Date of Award


Document Type


Degree Name

Master of Education (MEd)


College of Science and Mathematics


Mathematical Sciences

Thesis Sponsor/Dissertation Chair/Project Chair

Aihua Li

Committee Member

Andrew J. McDougall

Committee Member

Richard Wolfson


Research has suggested that the use of manipulatives may enhance students’ grasp of mathematical concepts. Manipulatives may be concrete (physical) objects or virtual computer-based images of objects similar to their three-dimensional counterparts. Studies with elementary and middle-school students generally have found virtual manipulatives to outperform concrete manipulatives in enhancing students’ conceptual understanding of mathematics. However, very little research on manipulative use has been conducted with secondary school students. Furthermore, few studies have investigated instruction using virtual or concrete manipulatives across more than one content area.

This study assessed the impact of virtual and concrete manipulatives in a group of 164 tenth grade math students across four different content areas (number sense, geometry, algebra, and discrete math). Students were enrolled in the Geometry lab (Geolab) curriculum, which is designed to reinforce mastery of core content standards and to prepare students for standardized tests. A crossover design was employed whereby seven classes of Geolab students were randomized to receive instruction in the first content area using only concrete manipulatives while seven other classes were taught the same topics using only virtual manipulatives. Each week a new content area was presented, and the type of manipulatives alternated between the classes each week. The study duration was four weeks.

Pre and post-tests were conducted for each content area. Difference scores between post-test and pre-test in each area served as the primary measures of achievement. Dependent variables were either the unit difference scores or subset scores. Manipulative subset scores were created by combining the unit difference scores across weeks where the same type of manipulative was used. Repeated-measures Analysis of Variance (ANOVA) was employed to determine any differences according to the type of manipulatives or the content area. Students’ sex and ethnicity were also included in the analyses as control variables.

Overall, student improvement was highest for algebra followed by number sense, geometry, data analysis. However, an interaction was observed between group and unit score. An attitudinal survey was also administered to students upon completion of the study to determine their opinions on the use of manipulatives. Both groups appeared to exhibit a slight preference for using virtual manipulatives, and stated that they learned better using such methods. This is despite the analysis findings showing a slight superiority for concrete manipulatives in terms of achievement scores.

Both types of manipulatives assisted instruction more so for the unit content areas of number sense and geometry, than for the areas of algebra and data analysis. Concrete manipulatives appeared to slightly outperform virtual manipulatives, although this difference emerged significantly only for the second subset (algebra & data analysis). This finding suggests that concrete manipulatives may outperform virtual ones only when the material is more difficult for students. Survey results of students’ preferences were not related to achievement scores.

This research was IRJB approved.

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