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An important assumption in the implementation of the National Council of Teachers of Mathematics (NCTM) Standards 2000 is that the mathematics classroom is a place where students are actively involved in the learning process. One way to foster such a learning environment is by using manipulatives. By their nature, manipulatives make the learning of mathematics a discovery-based activity. As computer use increases in the classroom, virtual manipulatives will become more important as instructional tools. Virtual manipulatives offer advantages over their traditional versions. Most important is their dynamic nature. Their dynamic capabilities provide two main benefits. First, the number of potential problems for students to explore is increased dramatically. Students can use this capability to explore an idea from many viewpoints using different problems. Second, problem states are updated instantly and results are displayed. That is, the user can see immediately the effect of changes in problem parameters. Both of these benefits promote student engagement in the learning of mathematics. This thesis consists of a collection of such virtual manipulatives, in the form of Java applets, along with a description of their use.

The study investigated the effects of two generative learning activities on students’ academic achievement of the part-whole representation of rational numbers while using virtual manipulatives. Third-grade students were divided randomly in two groups to evaluate the effects of two generative learning activities: answering-questions and generating-examples while using two virtual manipulatives related to part-whole representation of rational numbers. The study employed an experimental design with pre- and post-tests. A 2x2 mixed analysis of variance (ANOVA) was used to determine any significant interaction between the two groups (answering questions and generating-examples) and between two tests (pre-test and immediate post-test). In addition, a 2x3 mixed analysis of variance (ANOVA) and a Bonferroni post-hoc analysis were used to determine the effects of the generative strategies on fostering comprehension, and to determine any significant differences between the two groups (answering-questions and generating-examples) and among the three tests (pre-test, immediate post-test, and delayed posttest). Results showed that an answering-questions strategy had a significantly greater effect than a generating-examples strategy on an immediate comprehension posttest. In addition, no significant interaction was found between the generative strategies on a delayed comprehension tests. However a difference score analysis between the immediate posttest scores and the delayed posttest scores revealed a significant difference between the answering-questions and the generating-examples groups suggesting that students who used generating-examples strategy tended to remember relatively more information than students who used the answering-questions strategy. The findings are discussed in the context of the related literature and directions for future research are suggested. / Ph. D.

Educational games have been demonstrated to increase engagement and engagement has been demonstrated to increase achievement. Therefore, the researcher attempted to investigate how to better measure engagement and refine the measurement of engagement in this study. To frame the engagement, three domains of engagement – behavioral, cognitive, and emotional– are analyzed in detail to be able to examine the qualities of each type. Moreover, three game attributes –clear goals, immediate feedback, and balance between challenges and skills- are presented and discussed as fundamental features of virtual manipulatives and educational games used in this study to make an impact on students’ engagement. To measure effects of educational games and virtual manipulatives on three domains of engagement, the researcher designed an engagement survey that examines each domain separately with their sub-domains. The Cronbach’s alphas for engagement pre-test and post-test were found .89 and .91 respectively. In this pre-test and post-test quasi-experimental design, four fifth-grade classrooms (N=86) from four schools in southwest Virginia were assigned as three experimental groups and one control group. In the first experimental group, participants played an educational game called Candy Factory and in the second experimental group, the students played another educational game called Pearl Diver on iPod Touch for eight days consecutively, for 20 minutes each. In the third experimental group, participants performed activities with virtual manipulatives, whereas in the control group, participants did paper-and-pencil iii drills for the same duration. All of the groups studied on the same topic, fractions. According to the results of ANCOVA, experimental group students’ engagement scores were found significantly higher than control group students’, F(1,80)=11.568, p=.001. When three domains of engagement were analyzed, significant differences were found among all three domains between experimental and control groups. When the researcher conducted separate analysis for educational games group and virtual manipulatives group, students who played educational games were also found significantly different than control group students in terms of all three domains of engagement and general engagement, F(1, 58)= 8.883, p=.004. In addition to this, students who did activities with virtual manipulatives showed significantly higher engagement than students who did paper-and-pencil drills in control group, F(1, 46)= 7.967, p=.007. Statistical difference was found in emotional and cognitive engagement while the results showed no significant difference in behavioral engagement between virtual manipulatives and control group students. Therefore, the three game attributes were considered as the main determining factors to engage students more to the content. / Ph. D.

