
 * S-MD.3. Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.
 * S-MD.5. Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
 * a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.
 * b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
 * S-MD.6. Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
 * S-MD.7. Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Class Lesson Plan
This is a professional development plan to help high school math teachers understand the role that randomness plays in probability and variability. The class will develop understanding of randomness through activities they can later share with students

Outline: Three day workshop of two hours each (includes breaks each class, during which appropriate snacks (coffee, sodas, maybe pizza for evening) are provided, budget permitting.) This could be a workshop for three evenings during the school year or in the daytime during the summer. Breaking it into three sessions will give the attendees time to absorb and reflect between classes and for homework activities to reinforce ideas.

Collect emails and set up a forum or wiki to simulate community, duration, connections and support after the class (Garet, Porter, Desimone, Birman & Yoon, 2001).

Materials

 * Blank paper and pencils
 * A quarter for each attendee
 * Presentations/Worksheets/Handouts (in plan below)
 * Computer with projector and internet access
 * Computer for every two attendees with internet access and loaded with Probability Explorer and Schoolopoly files
 * Opaque jar or brown bag of M&Ms with 2/3 Red and 1/3 Yellow mixed together (or appropriate food and/or color substitute) and paper towels

Day 1 - Independence and variability Part 1 - Introduction - 30 mins
 * **Activity:** Coin toss - As the attendees come in the door, give them each a quarter and ask them to start tossing the coin and record the results. “I’m paying you to be here!” - loosen up exercise.
 * Have them sign in:[[file:SignupSheet.doc]] Ask for permission to share with class to potentially create a forum or wiki for continued contact, support and learning.


 * **Discussion:** Have everyone introduce themselves: what do they teach and why are they in this class. “Who are you and Why are you here? What do you want out of the class?" Take notes on their objectives.


 * **Presentation: [[file:Introduction.ppt]] **

Part 2 – 30 mins // Which of the above arguments are right? How would you explain the wrong answers? // (Batanero, Godina, and Roa, 2004)
 * **Activity:** Back to coin tosses – hand out Worksheet 1a: [[file:Worksheet1a.doc]]
 * Discuss in pairs and then in whole class. Compare to their actual recorded coin tosses. Discuss number and lengths of runs.
 * **Discussion:** Hand out Worksheet 1b: [[file:Worksheet1b.doc]]
 * Discuss the r easons given by the students to justify that Daniel or Diana were cheating:
 * 1) The sequence pattern is too regular to be random, results almost alternate;
 * 2) The frequencies of heads and tails are too different;
 * 3) There are too long runs; heads and tails should alternate more frequently.

Break – 15 mins


 * Presentation** (30 mins): [[file:Day1.ppt]]


 * Homework:** (15 mins to explain) [[file:Homework1.doc]]

Day 2 - Law of large numbers, equiprobability

Part 1 – 45 mins
 * **Activity**: (25 mins) From an opaque jar or brown bag (filled with 2/3 red and 1/3 yellow M&Ms for example), give each attendee some M&Ms and have them guess the ratio of the colors based on their sample.Then have them compare/combine theirs with someone else’s and then chart how many of each color everyone had on the board. Discuss variability in small samples, law of large numbers.


 * **Discussion:**(20 mins) Go over Homework1. For each question, discuss the answers and reasons. Categorize the reasonings. Focus on independence, equiprobability, law of large numbers, larger variability in smaller sample sizes.
 * Q1 addresses variation; independence
 * Q2 addresses independence; equiprobability
 * Q3 addresses larger variability in smaller sample sizes; law of large numbers; equiprobability
 * Q4 addresses larger variability in smaller sample sizes; law of large numbers

Break – 15 mins

Part 2 – 45 mins

Slides for Day 2:
 * **Activity:** Craps (15 mins) Compound events with 2 dice – (Day2: Slide 1) Go over basic rules and then have them play craps (@http://wizardofodds.com/play/craps/). See who can “win” the most money and ask what strategies they used.
 * **Activity:** What are the Odds? (10 mins) Have them complete Worksheet 2a: [[file:Worksheet2a.doc]]
 * Discuss distribution and connection with the “pay-out” amounts (Day2: Slide 2)
 * **Discussion:** Worksheet 2b (20 mins): [[file:Worksheet2b.doc]]
 * Discussion questions (Day2: Slide 3) Why does the house always win (law of large numbers) and individuals don't (large variability in small samples)?


