What is involved in consolidating a new mathematical abstraction? This paper examines the work of one student who was working on a task designed to consolidate two recently constructed absolute function abstractions. The study adopts an activity theoretic model of abstraction in context. Selected protocol data are presented. The initial state of…

This paper reports on a study that investigated the interpretation of media graphs developed by student teachers. The analyses of interviews indicate that cognitive, affective and contextual, aspects might constitute important components of the interpretation of graph such as those that we can find in print media. Particularly we discuss Critical…

Nonroutine problem-solving, too, involves elements of routine. Often, elementary routines such as solving for a variable are not carried out without error. In this paper, we use new findings from neuroscientific research to explain why even excellent students make mistakes in elementary routines. [For complete proceedings, see ED489597.]

Human nature had traditionally been the realm of novelists, philosophers, and theologicians, but has recently been studied by cognitive science, neuroscience, research on babies and on animals, anthropology, and evolutionary psychology. In this paper I will show--by surveying relevant research and by analyzing some mathematical "case studies"--how…

Numeracy can be defined as having the competence and disposition to use mathematics to meet the general demands of life at home, in paid work, and for participation in community and civic life (Willis 1992). An important aspect of developing the capacity to use mathematics in everyday life is, for students at school, to use mathematics to meet the…

Efforts to develop a mathematics curriculum that meets the needs of a modern society are reflected in reform recommendations across the developed world. A common requirement is for students to understand the calculation procedures they are taught and to develop "number sense". This paper will analyse students' strategies for calculating…

Our study concerns the analysis of teacher and student activities. Secondary school 6th grade students were confronted, for the first time, with solving equations. We used our cognitive models of students and experts (in algebraic calculations) for analyzing the teaching process and the cognitive functioning of students. Our model led us to…

The project's focus was to learn more about the ways to best assess and support the development of numeracy in the early grades. Project goals include involving teachers and researchers in: (1) the creation and use of performance-based tasks most appropriate for assessing numeracy in young learners; (2) the development and refinement of…

This is a qualitative study of 22 9th graders performing generalizations on a task involving linear patterns. Our research questions were: What enables/hinders students' abilities to generalize a linear pattern? What strategies do successful students use to develop an explicit generalization? How do students make use of visual and numerical cues…

Much advice about teaching for understanding implies that teacher should help children to develop connections between aspects of their experience, knowledge, and skills. This paper outlines points from the literature about different types of connections and describes relevant points from four case study teachers said and did. [For complete…

This study explores students algebraic and geometric approach in solving tasks in functions and the relation of these approaches with complex geometric problem solving. Data were obtained from 95 sophomore pre-service teachers, enrolled in a basic algebra course. Implicative statistical analysis was performed to evaluate the relation between…

This paper explores one student's attempt to apprehend an abstract mathematical structure (similar to Z[99]). We discuss Karmiloff-Smith's theory of representational redescription as a model for the development of structural understanding and contrast this with existing process-object theories. We use two cycles in Molly's movement from an action…

The major idea in this paper is the formulation of a theory of three distinct but interrelated worlds of mathematical thinking each with its own sequence of development of sophistication, and its own sequence of developing warrants for truth, that in total spans the range of growth from the mathematics of new-born babies to the mathematics of…

In our investigation of preservice secondary teachers' understanding of irrational numbers we focus on how different representations influence participants' responses with respect to irrationality. As a theoretical perspective we use the distinction between transparent and opaque representations, that is, representations that "show"…

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,…