The development of chance measurement

Abstract

This paper presents an analysis of three questionnaire items which explore students' understanding of chance measurement in relation to the development of ideas of formal probability. The items were administered to 1014 students in Grades 3, 6 and 9 in Tasmanian schools. The analysis, using the NUDIST text analysis software, was based on the multimodal functioning SOLO model. An analysis of the results and a developmental model for understanding chance measurement are presented, along with implications for curriculum and teaching practice.

Development of Student Understanding of Outcomes Involving Two or More Dice

“…On one hand such a phrase may have a similar intended meaning to Konold_s observed BI can_t tell what will happen next.^On the other hand, however, it might be considered at the opposite extreme: instead of not being able to predict, any possibility is predictedsomething will happen. In considering chance measurement in classical situations with random generators, Watson, Collis & Moritz (1997) found many such nebulous responses, refusing to put a measure on an event. They classified these responses as the first step beyond idiosyncratic responses, before qualitative or quantitative responses were attempted.…”

“…A significant addition was the observation of consideration of two compound events that were singularly Bmost likely^and concluding that together they were also Bmost likely.^Li & Pereira-Mendoza produced a single developmental pathway of five steps, from Prestructural illogical responses to Extended Abstract responses correct for compound events, based on sample spaces and suggestions for experimentation. Watson et al (1997) also employed the SOLO model to describe observed development for chance measurement based on three survey tasks used with students in grades 3, 6, and 9 in Australia. Their model suggested two cycles of the basic SOLO model (Unistructural-Multistructural-Relational), with proportional reasoning emerging in the second to cope with the most complex task involving comparisons for ratios of different numbers of objects.…”

A developmental scale of mental computation with part-whole numbers

“…They assessed number sense in terms of distinguishing larger from smaller numbers, part from whole, and top from bottom in fractions. The connections between proportional reasoning and ratios were illustrated in the developmental work of Noelting (1980aNoelting ( , 1980b with consequent links to concepts in probability contexts (Shaughnessy, 2003;Watson, Collis, & Moritz, 1997). Building conceptual understanding associated with proportional reasoning has been the focus of many studies across contexts that constantly reflect back to the requirement for multiplicative rather than additive features (Cobb, 1999;Harel & Confrey, 1994;Thompson & Saldanha, 2002).…”

An Overview of Research into the Teaching and Learning of Probability

“…With respect to knowledge of student cognitions, Cobb et al (1991) have suggested that teachers need cognitive frameworks for the various mathematical domains so that they will be aware not only of students' prior knowledge but how development of their mathematical knowledge is likely to occur. Much of the research on students' probabilistic thinking during this phase has attempted to distil students' prior knowledge and intuitions in the form of frameworks or profiles (e.g., Jones et al, 1997;Maher, Speiser, Friel, & Konold, 1998;Watson et al, 1997) that can be used by teachers to inform instruction and by teacher educators in the professional development of teachers.…”

“…This research included cognitive studies that profiled the probabilistic reasoning students brought to the classroom (e.g., Watson, Collis, & Moritz, 1997) and teaching experiments (e.g., Jones, Langrall, Thornton, & Mogill, 1999) including ones that incorporated technology . This was a period where research was more geared to the needs of curriculum and classroom instruction.…”