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…

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)

Discusses the development of visual thinking in students. Also presents a strategy that incorporates visualization exercises within the framework of traditional mathematics. The technique appears successful for students studying geometry, fractions, and problem solving. Visualization abilities increase understanding and therefore give students…

This volume attempts to bring together a collection of reports on the Geometric Supposer, a series of computer software environments which can be a tool for exploring particulars and generalizations in geometry. The book contains the following chapters: (1) "A Personal View of the Supposer: Reflections on Particularities and Generalities in…

A system for categorizing components of geometry class discussions is described. The unit of observation is the teacher question; responses are coded at four cognitive levels. The system is easier to use than many in the literature; use of it can help teachers develop questioning techniques. (SD)

These papers presented at a symposium describe how tenth grade students would be taught the slope of a line in geometry using three different approaches to teaching theory: behaviorism, Piagetian cognitive development, and information processing. Analyses of each approach focus on manner of learner cooperation and differing teacher role. (MBR)

Some geometric activities are described that teachers can use to give their students experiences that will influence their spatial abilities. It is noted that the goal is to improve spatial abilities, not to increase knowledge, so individual pupil responses should not be used to judge student achievement. (MP)

Compared are the van Hiele levels of geometric thinking and the geometry curriculum recommended by the National Council of Teachers of Mathematics Curriculum and Evaluation Standards for School Mathematics. Activities which illustrate the various levels are provided by grade level with procedures. (CW)

Examines aspects of Navajo cosmology relevant to understanding Navajo spatial representations. Compares Navajo children's spatial knowledge with Piaget's findings about the development of geometric concepts in Swiss children. Describes classroom activities whereby Navajo children explore the geometry inherent in their cultural and physical…

Discussed are supercalculator capabilities and possible teaching implications. Included are six examples that use a supercalculator for topics that include volume, graphing, algebra, polynomials, matrices, and elementary calculus. A short review of the research on supercomputers in education and the impact they could have on the curriculum is…

This book examines the Navajo system of spatial knowledge and describes a culture-based curriculum for the development of an intuitive geometry based on the child's experience of the physical world. Aspects of the Navajo cosmology relevant to spatial knowledge are discussed: the structure of the world; the dynamic nature of the universe;…

Mathematical fiction writing as a learning tool is discussed utilizing an actual example of a student's fictional text related to the field of geometry. Commentary about this example and how it relates to other areas of mathematics can provide teachers with insight into students' schema for organization and internalization of mathematical…

Describes one teacher's reflection concerning the quest to develop an understanding of school mathematics that promotes and sustains students' opportunities for exploration and conjecture. Recounts that a particular student's exploration of the features of parabolas eventually led to an understanding of the quadratic formula precisely because of…