The purpose of this embedded quasi-experimental mixed methods research was to use solving simple linear equations as the lens for looking at the effectiveness of concrete and virtual manipulatives as compared to a control group using learning methods without manipulatives. Further, the researcher wanted to investigate unique benefits and drawbacks associated with each manipulative. Qualitative research methods such as observation, teacher interviews, and student focus group interviews were employed. Quantitative data analysis techniques were used to analyze pretest and posttest data of middle school students (n=76). ANCOVA, analysis of covariance, uncovered statistically significant differences in favor of the control group. Differences in posttest scores, triangulated with qualitative data, suggested that concrete and virtual manipulatives require more classroom time because of administrative issues and because of time needed to learn how to operate the manipulative in addition to necessary time to learn mathematics content. Teachers must allow students enough time to develop conceptual understanding linking the manipulatives to the mathematics represented. Additionally, a discussion of unique benefits and drawbacks of each manipulative sheds light on the use of manipulatives in middle school mathematics.

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Understanding the relationship between fractions and decimals is an important step in developing an overall understanding of rational numbers. Research has demonstrated the feasibility of technology in the form of virtual manipulatives for facilitating students’ meaningful understanding of rational number concepts. This exploratory dissertation study was conducted for the two closely related purposes: first, to investigate a sample of fifth-grade students’ reasoning regarding the relationship between fractions and decimals for fractions with terminating decimal representations while using virtual manipulative incorporating parallel number lines; second, to investigate the affordances of the virtual manipulatives for supporting the students’ reasoning about the decimal-fraction relationship. The study employed qualitative methods in which the researcher collected and analyzed data from fifth-grade students’ verbal explanations, hand gestures, and mouse cursor motions. During the course of the study, four fifth-grade students participated in an initial clinical interview, five task-based clinical interviews while using the number line-based virtual manipulatives, and a final clinical interview. The researcher coded the data into categories that indicated the students’ synthetic models, their strategies for converting between fractions and decimals, and evidence of students’ accessing the affordances of the virtual manipulatives (e.g., students’ hand gestures, mouse cursor motions, and verbal explanations). The study yielded results regarding the students’ conceptions of the decimal-fraction relationship. The students’ synthetic models primarily showed their recognition of the relationship between the unit fraction 1/8 and its decimal 0.125. Additionally, the students used a diversity of strategies for converting between fractions and decimals. Moreover, results indicate that the pattern of strategies students used for conversions of decimals to fractions was different from the pattern of strategies students used for conversions of fractions to decimals. The study also yielded results for the affordances of the virtual manipulatives for supporting the students’ reasoning regarding the decimal-fraction relationship. The analysis of students’ hand gestures, mouse cursor motions, and verbal explanations revealed the affordances of alignment and partition of the virtual manipulatives for supporting the students’ reasoning about the decimal-fraction relationship. Additionally, the results indicate that the students drew on the affordances of alignment and partition more frequently during decimal to fraction conversions than during fraction to decimal conversions.

Virtual manipulatives are employed by both preservice and inservice teachers to enhance the instructional effectiveness of physical manipulatives and related tools by addressing limitations of access, cost, and adaptability. While research into the use of emerging technologies continues, there are several variables to consider when measuring the effects of virtual manipulative use. Research design, sampling characteristics, and the type of manipulative used may influence achievement. Variables that may influence the effectiveness of virtual manipulatives include: previous experience with computers, grade level, mathematical topic, treatment length, student attitudes toward mathematics, and computer-to-student ratio.

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