Martha W. Alibali, Ph.D. - US grants

Mathematical reasoning requires understanding connections between different representations of mathematical information. The way mathematical representations are linked in the classroom may determine whether students come to understand important mathematical principles and procedures. Our past research showed that teachers use various forms of visual scaffolding to link different mathematical representations. The purpose of this project is to understand how variations in teachers' visual scaffolding affect students' learning. Our specific focus is on the nonverbal supports that teachers produce in instructional episodes that link related representations of mathematical information. In particular, we examine those nonverbal supports that serve to ground ideas in the physical environment or in familiar actions, experiences or representations. The research has three aims: (1) to investigate whether students' learning is facilitated if teachers ground the to-be-linked ideas with hand gestures (as opposed to using speech alone); (2) to examine whether certain types of nonverbal supports are especially beneficial for learning (specifically, redundant vs. complementary gestures, and pointing vs. representational gestures); and (3) to examine whether gestures offer a "special" way to visually scaffold ideas, in the sense that they are more effective at doing so than other, non-gestural methods of visual scaffolding. We will address these aims in experiments with middle school students learning about linear equations. The experiments will involve video lessons that vary the teachers' gestures or the medium used to highlight aspects of the linked representations (hand gestures or digital icons). We will assess students' conceptual and procedural knowledge of linear equations before and after the lessons, so that we can evaluate how variations in teachers' visual scaffolding affect students' learning. We will also conduct a pilot study to prepare us to extend this line of inquiry to college students learning about statistics. This pilot study will investigate how teachers link representations using speech and gesture in instruction about confidence intervals.

This work will contribute to our scientific understanding of learning and instruction from an embodied cognition perspective. By experimentally manipulating the ways in which relations between mathematical ideas are conveyed, and exploring the consequences for learning, we will gain a deeper understanding of the cognitive processes involved in acquiring mathematical understanding. This work will provide an empirical basis for recommendations about how teachers can use visual scaffolding effectively.

This project will develop and test the effectiveness of a semester-long conceptually-based instruction for promoting understanding of mathematical equivalence and associated gains in algebraic thinking. Participants in the research will be elementary- and middle- school students. Although many brief, single-session laboratory studies have suggested effective ways to promote equal sign knowledge in the short term, these studies have generally failed to produce practical guidelines for use by regular classroom teachers. This project will apply findings from laboratory studies over a longer time period with the goal of packaging a practical approach to developing students' knowledge of the equal sign for classroom teachers. The hypothesis is that improved equal sign knowledge will lead to improved access to algebra, an important pathway into higher mathematics and science that provides access to Science, Technology, Engineering, and Mathematics (STEM) fields that help power modern society.

This project will explore the effectiveness of spaced, conceptually-based instruction for promoting understanding of mathematical equivalence and associated gains in algebraic thinking. Multiple experimental studies have shown that brief, conceptually-based instructional interventions can lead to improvements in children's equal sign knowledge. The proposed research will test whether spacing such interventions over time can lead to more substantial and long-term gains in equal sign knowledge, and whether such knowledge, in turn, fosters algebraic reasoning. One component of the research will aim to optimize the conceptually based intervention. The other component will be a study that investigates the effects of the intervention over time, using a crossover design. The research will employ a measure of equal sign knowledge that is more sensitive than most commonly used measures, allowing for detection of relatively fine-grained gains in response to the instructional intervention. In total, the work will contribute to the field's understanding of how to improve equal sign knowledge and understanding of the causal impact of equal sign instruction on student competence in algebra.

This project is co-funded by the Discovery Research preK-12 program (DRK-12) that seeks to significantly enhance the learning and teaching of science, technology, engineering, and mathematics (STEM) by preK-12 students and teachers, through research and development of innovative resources, models, and tools.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.