Learning probability with Fathom Óscar Chávez University of Missouri Columbia

Transcription

1 Learning probability with Fathom Óscar Chávez University of Missouri Columbia Interface 2007 Standards, Curriculum, Assessment, Instruction: Forces Aligned to Increase Student Understanding Osage Beach, Missouri, February 26 27, 2007

2 Overview What is probability? What does research say about the learning of probability? The basics: coins and dice, using Fathom. Children, with Fathom. Solving problems by doing simulations.

3 What is probability? One can hardly give a satisfactory definition of probability. H. Poincaré

4 What is probability? Probability is a degree of certainty, which is to certainty as a part is to the whole. J. Bernoulli

5 What is probability? The concept of probability is a generalization of the concepts of truth and falsehood. J. Lucas

6 What is probability? Probability is a feeling of the mind. A. de Morgan

7 What is probability? The probability of an event is the ratio of the number of cases that are favorable to it, to the number of possible cases, when there is nothing to make us believe that one case should occur rather than any other. P. Laplace

8 What is probability? The probability of an outcome is our estimate for the most likely fraction of a number of repeated observations that will yield that outcome. R. Feynman

9 Addressing Misconceptions [A]n initial formalistic approach to probability is unlikely to help students overcome misconceptions. However, if probability is introduced first through experiments, students appear to have more success in overcoming their probabilistic prejudices. Shaughnessy (1981)

10 Our goal today Get acquainted with Fathom and its tools to create simulations. While we can throw dice, and toss coins, Fathom allows us to perform a large number of experiments, or simulate complex phenomena, in ways that would not be possible with physical objects.

11 Coins and dice Coins and dice are defined by using random numbers (we are tossing the coins or throwing dice). Measures can be assigned to collections, and these measures can be collected (we are keeping record of repeated events).

12 Sample size matters, or does it? The chance that a baby will be a boy is about 1/2. Over the course of a year, would there be more days when at least 60 percent of the babies born were boys a) in a large hospital b) in a small hospital c) no difference

13 Using Fathom Modeling a hospital (simulating births). Counting the number of boys per day during the course of one year. We will create a collection, we will define a measure, and we will collect measures.

14 The Monty Hall Dilemma

15 One Son Policy In order to limit the growth of China s population, some politicians have suggested that families be limited to a single child. However, for a variety of cultural reasons, rural Chinese families highly value the birth of a male child. Konold (1994)

16 One Son Policy In a compromise, politicians are discussing implementation of the One Son policy. Under this policy, couples may continue to have children until the first son is born; thereafter, the couple is forbidden to have additional children.

17 One Son Policy Under full compliance of this policy, what will be the average number of children per family? What will be the ratio of girls to boys born under this policy?

18 Disease control If a disease affects 1% of the population, and a test to determine whether a person has the disease is accurate 97% of the time, what is the probability of a person having the disease, given that he tested positive?

19 References Konold, C. (1994). Teaching probability through modeling real problems. Mathematics Teacher, 87(4), p Stirzaker, D. (1999). Probability and random variables: A beginners guide. Cambridge, UK: Cambridge University Press. Shaughnessy, J. M. (1981). Misconceptions of probability: From systematic errors to systematic experiments and decisions. In A. P. Shulte & J. R. Smart (Eds.), Teaching Statistics and Probability (pp ). Reston, VA: NCTM.

20 Contact information Óscar Chávez University of Missouri