Development of Reasoning about Random Events

Article excerpt

Abstract

The development of school students' understanding of random events is explored in three related studies based on tasks well known in the research literature. In Study 1,99 students in Grades 3 to 9 were interviewed on three tasks and surveyed on two tasks about luck and random behaviors in chance settings. Four levels of response are identified across the five tasks, reflecting increasing structural complexity and statistical appropriateness. In Study 2,23 of these students were interviewed on the same interview protocol tasks, three or four years later to monitor developmental change. In Study 3, a different group of 15 students was interviewed with two of the tasks and prompted with conflicting responses of other students on video. The aim of Study 3 was to monitor the influence of cognitive conflict in improving student levels of response. Implications for teachers, educational planners, and researchers are discussed in the light of other researchers' findings.

Introduction

Many research studies over the years have explored people's understanding of random processes. The original studies mainly dealt with college students and their misunderstandings by describing from a psychological perspective how people reason in uncertain conditions, a psychological approach. Later studies were conducted by mathematics educators on data sets composed of school students, some showing little change in understanding across grades. The desire to improve students' statistical understanding and reasoning motivated this research, an educational approach. These two different research perspectives and the contributions they make to the literature on probability and statistics have been considered in some detail by Shaughnessy (1983, 1992). After a review of the place of the random concept in the school curriculum and of previous research on the concept, this study uses familiar tasks to extend previous research in three directions. First, a development model is proposed that displays increasingly complex understandings of the concept of random as involved in the tasks presented. Second, longitudinal interview data are used to monitor the developmental change in understanding over three or four years. Third, cognitive conflict from other students is introduced as a means of testing the tenacity of beliefs and their susceptibility to change. Background for these three avenues of research is also provided.

Random in the School Curriculum

As a key probability concept, random is notoriously difficult for students to grasp. Used as an adjective, more attention is often given to the associated noun, such as in "random sample," to give meaning to a complex idea (Batanero, Green, & Serrano, 1998). The word is often used colloquially in non-mathematical contexts to convey a meaning of haphazard. This is reflected to some extent in the nebulous nature of some dictionary definitions, for example, "Made, done without method or conscious choice" (Waite, 1998, p. 530). Indeed like many concepts that are difficult to define, it seems easier to define random by considering "antonyms to randomness" and exploring what is not random, for example, "order," "organization," and "predictability" (Falk, 1991). Curriculum documents are generally guilty of discussing random events, and random numbers, without specifically defining the term "random" (e.g., Australian Education Council [AEC], 1991; Department for Education, 1995; Ministry of Education, 1992; National Council of Teachers of Mathematics [NCTM], 2000). An exception is the Mathematics Guidelines K-8: Overview to Chance and Data of the Department of Education and the Arts [DEA] in the Australian state of Tasmania, which follows closely the model provided by Moore (1990).

The focus on chance in this Strand is to develop in students the ability
to describe randomness and to measure (quantify) uncertainty. Phenomena
or events which may individually have uncertain outcomes (for example,
tossing a coin), but that have a regular pattern of outcomes over the
long term, are referred to as being random. …