Children's understanding of Euclidian space was investigated using three Piaget-type tasks to examine the ability of children to quantitatively locate a point in one, two, and three dimensions. Among the findings were that a disagreement exists between Piaget's data and the results of this study. (BT)

This paper reports results related to the development of a consistent descriptive language for research on mathematical reasoning. Ways of reasoning deductively are highlighted, using examples drawn from observations of young students. One-step deductions versus multi-step deductions, known versus hypothetical premises, and single versus multiple…

The natural environment rings with mathematical sounds, as can be seen in the logarithmic spirals of elephant tusks, the hexagonal shapes of snowflakes and honeycomb cells, the rhythmic spouting of gysers, and in the circular growth of microscopic sea plants. Photographs reveal the mathematical relationships of familiar objects. (Author)

Utilizes a psychological-constructivist perspective to examine the transitions that students make from arithmetic to algebra in the context of problems that, from an expert's perspective, involve the concept of linear inequality. Reports that students who completed a transition to algebra operated at higher levels of reflective abstraction than…

Investigates the concepts of ratio and proportion constructed by grade 9 students by investigating their proportional reasoning schemes and procedures on three types of tasks: missing value, numerical comparison, and qualitative reasoning. Indicates that students frequently used additive reasoning--that is, a comparison of two numbers by…

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…

In teaching symbolic logic to students not majoring in mathematics, one can appeal to simple arithmetic examples, set theory examples, and the use of conjunction and disjunction as aids in presenting the conditional. (SD)

The relationship of particular learning styles and plausible reasoning ability of 110 subjects from 6th, 7th, and 8th grade was examined. Data were compiled on a plausible reasoning test and on a Learning Style Inventory with four subscales: (1) Concrete Experience (CE), (2) Reflective Observation (RO), (3) Abstract Conceptualization (AC), and (4)…

This is one of a series that is a collection of translations from the extensive Soviet literature of the past 25 years on research in the psychology of mathematics instruction. It also includes works on methods of teaching mathematics directly influenced by the psychological research. Selected papers and books considered to be of value to the…

Examples are given illustrating a facet of early learning involving insights into abstract thinking. Several learning theories examined fail to account for this phenomenon. (MP)

The relationship between a cognitive style variable, field-dependence-independence, and instructional treatments that differed in two dimensions of discovery learning, level of guidance and level of abstraction, was investigated with 116 prospective elementary teachers. As predicted, field-independent students did better with minimum guidance,…

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…

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. This article extends the scope of applicability of this…

This paper poses and answers 20 questions about mathematical reasoning. Questions include: (1) Is mathematical reasoning mathematical? (2) Is mathematical reasoning useful? (3) Is mathematical reasoning an appropriate goal of school mathematics? (4) Can teachers teach mathematical reasoning? (5) Can mathematical reasoning be taught? (6) Do skills…

An experimental study of children's confusion between if" and if and only if" statements when phrased in different language and used in different contexts (grades 4, 6, 8, and 10). (MM)