The notion of definition is a central building block of mathematics. In addition to specific definitions, it is desirable that students learn about meta-mathematical aspects of definitions such as their role of classification, and about characteristics of definitions such as arbitrariness. We investigated how such meta-mathematical aspects emerge…

The multiplicative reasoning that students should develop in elementary school is a key area of research in contemporary mathematics education. Researchers employ various views including multiplication as arithmetic operation, multiplicative structures, and multiplicative relationships. They also propose various classifications of multiplicative…

In this paper, we take a semiotic/dialogic approach to investigate how a group of UK 12-13-year-old students work with hierarchical defining and classifying quadrilaterals. Through qualitatively analysing students' decision-making processes, we found that the students' decision-making processes are interpreted as transforming their…

Reversibility of concepts, a key aspect of mathematical development, is often problematic for learners. In this theoretical paper, we present a typology we have developed for categorizing the different reverse concepts that can be related to a particular initial concept and explicate the relationship among these different reverse concepts. We…

A broad categorisation of different reading styles for mathematics text is generated in this research. The styles derive from those found in literature around academic reading skills. These styles are inductively refined using video transcripts of five specially chosen students studying out loud from a prescribed mathematics textbook. The context…

This paper describes a study in which we investigated relationships between defining mathematical concepts--increasing and decreasing infinite sequences--explaining their meanings and classifying consistently with formal definitions. We explored the effect of defining, explaining or studying a definition on subsequent classification, and the…

The paper introduces an interpretative framework that contains a characterization of "epistemic schemes" (constructs that are used to explain how class agents themselves are able to gain convincement in or promote convincement of mathematical statements) and "epistemic states" (a person's internal states, such as…

In this paper, we propose an approach to analysing teacher arguments that takes into account field dependence--namely, in Toulmin's sense, the dependence of warrants deployed in an argument on the field of activity to which the argument relates. Freeman, to circumvent issues that emerge when we attempt to determine the field(s) that an argument…

This study examines patterns in middle-grade boys' and girls' written problem solving strategies for a mathematical task involving proportional reasoning. The students participating in this study attend a coeducational charter middle school with single-sex classrooms. One hundred nineteen sixth-grade students' responses are analyzed by gender…

This is a theoretical article proposing a way of organizing and structuring the discussion of why and how to use the history of mathematics in the teaching and learning of mathematics, as well as the interrelations between the arguments for using history and the approaches to doing so. The way of going about this is to propose two sets of…

A previous study shows how a twelve-year-old girl discovers angles in her narrative from a climbing trip. Based on this research, the girl's class takes part in one day of climbing and half a day of follow-up work at school. The students mathematise their climbing with respect to angles and they express themselves in texts and drawings. Their…

Understanding that mathematics is founded on reasoning and is not just a collection of rules to apply is an important message to convey to students. Here we examined the reasoning presented in seven topics in nine Australian eighth-grade textbooks. Focusing on explanatory text that introduced new mathematical rules or relationships, we classified…

The development and use of a taxonomy for describing the way distance students plan work and use course materials to complete mathematics exercises is detailed. The taxonomy was developed from an exploratory study but the initial results are viewed as having implications for educational use in many situations. (Author/MP)

Proposes a four-category classification of semantic structures of one-step word problems involving multiplication and division: forming the n-th multiple of measures, combinatorial multiplication, composition of operators, and multiplication by formula. This classification is compatible with semantic structures of addition and subtraction word…

Discusses some aspects of the defining process in geometrical context in the reference frame of the theory of "figural concepts." Presents analysis of some examples taken from a teaching experiment at the sixth-grade level. Contains 30 references. (Author/ASK)