Ernesto Sánchez

WHAT IS THE NATURE OF HIGH SCHOOL STUDENTS' CONCEPTIONS AND MISCONCEPTIONS ABOUT PROBABILITY?

But it is important to stress that the relationship between intuition and logical structures plays an essential part in the domain of probability, perhaps more conspicuously and strikingly than it does in other domains of mathematics. (Fischbein, 1975, p.5) In high school, students are expected to determine the likelihood of an event by constructing probability distributions for simple s ample spaces, compute and interpret the expected value of random variables in simple ca ses, and describe sample spaces in compound experiments. They are also expected to learn to identify mutually exclusive and joint events, understand conditional probabilit y and independence, and draw on their knowledge of combinations, permutations, and counting principles to compute these different probabilities. By the end of high s chool, students should understand how to draw inferences about a population from random s amples; a process that involves understanding how these samples might be distributed. Such an understanding can be developed with the aid of simulations, that enable students to explore the variability of sample statistics from a known population and to ge nerate sampling distributions (NCTM, 2000; Pfannkuch, Chapter 11 in this book). Borovcnik and Peard (1996) remark that probabilisti c reasoning is different from logical or causal reasoning and that counterintuiti ve results are found in probability even at very elementary levels. By way of contrast, in o ther branches of mathematics counterintuitive results are encountered only when working at a high degree of

Research on Teaching and Learning Probability

Research in probability education is now well established and tries to improve the challenges posed in the education of students and teachers. In this survey on the state of the art, we summarise existing research in probability education before pointing to some ideas and questions that may help in framing a future research agenda.

Levels of Probabilistic Reasoning of High School Students About Binomial Problems

In this chapter, some aspects of the process in which students come to know and use the binomial probability formula are described. In the context of a common high school probability and statistics course, a test of eight problems was designed to explore the performance of students in binomial situations. To investigate the influence of instruction to overcome some common cognitive bias or their persistency, the first three problems are formulated in a way that may induce bias. Each one is structurally equivalent to another problem phrased to avoid any bias that was included in the test. Also, the second and third problems were administered before and after the course to assess the changes produced by instruction. A hierarchy of reasoning, designed in a previous study, was adapted and used to classify the answers of the students in different levels of reasoning. The classification of these answers points out that the components of knowledge, the classical definition of probability, the rule of product of probabilities, combinations, and the binomial probability formula, are indicators of transitions between levels. The influence of the phrasing of the problems is strong before instruction, but weak after it.

Determinism and Empirical Commitment in the Probabilistic Reasoning of High School Students

In this paper, we explore students’ responses to two binomial tasks, one related to prediction and another to distribution, in an effort to understand how students express variability in their predictions before and after simulation activities. To collect data, a four-stage study was conducted with two student groups (one with instruction in probability and the other without it). The first and fourth stages consisted of administering a test that included questions about a binomial experiment, B(x, 2, ½). During the second and third stages, the students conducted simulations with manipulatives and with software. The SOLO taxonomy was used to analyse their progress in reasoning based on their responses to two questions on the test. By analysing the students’ responses in a double-entry table to both questions, we were able to ascertain the difficulties that students face in integrating variability into their reasoning, despite their experience with simulation. Two patterns of student responses are salient from the set of answers: Determinism and Empirical commitment.