Mathematical objects are artifacts of abstract thinking achieved by both conceptual and procedural knowledge. Mathematical thinking is usually communicated and constructed with external representations such as texts, symbols, or diagrams. Since it is difficult to demonstrate the dynamic nature of mathematical objects using static representations, teachers and instructional designers often apply dynamic visualization to present abstract or complex content in digital learning materials. While this illustrates abstract concepts and procedures with mathematical discipline, the accompanying transient effects may impede perceptual and cognitive processing over the short term, such that the intended learning effect may not be achieved. These transient effects are more pronounced for mathematical content with a high intrinsic cognitive load which require the integration of concepts and procedures. Relevant studies have suggested that the learning effect of dynamic visualization can be enhanced using physical experiences such as moving the body while learning. This includes demonstrating learning content using embodied simulation, which means combining gestures and actions in the process of dynamic visualization. This approach is termed embodied dynamic visualization.

To facilitate the connection between physical gestures and procedural knowledge in the creation of mathematical meaning, this study adopted conceptual metaphor theory to develop related research instruments, including learning materials for embodied simulation. In addition to determining whether students can successfully integrate mathematical concepts and the processes of calculation in the cognitive process of procept formation, we investigated affective factors. Affective factors which have been shown to affect learning outcomes include aspects of cognitive load, mental effort, self-efficacy, learning strategies, and degree of initiative. Therefore, this study applied a quasi-experimental design to explore the effects of different demonstration methods (i.e., embodied simulations, instructional animations, and static illustrations) on the cognition and emotions of seventh-grade students learning algebraic manipulation.

Criteria for inclusion in the experiment were participation in the whole process of the instructional experiment and a score of less 85% in the pre-test (to ensure the subjects had not already mastered the basic concepts and problem-solving skills relevant to algebraic manipulation). Based on these criteria, we recruited 96 seventh grade students, who were randomly provided with embodied simulations, instructional animations, or static illustrations. They were expected to engage in self-regulated learning of content related to algebraic manipulation. Data on both cognitive and affective aspects was collected and analyzed. For the former, students’ comprehension of algebraic manipulation was evaluated using a pre-test and a post-test. For the latter, students completed a self-reported questionnaire on their learning perceptions. Cronbach’s α for these two aspects were .743 and .887, respectively, thereby confirming reliability.

Our results were as follows: (1) Students’ overall learning outcomes can be significantly enhanced by including embodied simulation, instructional animation, or static illustration. All three types of demonstration helped students solve problems related to basic algebraic manipulation. They also increased the positive learning effects on misconceptions and near-transfer problems as well as far-transfer problems. However, static illustrations did not have a significant effect on the learning performances of the seventh-grade students in solving far-transfer problems. This type of demonstration illustrates the algorithm and structure of algebraic manipulation; however, the procedure and approach to calculation are not shown. Therefore, students must depend on their own imagination to determine how these algebraic expressions are calculated based on the result alone. This places a high cognitive demand on students and may thus hinder the learning effects on far-transfer and other applications relevant to algebraic manipulation. (2) For high-performing students, exposure to embodied simulation enabled the students to perform significantly better on basic questions related to algebraic manipulation than did exposure to instructional animation. Moreover, students exposed to embodied simulation performed significantly better on far-transfer problems than did students exposed to static illustration. This indicates the value of using different gestures to simulate the operations involved in the calculation process, such as distribution expansion or the merging of similar items. This enabled high-performing students to acquire basic operating rules and calculation skills more effectively. In addition, embodying the process of the calculation exerted a more positive learning effect for high-performing students solving far-transfer problems than did exposure to static illustrations. This is because embodied simulation not only offers a form of dynamic visualization, but also connects the elements of the calculation based on operation processes. In addition, the simulated gestures help students understand the meaning of the operations, thereby generating meaningful dynamic mental images. (3) For low-performing students, exposure to either embodied simulation or instructional animation resulted in significantly better performance in the solution of basic questions related to algebraic manipulation than did exposure to static illustration. This means that dynamic visualization methods such as embodied simulation or instructional animation help improve understanding of procedural knowledge of algebraic manipulation for low-performing students. However, a lack of sufficient prior knowledge or proficiency at solving basic problems hampered the learning effects for low-performing students in the topic of far-transfer problems. For these, the results of exposure to embodied simulation or static instruction did not differ from those of exposure to static illustration. (4) We found significant interaction effects in terms of students’ learning perceptions among the different forms of demonstration and high and low performance groups. That is, for different forms of demonstration, students’ learning perceptions differ according to their level of mathematics learning achievement. To be specific, for high-performing students, exposure to instructional animation enable the students to perceive significantly better on the degree of initiative than did exposure to embodied simulation. However, for low-performing students, exposure to instructional animation showed significantly better on the self-efficacy and learning strategies than did exposure to static illustration.

Based on these results, we make the following recommendations for future studies into the use of dynamic visualization in mathematical instruction: (1) The current study focused on embodied simulation, instructional animation, and static illustration. It would be worth exploring whether other forms of dynamic visualization (such as an instructional video with a human instructor) could help students in the learning of algebraic manipulation, enabling them to successfully integrate the concept and process of algebraic manipulation. (2) In the current study, students studied algebraic manipulation using non-interactive demonstration. In future studies, it would be worth investigating the effects of interactive gestures on students’ learning performances and perceptions. (3) As the complexity of algebraic manipulation problems may have been too low for high-performing students, it would be worth exploring the effects of embodied simulation or interactive gestures on other algebraic topics of higher cognitive complexity, such as solving square-root equations or completing the square, as well as geometric calculation and geometric reasoning.