The study examined first-year engineering students' responses to a statistics task that asked them to generate a procedure for quantifying variability in a data set from an engineering context. Teams used technological tools to perform computations, and their final product was a ranking procedure. The students could use any statistical measures,…

This article examines the notion of informal mathematical products, in the specific context of teaching mathematics to low achieving students at the secondary school level. The complex and relative nature of this notion is illustrated and some of its characteristics are suggested. These include the use of ad-hoc strategies, mental calculations,…

Through a case study of four elementary education undergraduates, we seek new analytic constructs that could help make clearer how arguments discovered or tested in quite special cases might come to support assertions that are understood to hold in general. We began analysis from a particular standpoint: to focus fundamentally on learners'…

This study is a part of a research project that seeks to characterize the relationship between mathematics teachers' knowledge and their practice. In this paper, we focus on identifying the characteristics of subject matter knowledge and pedagogical content knowledge that two teachers integrate in decisions they make about the introduction of…

A cross-curricular structured-probe task-based clinical interview study with 44 pairs of third-year high-school mathematics students, most of whom were high achieving, was conducted to investigate their approaches to a variety of algebra problems. This paper presents results from one problem that involved solving a set of three linear equations of…

This classroom scholarship report presents a group of elementary students' experiences learning the traditional long division algorithm. The traditional long division algorithm is often taught mechanically, resulting in the student's performance of step-by-step procedures with no or weak understanding of the concept. While noting some initial…

An attempt is made to analyze mathematical behavior from more general psychological perspectives. The mathematical language is a special case of the human language, which is a form of expression. Many people use common language in a meaningless way. The same is true about the mathematical language. Rituals are other forms of expression. Many…

This paper describes a detailed analysis of verbal discourse within an exemplary mathematics lesson--that is, George Polya teaching in the Mathematics Association of America [MAA] video classic, "Let Us Teach Guessing" (1966). The results of the analysis reveal an inductive model of teaching that represents recursive cycles rather than linear…

This study investigates two sixth grade students' dilemmas regarding the parity of zero. Both students originally claimed that zero was neither even nor odd. Interviews revealed a conflict between students' formal definitions of even numbers and their concept images of even numbers, zero, and division. These images were supported by practically…

In the late seventies, Guy Brousseau set himself the goal of verifying experimentally a theory he had been building up for a number of years. The theory, consistent with what was later named (non-radical) constructivism, was that children, in suitable carefully arranged circumstances, can build their own knowledge of mathematics. The experiment,…

An enduring challenge in mathematics education is to create learning environments in which students generate, refine, and extend their intuitive and informal ways of reasoning to more sophisticated and formal ways of reasoning. Pressing concerns for research, therefore, are to detail students' progressively sophisticated ways of reasoning and…

The aim of this study is to describe and analyze students' levels of understanding of exponents within the context of procedural and conceptual learning via the conceptual change and prototypes' theory. The study was conducted with 202 secondary school students with the use of a questionnaire and semi-structured interviews. The results suggest…

The significance of discussion in mathematics classes has been prominently debated in the research literature. Different studies have stressed the importance of teacher flexibility in orchestrating the discussion. We introduce an operational definition of teacher flexibility. In a case study with one secondary-school mathematics teacher, we…

The purpose of this paper is to present evidence supporting the conjecture that graphs and gestures may function in different capacities depending on whether they are used to develop an algorithm or whether they extend or apply a previously developed algorithm in a new context. I illustrate these ideas using an example from undergraduate…

Empirical research shows that students often use reasoning founded on copying algorithms or recalling facts (imitative reasoning) when solving mathematical tasks. Research also indicate that a focus on this type of reasoning might weaken the students' understanding of the underlying mathematical concepts. It is therefore important to study the…