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…

This paper reports on data from investigations on learners' understanding of inclusion relations of quadrilaterals, building on the ideas from our earlier study (Fujita & Jones, 2007). By synthesising past and current theories in the teaching of geometry (van Hiele's model, figural concepts, prototype phenomenon, etc.), we propose a theoretical…

This dissertation documents a new way of conceptualizing vectors in college mathematics, especially in geometry. First, I will introduce three problems to show the complexity and subtlety of the construct of vectors with the classical vector representations. These highlight the need for a new framework that: (1) differentiates abstraction from a…

To better understand the development of children's thinking in three-dimensional geometry, we conducted a teaching experiment with 8- and 9-year olds in which children built and described polyhedra during several lessons. Analysis of pre-/post-assessments showed that children advanced in their geometric reasoning and began to identify, enumerate,…

Four- to 10-year-old children (n = 50) participated in a 2D search task that included geometry (with- and without lines) and feature conditions. During each of 27 trials, participants watched as a cartoon character hid behind one of three landmarks arranged in a triangle on a computer screen. During feature condition trials, participants could use…

This study investigated the geometric cognitive growth of pre-service mathematics teachers in terms of the Van Hiele levels in a technology-enriched environment, as opposed to that of students in a learning environment without any technological enhancements. In order to investigate this, a quasi-experimental non-equivalent comparison group design…

Two experiments tested whether 4-year-old children extract and use geometric information in simple maps without task instruction or feedback. Children saw maps depicting an arrangement of three containers and were asked to place an object into a container designated on the map. In Experiment 1, one of the three locations on the map and the array…

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…

This study examined the knowledge and strategies that young children used for comparing sizes of geometric figures. Sixty-nine children from the ages 3 to 6 years were asked to compare sizes of geometric figures and their placement and adjustment strategies were observed. The children were also presented with strategies for comparing sizes and…

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)

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)

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)

Theory and research concerning the development of spatial concepts by young children are discussed. Results of Martin and Kidder are compared with those of Piaget. (SD)

This article describes how children build increasingly abstract knowledge of linear measurement, emphasizing ways they relate space and number. Assessments indicate children struggle to understand measurement, especially concepts related to complex paths as in perimeter tasks. This article draws on developmental accounts of children's knowledge of…