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This article is an attempt to place mathematical thinking in the context of more general theories of human cognition. We describe and compare four perspectives—mathematics, mathematics education, cognitive psychology, and evolutionary psychology—each offering a different view on mathematical thinking and learning and, in particular, on the source of mathematical errors and on ways of dealing with them in the classroom. The four perspectives represent four levels of explanation, and we see them not as competing but as complementing each other. In the classroom or in research data, all four perspectives may be observed. They may differentially account for the behavior of different students on the same task, the same student in different stages of development, or even the same student in different stages of working on a complex task. We first introduce each of the perspectives by reviewing its basic ideas and research base. We then show each perspective at work, by applying it to the analysis of typical mathematical misconceptions. Our illustrations are based on two tasks: one from statistics (taken from the psychological research literature) and one from abstract algebra (based on our own research). [ABSTRACT FROM AUTHOR]

This article focuses on the similarities and differences in the academic education of software engineers and architects. The rationale for this work stems from our observation, each from the perspective of her or his own discipline, that these two professional design and development processes share some similarities. A pilot study was performed, in which 24 practitioners (12 software engineers and 12 architects) were interviewed concerning the main characteristics of the two professions. The characteristics were classified according to four categories: discipline-related characteristics, professional skills, characteristics of typical problems and processes, and characteristics of a typical product. Data analysis indicates that according to the practitioners, while the professional skills required of professionals in the two disciplines are mostly common, the difference between the two disciplines is reflected most significantly in the features of the product created in the two professional processes. Based on this observation, some conclusions are outlined with respect to the academic education in the two fields. [ABSTRACT FROM AUTHOR]

Research in mathematics education usually attempts to look into students’ learning and other mental processes. It could therefore be expected to build on knowledge acquired within the academic discipline of cognitive psychology. Our aim in this paper is to show how some recent developments in cognitive psychology can help interpret empirical results from mathematics education. In particular, we will be looking into the heuristics-and-biases research by Kahneman and Tversky, the alternative views by Gigerenzer et al., and the more recent dual-process theory that has come to play a central role in interpreting this research. We first introduce the relevant background from cognitive psychology and survey its connections to previous work in mathematics education; then we apply this theoretical framework for re-interpreting previously-published empirical data from mathematics education research. We conclude with a discussion of potential theoretical and practical benefits of such synthesis. [ABSTRACT FROM AUTHOR]

There is a growing interest in the mathematics education community in the notion of abstraction and its significance in the learning of mathematics. “Reducing abstraction” is a theoretical framework that examines learners’ behavior in terms of coping with abstraction level. It refers to situations in which learners are unable to manipulate concepts presented in a given problem; therefore, they unconsciously reduce the level of abstraction of the concepts involved to make these concepts mentally accessible. This framework has been used for explaining students’ conception in different areas of undergraduate mathematics and computer science. This article extends the applicability scope of this framework from undergraduate mathematics to school mathematics. We draw on recently published research articles and exemplify how students’ behavior can be described in terms of various interpretations of reducing abstraction level. [ABSTRACT FROM AUTHOR]

How do undergraduate students cope with abstract algebra concepts? How should we go about researching this question? Based on interviews with undergraduate students and on written questionnaires, a theoretical framework evolved which could coherently account for most of the data. According to this theoretical framework, students‘ responses can be interpreted as a result of reducing the level of abstraction. In this paper, the theme of reducing abstraction is examined, based on three interpretations for levels of abstraction discussed in mathematics education research literature. From these three perspectives on abstraction, ways in which students reduce abstraction level are analyzed and exemplified. [ABSTRACT FROM AUTHOR]

Students in mathematical problem-solving situations often experience confusion and loss of meaning. In these situations, affective and social factors are as much part of the student‘s thinking and behavior as are cognitive factors. These additional factors might include, for example, the need to make sense and the need to meet expectations of the authority figure involved (e.g., teacher or researcher). In this paper we attempt to analyze students‘ productions, taking into account such additional affective and social factors. To this end, we have tried to reproduce the student‘s voice in what we call {\it virtual monologue}. It consists of a monologue in the student‘s voice given in first person language, in which we try to describe as vividly and as faithfully as we can our picture of what might be going on in the student‘s mind during such situations. [ABSTRACT FROM AUTHOR]

Focuses on how undergraduate students in the first abstract algebra course learn the concept of group isomorphism. Various faces of isomorphism; Survey of literature; Research protocol and results; Naive isomorphism; Object isomorphism.

The article presents a paper from the Research Conference in Collegiate Mathematics Education (RCCME) held at Central Michigan University in September 1996. It attempted to define abstraction but it resulted to a dialogue which gathered different perspectives and written statements. Cited are the conversations of two mathematics instructors at a conference, Allie and Bert, discussing topics such as locating abstraction in mathematics, and relationships between the person and the object about which she or he thinks.

To make optimal use of computational environments, one must understand how students interact with the environments and how students' mathematical thinking is reflected and affected by their use of the environments. Similarly, to make sense of research on students' thinking and learning, one must understand how the environments and contexts used in the research may affect the conclusions one derives. The research on students' learning of functions has approached the topic in terms of symbols and graphs (see, for example, Leinhardt et al. (1990) for a review of work up to that point; Harel and Dubinsky (1992) for a collection of research; and Dugdale et. al. (1995), for some recent thinking about implications for curriculum reform using technology). Dynamic geometry environments (DGEs) like Cabri Geometry or Geometer's Sketchpad, offer us an opportunity to get a new perspective on these old and important issues. DGEs let students build geometrical constructions and then drag certain objects around the screen in a continuous manner while observing how the entire construction responds dynamically. In this way DGEs model functional relationships that are not specified by symbols or represented by graphs. Based on interviews with undergraduate mathematics majors, this paper presents preliminary observations that confirm some old results and raise some new questions about students' notions of function. [ABSTRACT FROM AUTHOR]