This article shows that the recognition of the dualities in equality (operator-equivalent) of the minus sign (unary-binary) and the zero (nullity-totality) during the transitional process from arithmetic to algebra by 12-13 year-old students constitutes a possible way to achieve the extension of the natural number domain to the integers. (Contains…

This account of my extended conversation with a high school mathematics class focuses on voice and agency. I prompted the students daily to become ever more aware of their language practices in class. The tensions in this conversation proved parallel to the tensions in mathematics between individual initiative and convention, a tension that…

Despite mathematicians valuing the ability to visualize a problem and psychologists finding positive correlations of visual-spatial ability with success in mathematics, many educationists remain unconvinced about the benefits of visualization for mathematical understanding. This paper describes research that compared a "visual" to a "verbal"…

The Purposeful Algebraic Activity Project is concerned with the development of algebraic activity in pupils in the early years of secondary schooling. Here we report on the different ways in which we have observed children articulating generalisations during semi-structured interviews which are being used to gain snap-shots of this development. We…

Fairly large amount of research on students' conceptions of probability has been reported in the literature. Most of this work represents school students' reasoning of probability in individual setting. This study provides 8 secondary school mathematics preservice teachers' reasoning of probability in both individual and pair settings. Consistent…

Seventh and eighth grade students identified as holding an incomplete fraction conception of decimals were tested on related fraction knowledge. Most of these students (78%) had a problem in coordinating the size of the parts and the number of parts in comparing fractions. These students underwent several instructional sessions. Half of them…

The study analyzed gender attitudes of students in learning mathematics by using a calculator, in one of Botswana's Junior Secondary Schools. Students' attitudes were sought using a questionnaire and data was analyzed by both quantitative and qualitative methods. Attitude variables used were usefulness of calculators, enjoyment and anxiety in…

Starting from a situated cognition perspective, this paper reports on the activity of 9th grade students who are interpreting the shape of a graph arising from the motion of a bouncing ball. In an unfamiliar context, informed by previous knowledge of similar experiments, the obstacle of understanding why the graph does not start from the origin is…

The teaching and learning of early algebra draw heavily on arithmetic and the relationship between these two forms of activity is much debated. Drawing on interviews with 12 year old pupils we consider the ways in which some pupils used substitution of numbers for letters to extend their ability to manipulate algebraic expressions. We argue that…

This paper presents the design and some results of a series of teaching experiments. The design was created to develop a model for everyday math lessons, that describes the conditions which foster or hinder the construction of new mathematical meanings. The development process includes the students' epistemic processes, their social interactions,…

During several years of experimentation, we accumulated a large collection of data on the analysis of problems generally encountered in algebra and on the reasoning developed by students at different levels when confronted with these problems. Analysis of this data led us to a new way of looking at differences among word problems, and to the…

This paper first postulates the existence of co-constructed patterns of participation in the classroom and then documents one such pattern of participation: Kikan-shido (instruction between desks), both as it is enacted and as it is perceived by the classroom participants. In the course of detailing the use of kikan-shido in three Australian…

This report focuses on teachers' individual curricula. An individual curriculum includes contents and reasoning and can be structured in a quasi-logical system of goals and methods, which is the result of teachers' planning of mathematics instruction. There is consent that the planning of individual curricula or the instructional practice is a…

Previous research has shown that many secondary school students improperly apply linear models when solving non-linear problems involving lengths, area and volume of similar plane figures and solids. This phenomenon is called the "illusion of linearity". This paper presents a teaching experiment in which we developed and tested a learning…

This paper discusses problems many tenth grade students have when asked to apply a familiar formula in an unfamiliar context: specifically factoring a compound expression. Some of their attempts at solution are presented and discussed in terms of structure sense. (Contains 1 table.) [For complete proceedings, see ED496848.]