Michael Mitchelmore

Young Children's Intuitive Models of Multiplication and Division

In this study, an intuitive model was defined as an internal mental structure corresponding to a class of calculation strategies. A sample of female students was observed 4 times during Grades 2 and 3 as they solved the same set of 24 word problems. From the correct responses, 12 distinct calculation strategies were identified and grouped into categories from which the children’s intuitive models of multiplication and division were inferred. It was found that the students used 3 main intuitive models: direct counting, repeated addition, and multiplicative operation. A fourth model, repeated subtraction, only occurred in division problems. All the intuitive models were used with all semantic structures, their frequency varying as a complex interaction of age, size of numbers, language, and semantic structure. The results are interpreted as showing that children acquire an expanding repertoire of intuitive models and that the model they employ to solve any particular problem reflects the mathematical structure they impose on it.

Awareness of Pattern and Structure in Early Mathematical Development

Recent educational research has turned increasing attention to the structural development of young students’ mathematical thinking. Early algebra, multiplicative reasoning, and spatial structuring are three areas central to this research. There is increasing evidence that an awareness of mathematical structure is crucial to mathematical competence among young children. The purpose of this paper is to propose a new construct, Awareness of Mathematical Pattern and Structure (AMPS), which generalises across mathematical concepts, can be reliably measured, and is correlated with general mathematical understanding. We provide supporting evidence drawn from a study of 103 Grade 1 students.

Young children's intuitive understanding of rectangular area measurement

The focus of this article is the strategies young children use to solve rectangular covering tasks before they have been taught area measurement. One hundred fifteen children from Grades 1 to 4 were observed while they solved various array-based tasks, and their drawings were collected and analyzed. Childrens solution strategies were classified into 5 developmental levels; we suggest that children sequentially learn 4 principles underlying rectangular covering. In the analysis we emphasize the importance of understanding the relation between the size of the unit and the dimensions of the rectangle in learning about rectangular covering, clarify the role of multiplication, and identify the significance of a relational understanding of length measurement. Implications for the learning of area measurement are addressed.

Development of angle concepts by progressive abstraction and generalisation

This paper presents a new theory of the development of angle concepts. It is proposed that children progressively recognise deeper and deeper similarities between their physical angle experiences and classify them firstly into specific situations, then into more general contexts, and finally into abstract domains. An angle concept is abstracted from each class at each stage of development. We call the most general angle concept the standard angle concept. To investigate the role of the standard abstract angle concept in conceptual development, 192 children from Grades 2 to 8 were tested to find how they used it in modelling 9 physical angle situations and in expressing similarities between them. It was found that the standard angle concept first develops in situations where both arms of the angle are visible. Even at Grade 8, there are still significant proportions of students who do not use standard angles to represent turning and sloping situations. Implications for theory and practice are explored.

Children's informal knowledge of physical angle situations

Abstract There has been very little research on childrens informal knowledge of familiar situations from which the angle concept could be abstracted. The present study investigated 7-year-old childrens situated knowledge of turns, slopes, crossings, bends, rebounds and corners, as well as how the children classified angle situations, how they represented each situation using abstract angle models, and how well they recognised the similarity between different angle situations. It was found that children had an excellent knowledge of all situations presented, but that specific features of each situation strongly hindered recognition of the common features which define the angle concept. Implications for teaching are discussed.

Development of Angle Concepts: A Framework for Research.

Abstract36 Grade 4 children were interviewed to find how they interpreted the angles implicit in six realistic models. Angle recognition varied considerably across the six situations, as did children’s tendency to recognise angular similarities between them. Based on these and other results, a framework for further research on the development of children’s angle concepts is proposed. It is suggested that children first classify their everyday angle experiences intophysical angle situations; they then group situations to formphysical angle contexts; and then they gradually group contexts intoabstract angle domains. Each classification step leads to the formation of a corresponding angle concept.

Abstraction in Mathematics: Conflict, Resolution and Application.

Everyday usage of the term “abstract” has been shown to lead to a conflict in which abstract mathematics is seen to be both easier and more difficult than concrete mathematics. A literature review undertaken to identify the source of this conflict has revealed that the term “abstraction” may be used to denote either a process or a product. Two meanings of “abstract” are also identified. The first meaning, calledabstract- apart, refers to ideas which are removed from reality; the second meaning, calledabstract- general, refers to ideas which are general to a wide variety of contexts. It is argued in this paper that, whereas mathematics isabstract- general, mathematics teaching often leads toabstract- apart ideas. The initial conflict has been resolved by noting that abstract-apart ideas are adequate when a mathematical problem can be solved within a single level of abstraction; such problems are relatively easy. On the other hand, abstract-general ideas are essential for the successful solution of problems which require links between levels of abstraction; these problems are relatively difficult. The concepts of abstract-general and abstract-apart have then been applied to re-interpret two research studies (on letters in algebra and rates of change). It is suggested that greater interest in abstraction as a process would be beneficial to both the theory and practice of mathematics education.

Early Awareness of Mathematical Pattern and Structure

This chapter provides an overview of the Australian Pattern and Structure Project, which aims to provide new insights into how young students can abstract and generalize mathematical ideas much earlier, and in more complex ways, than previously considered. A suite of studies with 4- to 8-year old students has shown that an awareness of mathematical pattern and structure is both critical and salient to mathematical development among young students. We provide a rationale for the construct, Awareness of Mathematical Pattern and Structure (AMPS), which our studies have shown generalizes across early mathematical concepts, can be reliably measured, and is correlated with mathematical understanding. A study of Grade 1 students and follow up case studies enabled us to reliably classify structural development in terms of a five structural levels. Using a Pattern and Structure Assessment (PASA) interview involving 39 tasks, students identified, visualized, represented, or replicated elements of pattern and structure. Students with high AMPS are likely to have a better understanding of Big Ideas in mathematics than those with low AMPS. They are likely to look for, remember and apply spatial and numerical generalizations and in particular are likely to grasp the multiplicative relationships that underlie the majority of the concepts in the elementary mathematics curriculum.

Students’ difficulties in operating a graphics calculator

We investigated how students interpret linear and quadratic graphs on a graphics calculator screen. Clinical interviews were conducted with 25 Grade 10–11 students as they used graphics calculators to study graphs of straight lines and parabolas. Student errors were attributable to four main causes: a tendency to accept the graphic image uncritically, without attempting to relate it to other symbolic or numerical information; a poor understanding of the concept of scale; an inadequate grasp of accuracy and approximation; and a limited grasp of the processes used by the calculator to display graphs. Implications for teaching are discussed.

Teaching for Abstraction: A Model

This article outlines a theoretical model for teaching elementary mathematical concepts that we have developed over the past 10 years. We begin with general ideas about the abstraction process and differentiate between abstract-general and abstract-apart concepts. A 4-phase model of teaching, called Teaching for Abstraction, is then proposed that is explicitly designed to promote abstract-general learning. Studies investigating the model with 4 different topics (angles, decimals, percentages, and ratios) are reported, and the 4 phases are further elucidated. The article concludes with a discussion of the effectiveness of the model and its applicability to other mathematical concepts.