Date of Award


Document Type


Degree Name

Doctor of Philosophy (PhD)


Educational Theory, Policy, and Practice

First Advisor

David Kirshner


Children encounter and recognize similar figures in their everyday experiences with such things as basketballs, soccer balls, tennis balls, ping-pong balls; or a candy bar that comes in various sizes of the same shape. Yet their school experience with the mathematics of similarity generally does not build on these perceptual intuitions. Traditional mathematics curricula bypass students' visual intuitions and their quantitative understandings, proceeding directly to set piece problems solved by formal algebraic methods. The result for many students is that the topic of similarity contributes to their evolving view of mathematics as a domain of complex procedural methods divorced from their intuitive sense of quantity and space. The purpose of this study was to explore how to develop students' mathematical understandings of similarity by having quantitative methods evolve from students' visual intuitions about similar figures. The foundation of this curricular approach was a perceptual analysis of similarity consisting of within relationships which are the static relationships within one figure that may be recognized in another, and between relationships which stem from the dynamic perception of one figure as resulting from uniform growth of the other. The curricular strategy encouraged students to verbalize and quantify these perceptual attributes of similar figures, eventually applying quantitative methods to the standard problems encountered in traditional curricula. The subjects were secondary school students enrolled in two separate geometry classes (one classified as college-bound) which were taught by the researcher over a period of three weeks. Qualitative data were collected and analyzed from video recordings, students' work, and journals. The results indicated that the students were generally able to represent their perceptual orientations quantitatively, and to utilize quantitative methods to solve problems. However, the between representation was selected by the plurality of students even when a within strategy would have been computationally more convenient. This highlights the observations of other researchers concerning the difficulty of static representations, and suggests a developmental model in which the more accessible dynamic representations of similarity should precede the static approaches. The students in the college-bound class exhibited more overall flexibility than the students in the non-college-bound class.