The increased attention given to probability and statistics in the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) and other national curriculum statements around the world (e.g., Department of Education and Science [DES] 1991; Australian Education Council [AEC] 1991) has been applauded by most mathematics educators and many teachers. When they leave school, students need to make decisions when presented with claims about data. The fact that these challenges will arise is illustrated every day in reports in newspapers, magazines, and television news broadcasts.
Jane M. Watson
The Keller Plan for a personalized system of instruction as applied to college-level courses is usually characterized by (a) individual pacing, (b) a mastery orientation, (c) the presence of student tutors, (d) the use of printed study guides, and (e) the inclusion of a few lectures for enrichment (Kulik, Kulik, & Carmichael, 1974). In mathematics courses the plan has led to positive reactions from students, lower dropout rates, and higher final examination scores (Anderson & Pritchett, 1977; Peluso & Baranchik, 1977; Rogers & Young, 1977; Struik & Flexer, 1977; Waits & Riner, 1975; Weir, 1977). The issue of the long-term retention of concepts in personalized mathematics courses, however, appears not to have been addressed.
Jane M. Watson
Judging statistical claims in social contexts is fundamental to statistical literacy. This article uses a particularly contentious newspaper report that makes a cause-and-effect claim as the basis for discussing this important aspect of statistical understanding. The issue's relevance across the school curriculum is shown by extracts from curriculum documents. Teachers need to structure experiences to build ability to question claims made without proper justification. This article suggests a hierarchy to help teachers plan for and assess student learning in this area, and it closes with a plea for teachers to cooperate across subjects to achieve results.
Jane M. Watson
Often there are not enough data available for a model.
Jane M. Watson and Jonathan B. Mortiz
One hundred eight students in Grades 3, 5, 6, 7, and 9 were asked about their beliefs concerning fairness of dice before being presented with a few dice (at least one of which was “loaded') and asked to determine whether each die was fair. Four levels of beliefs about fairness and four levels of strategies for determining fairness were identified. Although there were structural similarities in the levels of response, the association between beliefs and strategies was not strong. Three or four years later, we interviewed 44 of these students again using the same protocol. Changes and consistencies in levels of response were noted for beliefs and strategies. The association of beliefs and strategies was similar after three or four years. We discuss future research and educational implications in terms of assumptions that are often made about students' understanding of fairness of dice, both prior to and after experimentation.
Jane M. Watson and Jonathan B. Moritz
A key element in developing ideas associated with statistical inference involves developing concepts of sampling. The objective of this research was to understand the characteristics of students' constructions of the concept of sample. Sixty-two students in Grades 3, 6, and 9 were interviewed using open-ended questions related to sampling; written responses to a questionnaire were also analyzed. Responses were characterized in relation to the content, structure, and objectives of statistical literacy. Six categories of construction were identified and described in relation to the sophistication of developing concepts of sampling. These categories illustrate helpful and unhelpful foundations for an appropriate understanding of representativeness and hence will help curriculum developers and teachers plan interventions.
Lyn D. English and Jane M. Watson
We analyzed the development of 4th-grade students' understanding of the transition from experimental relative frequencies of outcomes to theoretical probabilities with a focus on the foundational statistical concepts of variation and expectation. We report students' initial and changing expectations of the outcomes of tossing one and two coins, how they related the relative frequency from their physical and computersimulated trials to the theoretical probability, and how they created and interpreted theoretical probability models. Findings include students' progression from an initial apparent equiprobability bias in predicting outcomes of tossing two coins through to representing the outcomes of increasing the number of trials. After observing the decreasing variation from the theoretical probability as the sample size increased, students developed a deeper understanding of the relationship between relative frequency of outcomes and theoretical probability as well as their respective associations with variation and expectation. Students' final models indicated increasing levels of probabilistic understanding.
Jane M. Watson, Noleine E. Fitzallen, Karen G. Wilson and Julie F. Creed
The literature that is available on the topic of representations in mathematics is vast. One commonly discussed item is graphical representations. From the history of mathematics to modern uses of technology, a variety of graphical forms are available for middle school students to use to represent mathematical ideas. The ideas range from algebraic relationships to summaries of data sets. Traditionally, textbooks delineate the rules to be followed in creating conventional graphical forms, and software offers alternatives for attractive presentations. Is there anything new to introduce in the way of graphical representations for middle school students?