The Development of Seventh Graders' Conceptual Understanding of Geometry and Spatial Visualization Abilities Using Mathematical Representations with Dynamic Models

Date of Award


Document Type


Degree Name

Doctor of Education (EdD)


College of Science and Mathematics


Mathematical Sciences

Thesis Sponsor/Dissertation Chair/Project Chair

Evan Maletsky

Committee Member

Max Sobel

Committee Member

Tamara Lucas

Committee Member

James Fey


For many middle school students, representing and solving mathematical problems and internalizing their meanings can be a difficult task. The purpose of the present study was to explore the role of representations in mathematical learning and its relationship to middle school students' conceptual understanding of geometry and measurement. This exploration focused on two issues: (1) the impact of an instructional unit in which area and perimeter were treated as dynamic geometric concepts on the spatial visualization abilities and representation abilities of students, and (2) the thought processes that students used as they sought to build mathematical representations using dynamic geometric models. The study took place during the 2002–2003 school year in a small suburban middle school in Northeastern New Jersey. Two seventh grade teachers, with two enriched classes and four regular classes, participated in the study. Quantitative research was used to examine the impact of an originally designed experimental three-week geometry unit on the spatial visualization abilities and representation abilities of high, average, and low ability middle school students. The design was a quasi-experimental pretest-posttest, control-group design. Using an analysis of covariance (ANCOVA), there was a significant difference among population means at the 0.05 level on one of the three tests of spatial visualization abilities in favor of the experimental treatment. The results of the impact of the instructional unit on representation abilities revealed an increase in the number of external representations used and in the level of correctness in the experimental treatment groups. The qualitative design explored the thought processes of six randomly selected students through think-aloud protocols. All students engaged in the use of internal representations in the theoretical cognitive categories of imagistic, formal notation, verbal/syntactic, and strategic/heuristic processes; two other categories emerged: error reflection, a student's strong sense to check for a correspondence between the physical, external image and their own mental assessment of the situation, and affective systems of representation. The results have implications for curriculum design and practice using dynamic models in middle school mathematics in geometry and measurement.


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