Rosemary Callingham

A Model for Assessing Higher Order Thinking in Statistics

ABSTRACT As in other areas of the school curriculum, the teaching, learning and assessment of higher order thinking in statistics has become an issue for educators following the appearance of recent curriculum documents in many countries. These documents have included probability and statistics across all years of schooling and have stressed the importance of higher order thinking across all areas of the mathematics curriculum. This paper reports on a pilot project which applied the theoretical framework for cognitive development devised by Biggs and Collis to a higher order task in data handling in order to provide a model of student levels of response. The model will assist teachers, curriculum planners and other researchers interested in increasing levels of performance on more complex tasks. An interview protocol based on a set of 16 data cards was developed, trialed with Grade 6 and 9 students, and adapted for group work with two classes of Grade 6 students. The levels and types of cognitive function...

Students' Appreciation of Expectation and Variation as a Foundation for Statistical Understanding

Abstract This study presents the results of a partial credit Rasch analysis of in-depth interview data exploring statistical understanding of 73 school students in 6 contextual settings. The use of Rasch analysis allowed the exploration of a single underlying variable across contexts, which included probability sampling, representation of temperature change, beginning inference, independent events, the relationship of sample and population, and description of variation. Interpretation of the demands of increasing code levels for the resulting variable revealed an increasing appreciation of and interaction between the ideas of variation and expectation. Student progression in understanding is illustrated with kidmaps, and educational implications are considered.

Research in mathematics education and rasch measurement

A glance through the titles of research reports in current mathematics education journals might cause one to wonder why researchers in mathematics education eschew the very Queen of the sciences in representing the results of their research. Why do qualitative approaches appear to dominate this field? Many could claim that it is because the usual quantitative methods lose the important qualitative aspects of good mathematics education research. But, what if one quantitative research methodology in education incorporated the same genuine scientific measurement principles that mathematicians routinely expect from the metric system of measures and, at the same time, remained sensitive to those significant qualitative aspects of good educational research? What if this technique was an analytical model in which Australians are world leaders? What if applications of the model to research in mathematics education were already showing very promising results – both in Australia and internationally? Rasch measurement is being used increasingly as a research tool by “mainstream” researchers rather than merely by the sophisticated psychometricians involved in large-scale achievement testing. Using the performance interactions between persons and items, it is possible to produce an ordered conjoint measurement scale of both people and items. This allows researchers to examine the behaviour of persons (e.g., students, markers, teachers) in relation to a particular set of items (e.g., test questions, curriculum outcome indicators, problem-solving methods, attitude surveys.) This permits the identification and examination of developmental pathways, such as those inherent in the development of mathematics concepts as well as the developing capacities of the students. In addition, the behaviour of sets of items can be examined in relation to particular sub-groups of persons (e.g., age cohorts of students) in order to identify the extent to which the chosen items measure the core mathematical constructs the researcher was intending to measure. However, the features of the family of Rasch models make them useful tools for other kinds of research in mathematics education. We might reasonably ask: Is this sequence of the mathematics curriculum appropriate for the children who learn it, and not just appropriate in the eyes of the consultants who wrote it? The Rasch rating scale model allows Likert scale attitude data to be thought about in developmental rather than merely descriptive ways. The Rasch partial credit model provides for the

Teachers’ multimodal functioning in relation to the concept of average

Average is a concept encountered in a wide variety of situations. In this paper, responses of 136 pre- and in-service teachers to a series of graded questions about average are analysed. The theoretical model used as the basis of the analysis is the SOLO Taxonomy with multimodal functioning developed by Biggs and Collis (1991).Information presented in graphical form requiring respondents to compare data sets induced responses in ikonic and concrete symbolic modes, demonstrating multimodal functioning. These responses are mapped onto a model of problem solving proposed by Collis and Romberg (1990). Cycles of response in relation to the concept of average are proposed.

