It is a challenge for mathematics teachers to provide activities for their students at a high level of cognitive demand. In this article, we explore the possibilities that history of mathematics has to offer to meet this challenge. History of mathematics can be applied in mathematics education in different ways. We offer a framework for describing…

This paper is a commentary on the theoretical formulations of the five empirical papers in this special issue. All five papers use aspects of the theory of commognition as presented by Anna Sfard; however, even when the same notions (e.g., rituals or explorations) are incorporated into theoretical frameworks undergirding the research, these…

In 2008, the research center "Individual Development and Adaptive Education" was constituted by the Goethe University, the German Institute for International Educational Research, and the Sigmund Freud Institute, all located in Frankfurt am Main, Germany (http://www.idea-frankfurt.eu). The research of the center focuses on the…

The goal of this paper is to explore qualities of mathematical imagination in light of a classroom episode. It is based on the analysis of a classroom interaction in a high school Algebra class. We examine a sequence of nine utterances enacted by one of the students whom we call Carlene. Through these utterances Carlene illustrates, in our view,…

This conceptual research framework addresses the problem of rote learning by characterising key aspects of the dominating imitative reasoning and the lack of creative mathematical reasoning found in empirical data. By relating reasoning to thinking processes, student competencies, and the learning milieu it explains origins and consequences of…

In this paper we describe learners being asked to generate examples of new mathematical concepts, thus developing and exploring example spaces. First we elaborate the theoretical background for learner generated examples (LGEs) in learning new concepts. The data we then present provides evidence of the possibility of learning new concepts through…

Learning is better than knowing, generalization is more illuminating than abstract generality or universality because we perceive and thus become conscious of change or development only. Signs and representations establish the dialectic of fixation on the one hand and transformation on the other, which is so essential to learning and cognition.…

Analyses several examples of tacit influences exerted by mental models on the interpretation of various mathematical concepts in the domain of actual infinity. Specifically addresses the unconscious effect of the figural-pictorial models of statements related to the infinite sets of geometrical points related to the concepts of function and…

Recent research suggests that the examination of students' work may lead to changes in teaching practice that are more effective in terms of students' mathematical learning. However, the link between the examination of students' work and the teachers' actions in the classroom is largely unexamined, particularly at the secondary level. In this…

This is Part 2 of a two-part study of how APOS theory may be used to provide cognitive explanations of how students and mathematicians might think about the concept of infinity. We discuss infinite processes, describe how the mental mechanisms of interiorization and encapsulation can be used to conceive of an infinite process as a completed…

Listening to students in productive ways seems to be at the core of teaching practices aligned with the basic tenets of the constructivist world view. We present a definition for productive ways of listening, discuss the challenges involved in implementing it, and propose a way to support the "decentering" needed to learning to listen for teacher…

Examines the role that memory plays in the understanding of mathematics. Areas discussed include: retention research; memory and teaching; structure and organization in memory; and transformations in memory. Relevant research and implications for instruction are addressed. (JN)

Proposes to base research on a metaphor of thinking-as-communicating. Mathematical discourse is made special by two main factors: (a) exceptional reliance on symbolic artifacts as communication-mediating tools; and (b) particular meta-rules that regulate this type of communication. There is a better chance of accounting for some of the phenomena…

The framework for this paper is a recently developed theory of abstraction in context. The paper reports on data collected from one student working on tasks concerned with absolute value functions. It examines the relationship between mathematical constructions and abstractions. It argues that an abstraction is a consolidated construction that can…

A socio-constructivist account of learning and emotions stresses the situatedness of every learning activity and points to the close interactions between cognitive, conative and affective factors in students' learning and problem solving. Emotions are perceived as being constituted by the dynamic interplay of cognitive, physiological, and…