The Effect of Visually Enhanced Instructional Units on High School Calculus Students' Visualization Ability and Their Understanding of the Limit Concept

Date of Award


Document Type


Degree Name

Doctor of Education (EdD)


College of Science and Mathematics


Mathematical Sciences

Thesis Sponsor/Dissertation Chair/Project Chair

Kenneth Wolff

Committee Member

Evan Maletsky

Committee Member

Mark Weinstein

Committee Member

Mika Munakata


Calculus--Study and teaching, Calculus--Study and teaching (Secondary)--New Jersey, Mathematics--Study and teaching (Secondary)--New Jersey, Visualization


The number of high school students studying introductory calculus is increasing (Bressoud, 2005), so is the need to prepare them adequately for future study of mathematics and the sciences. The limit concept plays a pivotal role in the study of introductory calculus and visualization is an important tool for exploring this concept. Research has identified difficulties related to both visualization and limits. Students have misconceptions associated with limits, and they are reluctant to use visualization as a problem-solving tool. This study sought to address these difficulties with a group of high school calculus students.

A quasi-experimental, pretest-posttest, control group design was used for this study. Instructional units developed by the researcher were used to supplement three weeks of classroom instruction in the experimental group. The units were explorations of limits using geometric figures and functions; they relied on visualization as a primary problem-solving tool. The effectiveness of these units on students' understanding of the limit concept and their visualization ability was analyzed quantitatively and qualitatively. Pretests and Posttests were used to compare visualization ability before and after the three-week period for the control and experimental groups. Students' performance on a calculus assessment on limits was used to compare the two groups. Finally, students from the experimental group were interviewed for a qualitative assessment of their understanding of limits and their use of visualization.

The experimental group in this study performed significantly better than the control group on questions about limits that involved visualization. Although the two groups showed no growth in their visualization ability in the general cognitive context, the experimental group demonstrated a more significant gain than the control group in visualization ability in the general mathematics content-based portion of the posttest.

The qualitative analysis revealed that students were reluctant to use visualization in general mathematics questions during the interview, even when asked to use a figure. However, the same students approached the limit questions using graphs and visualization, even when graphs were not given.

These observations suggest that it is possible to enhance students' use of visualization in specific content areas through instructional strategies that incorporate visual representation.


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