Levels of use of Interactive Whiteboard technology in the primary mathematics classroom

Despite the availability of Interactive Whiteboard (IWB) technology in a large number of Australian primary schools, many teachers focus only on technical issues as opposed to pedagogical engagement in an attempt to incorporate the technology. Previous research suggests that the technology is being used for sophisticated transmission-style teaching as opposed to constructivist approaches. This article presents findings of a project that considered the implementation of IWB technology in three Victorian primary mathematics classrooms (5 to 12 years of age). The study analysed the teaching strategies adopted by three teachers as they embarked on the use of IWB technology as an integral component of mathematical activities with the support of professional development. Teacher use of IWB technology in the primary mathematics classroom was aligned against Beauchamp’s generic transitional framework for viewing the development of teacher use of IWB technology. Through this alignment, a transitional framework emerged which is specific to the introduction of IWB technology in the mathematics classroom.

A Developmental Scale of Mental Computation with Part-Whole Numbers

In this article, data from a study of the mental computation competence of students in grades 3 to 10 are presented. Students responded to mental computation items, presented orally, that included operations applied to fractions, decimals and percents. The data were analysed using Rasch modelling techniques, and a six-level hierarchy of part-whole computation was identified. This hierarchy is described in terms of the three different representations of part-whole reasoning — fraction, decimal, and percent — and is elaborated by a consideration of the likely cognitive demands of the items. Discussion includes reasons for the relative difficulties of the items, performance across grades and directions for future research.

Measuring Levels of Statistical Pedagogical Content Knowledge

The introduction of statistics and probability into the school curriculum has raised awareness of the expectations on teachers who have to teach it. A review of the related field of mathematics education indicates that teachers need more than content knowledge. They must also respond to their students’ statistical understandings in ways that move students’ current understanding to higher levels. Efforts to measure such statistical pedagogical content knowledge are still in their infancy. Findings from a large-scale Australian study are reported to exemplify these efforts, and the implications for future research are discussed.

Understanding Mathematical Giftedness: Integrating Self, Action Repertoires and the Environment

A modern conceptualization of mathematical giftedness must take into account different cultural views of the nature of mathematics as well as conform to Ziegler and Heller‘s (2000) four postulates of giftedness (temporal precedence, fulfillment of the “inus” condition, personal characteristic and theoretical significance). With the Actiotope Model of Giftedness (Ziegler,2005) as a conceptual framework, this chapter will focus on recent research in neuropsychology, cognition, personal factors, language and the environment as applied to the development of mathematical excellence. The first section will review recent research that examines aspects of self in the development of mathematical thinking. The next section examines the role of the environment in the growth of mathematical expertise, particularly in the selection and adaptation of action repertoires and the possibility for a collective mathematical cognition. Finally, this chapter will expose some of the emerging pedagogical and political issues in the development of mathematical excellence within the context of increasing globalization.

Two-Way Tables: Issues at the Heart of Statistics and Probability for Students and Teachers

Some problems exist at the intersection of statistics and probability, creating a dilemma in relation to the best approach to assist student understanding. Such is the case with problems presented in two-way tables representing conditional information. The difficulty can be confounded if the context within which the problem is set is one where students have preconceived opinions on the direction of the potential association present. This article considers school students’ responses to two problems of association, with data presented in 2 × 2 tables. A hierarchical rubric is presented to document students’ understandings. Teachers’ pedagogical content knowledge is also considered in relation to the same two problems. Findings include a surprising relationship of outcomes for students across the problem contexts and some concern about teachers’ pedagogical content knowledge in this area of the curriculum.

Measuring Middle School Students' Interest in Statistical Literacy

The following paper describes the development of an instrument designed to assess middle school students’ interest in statistical literacy. The paper commences with a review of the literature as it relates to interest in this context and then proposes a theoretical model upon which the proposed instrument is based. The Rasch Rating Scale model is then applied to student responses to items in the instrument and fit statistics are analysed in order to assess the extent to which these responses conform to the requirements of the measurement model. The paper then presents evidence, including interview data, to support the validity of interpretations that can be made from the proposed instrument. The findings suggest that the proposed instrument provides a theoretically sound measure of middle school students’ interest for statistical literacy that will be useful for the evaluation of interventions aimed at developing these students’ statistical literacy.