Geometry is one of the highest achievements of our species, but its foundations are obscure. Consistent with longstanding suggestions that geometrical knowledge is rooted in processes guiding navigation, the present study examines potential sources of geometrical knowledge in the navigation processes by which young children establish their sense…

In recent years, there has been increased interest in improving early mathematics curricula and instruction. Subsequently, there has also been a rise in demand for better early mathematics assessments, as most current measures are limited in their content and/or their sensitivity to detect differences in early mathematics development among young…

In the field of human cognition, language plays a special role that is connected directly to thinking and mental development (e.g., Vygotsky, "1938"). Thanks to "verbal thought", language allows humans to go beyond the limits of immediately perceived information, to form concepts and solve complex problems (Luria, "1975"). So, it appears language…

Describes the van Hiele levels of reasoning in geometry according to responses to clinical interview tasks concerning triangles and quadrilaterals. Subjects were 13 students from grades 1 through 12 plus a university mathematics major. Students' behavior on tasks was consistent with the van Hiele original general description of the levels.…

The aim of this study was to examine the views of pre-service mathematics teachers on the use of webquests in teaching and learning geometry with reference to a theoretical framework developed by Dodge in 1995. For this study the researcher identified four groups containing nineteen pre-service mathematics teachers, which were then assigned to…

Uses manipulative materials to build and examine geometric models that simulate the self-similarity properties of fractals. Examples are discussed in two dimensions, three dimensions, and the fractal dimension. Discusses how models can be misleading. (Contains 10 references.) (MDH)

Presented is an informal technique for counting the number of different-sized equilateral triangles on an isometric grid. Five activities that relate to this topic are included. (KR)

Examples from some area tasks indicate that misconceptions about linear measurement can be carried over to area measurement; children will occasionally count points rather than length units or area units. (MP)

Presents 13 examples in which the intuitive approach to solve the problem is often misleading. Presents analysis of these problems for five different sources of misleading intuitive generators: lack of analysis, unbalanced perception, improper analogy, improper generalization, and misuse of symmetry. (MDH)

Describes a geometry course that integrates transformation geometry into traditional high school geometry. Discussion of the scope and sequence of the course includes the topics of proof, congruence, translations, rotations, reflections, dilations, quadrilaterals, parallel lines, and similarity. (MDH)

Analyzes one student's thinking using the Van Hiele levels of geometric thinking. (KHR)

Defines spatial structuring as the mental operation of constructing an organization or form for an object/set of objects. Examines in detail students' structuring and enumeration of two-dimensional rectangular arrays of squares. Concludes that many students do not see row-by-column structure. Describes various levels of sophistication in students'…

This essay discusses the sketching and transformation of parabolas and other curves as areas where learners can exercise and develop control of their mental imagery. (MN)

Examines the nature of prior mathematical knowledge that facilitates the construction of useful problem representations in the domain of geometry. Indicates that high achievers build schemas that are qualitatively more sophisticated than low achievers, which in turn helps them construct representations that are conducive to understanding the…