 * Homework** (15 mins): Homework 2: [[file:Homework2.doc]]

Day 3 - Unknown probabilities - connecting theoretical and empirical probability through simulation

Before class, set up the weight tool in //Probability Explorer// on each computer with the weights corresponding to the company for that computer (see below). Then hide and password protect the weight tool so the students will not be able to access it. Label each computer with one of the fictitious company names. Try to have at least one computer for each company and you may duplicate companies if you have more computers available. Weights and probabilities for each event by company (Tarr et al., 2006, p. 142) Part 1 - 20 mins
 * **Discussion:** Go over Homework 2. Have volunteers present their findings on their favorite games. Discuss winning strategies and the reasonings behind them. Discuss risk and reward. See if there is (correct) consensus of the impact of randomness and probabilities in their decisions and strategies.

Part 2a - 35 mins
 * **Activity:**Schoolopoly
 * Introduction (5 mins):
 * Premise: “We’re creating a board game like //Monopoly// called //Schoolopoly// and, like //Monopoly//, it will be played with dice. Because many copies of the game expect to be sold, companies are competing for the contract to supply dice for //Schoolopoly//. Some companies have been accused of making poor quality dice and these are to be avoided since players must believe the dice they are using are actually “fair.” Each company has provided dice for analysis and you will be assigned one company to investigate.”
 * Ask the students how they think they would be able to tell if the die was fair. Appropriate responses would be “the results of each number are roughly the same” or “they would all have the same probability.”
 * Open //Probability Explorer// so the class can see it and go over the basic directions (how to chose one die, open graphs, set the number of trials, hit the “running man” to start the trials, and how to use the Notepad function) to get them started.
 * Exploration (30 mins) Hand out Worksheet 3: [[file:Worksheet3.doc]] and direct them to pair up on the computers depending on how many computers are available.

Break - 15 mins

Part 2b - (35 mins) Schoolopoly continued
 * ** Discussion ** (25 mins):
 * Ask each group, in turn, to present their findings on whether their die was fair or biased and why they thought that. If possible, have the students show their “compelling evidence” such as the graphs they gathered in Notepad by having the other students gather around their computer. Some questions to ask each group are “How many trials did you run?” and “Was that enough data to draw that conclusion?” Let the other students answer as well.
 * After each group has presented their finding of fair or biased, have each group share their estimated probability of each outcome (1-6), question #3. If more than one group did the same company, compare their answers and then show the actual probabilities set in the weight tool (see Figure 13) to see how close they were to each other and to the correct probabilities.
 * If there are differences, discuss what might have caused them, e.g. different sample sizes, misinterpretation of the data or graphs, or the variability of the different trials.
 * End with everyone voting for which company should win the bid. Presumably, they would pick “Pips and Dots” which is the only company with a fair die.


 * **Summary** (10 mins)**:** W ith //Probability Explorer// open, demonstrate how small numbers of trials, e.g. ten, show variability and point out the concept of independence of each throw. Discuss “the unpredictability of random phenomenon in the short-run but predictability in the long-run trends in data” (Tarr et al., 2006, p. 144). Explain this as the law of large numbers “which states that, for a given event, the empirical probability is more likely (although not certain) to approximate the theoretical probability as sample size increases” (Tarr et al., 2006, p. 143). For example, show two different data sets from 100 rolls of a fair die and compare to show the variability in small samples. Discuss the danger of making inferences from small numbers of trials. Run a large number of trials (about 500) live on //Probability Explorer// in fast mode to show how the graphs “settle down” after the large number and the theoretical probabilities are reached.

Part 3 - Wrap-up (15 mins):
 * Big Ideas
 * Tools and Activities to use in your classroom