The notions of "abstract "and "concrete" are central to the conceptualization of mathematical knowing and learning. It is generally accepted that development goes from concrete toward the abstract; but dialectical theorists maintain just the opposite: development consists of an ascension from the abstract to the concrete. In this article, we…

Since the game SET[R] was first introduced to the public in 1993, it has stimulated some interesting studies. While the game itself is rather straightforward, a plethora of decent mathematical questions lies beneath the surface. It is perhaps because the game ties in so closely with such an underlying mathematical term that its implications can be…

Abstract algebra courses tend to take one of two pedagogical routes: from examples of mathematics structures through definitions to general theorems, or directly from definitions to general theorems. The former route seems to be based on the implicit pedagogical intention that students will use their understanding of particular examples of an…

It is claimed that, since mathematics is essentially a self-contained system, mathematical objects may best be described as "abstract-apart." On the other hand, fundamental mathematical ideas are closely related to the real world and their learning involves empirical concepts. These concepts may be called "abstract-general" because they embody…

In this paper, we present a didactic analysis of the mathematical concept of implication under three points of view: sets, formal logic, deductive reasoning. For this study, our hypothesis is that most of the difficulties and mistakes, as well in the use of implication as in its understanding, are due to the lack of links in education between…

This paper is concerned with the use of spreadsheets within mathematical investigational tasks. Considering the learning of both children and pre-service teaching students, it examines how mathematical phenomena can be seen as a function of the pedagogical media through which they are encountered. In particular, it shows how pedagogical apparatus…

One major purpose of this investigation was to determine and compare the effects of three different perceptual situations (screening, neutral, and conflict) on kindergarten and first grade children's ability to use the transitive property of the equivalence relation "same length as" and the two order relations "longer than" and…

Investigated distribution of spatial visualization scores (Space Relations test of the Differential Aptitude Test) and mathematical problem-solving scores (Mental Arithmetic Problems) obtained by 161 male and 152 female 9th grade students. No significant sex-related differences were found between mean scores of tests of spatial visualization or…

Discusses three problems preschool teachers should consider when aiming to facilitate children's mathematical activity: (1) How may young children relate to the necessarily abstract basis of mathematics? (2) When is a child's activity mathematical? (3) How may "mathematical" language, concepts, and experiences be appropriately defined?…

Outlines a teaching sequence to help students move from finger counting to abstract levels of computation. (Author/SP)

Results showed that only 32 percent of adult female shoppers in a supermarket were able to use a proportional reasoning strategy to determine which of two sizes of a common item (size ratio 2:3) was the better buy. Performance declined when the ratio was more complex. (JMB)

Meaningful educational activities and cognitive tools might improve students' active involvements in the teaching-learning process and encourage their reflections on the concepts and relations to be investigated. It is claimed that usage of manipulatives not only increase students' conceptual understanding and problem solving skills but also…

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…

The concern of the experiment is to find out the roles of abstraction and generalization in the learning of mathematical structures. The basic question is whether to generalize before abstracting or vice-versa in order to maximize transfer. The experiment involves four mathematical tasks and a transfer of activity. Experimental procedures are…