Date of Award

8-2008

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Dr. Steve Ziebarth

Second Advisor

Dr. Jane-Jane Lo

Abstract

The concept of similarity is uniquely situated at the crossroads of geometric and numerical proportional reasoning. Although studies have documented the existence and nature of student difficulties with this topic, there exists a gap between documented visual insights of younger children and the quantitative inadequacies of older ones. Using a revised version of the Similarity Perception Test followed by 21 clinical interviews, this study investigated the visual and analytical strategies that are used by middle-school students to differentiate and construct similar figures.

New strategies for construction and differentiation were identified, and three overarching conclusions were drawn from the work. First, there is value in teaching students to reason both visually and analytically. What was observed in this study suggested that visual perception is not entirely guess-related or primitive. The consideration of visual perception as a powerful indicator and supportive extender of conceptual understanding in this area might be warranted. Second, distortion-detection is a skill that enabled students to reflect upon and evaluate the validity and accuracy of differentiation and construction strategies. Forexample, conflict between numeric strategies and a student's visual expectations provoked students to adopt a strategy of Betweening , which provided a powerful connection between visual judgment and numerical methods. Third, student strategies could be interpreted in light of hypothesized van Hiele sublevels in the context of similarity. When mapped onto proposed sublevels, the mapping was not one-to-one and some strategies spanned multiple levels.

Implications of these conclusions and the identified strategies are also discussed. The implications for curriculum design include the importance of including complex figures and various types of distortion when designing tasks to instruct and assess student conceptions of similarity. The implications for instruction include a recommendation for teachers to build on visual reasoning and distortion detection rather than replace them with analytic algorithms for proving two shapes are similar. Directions for possible research extensions are also discussed.

Access Setting

Dissertation-Open Access

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