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IES Grant

Title: Learning the Visual Structure of Algebra Through Dynamic Interactions with Notation
Center: NCER Year: 2011
Principal Investigator: Landy, David Awardee: University of Richmond
Program: Cognition and Student Learning      [Program Details]
Award Period: 3 years Award Amount: $1,117,614
Goal: Development and Innovation Award Number: R305A110060

Co-Principal Investigator: Robert Goldstone, Indiana University

Purpose: Algebra is a foundational domain for understanding advanced mathematics. However, many middle and high school students fail to understand basic algebraic concepts. Much of the difficulty stems from failures to achieve algebraic literacy: a robust, intuitive understanding of how to create, transform, and interpret statements expressed in standard algebraic notation. The goal of this project is to develop and explore the feasibility of an intervention that uses a set of verbal, physical, and computer-based visualizations to enhance middle, high school, and remedial college students' understanding of algebraic notations. The goal is to better prepare these students for future mathematics.

Project Activities: The researchers will follow an iterative development process using experiments to drive the design of a software package, Pushing Symbols, intended to help students acquire algebraic notations. The experiments will study the role of perception-action grounding (using the perceptual properties, such as the proximity of numbers, to help students build abstract representations of relations in equations) in acquiring symbolic reasoning skills. Researchers will use these results to develop prototypes of the system and to test the prototypes in various stages of completeness.

Products: Products will include a set of in-class discussions, paper-and-pencil activities, and a computer-based visualization tool. Publications of findings will also be produced.

Structured Abstract

Setting: The project will take place in Indiana and Virginia.

Population: The sample population will include about 520 students taking 9th or 10th grade algebra classes at a rural high school, as well as 300 college undergraduates taking remedial algebra courses.

Intervention: The intervention being developed is Pushing Symbols, a software tool that allows students to interact physically and dynamically with expression elements (e.g., by dragging a number on an interactive screen to the other side of an equation, causing the addition operator to become a subtraction operator) to provide a potentially powerful source of perceptual-motor experiences. This will then lead to rapid acquisition of object-centered understandings of algebra. Rather than simply rewriting algebraic expressions in a step-wise fashion, in Pushing Symbols learners can directly interact with dynamic expression objects. In addition to the computer-based visualization tool, the intervention will also include a set of in-class discussion guides and paper-and-pencil activities.

Research Design and Methods: The research team will follow an iterative process in which they generate materials and software and test prototypes of the system in various stages of completeness. The data collected in each round of collection will be used for design improvements. First, prototypes will be tested for usability and technical soundness with college undergraduates at the University of Richmond in remedial mathematics courses. Second, the team will build prototypes of the intervention components using a variety of different, cognitively plausible models of learning, such as perception or visually based learning. To determine which models are most valid and useful, the researchers will conduct a series of experimental studies with both college and high school students, focusing on the kinds of rules and transformations that are most problematic for students. In these experiments, the researchers will study the role of perception-action grounding in acquiring symbolic reasoning skills. Alternative versions of the Pushing Symbols system will explore the role of perceptual scaffolding, visual hints to structure, and sequencing instruction to introduce or remove perceptual supports. For example, perceptual scaffolding, such as the spacing between operators and numbers (e.g., 3+2 * 4 versus 3 + 2*4), may help students to perform operations in the correct order. These experiments will inform not only the development of Pushing Symbols but also other instructional systems that can strategically introduce or remove perceptual, conceptual, and interaction supports. During the final stage in development (Year 3), the software and supporting materials will be piloted in a small town high school in Indiana. This piloting will help assess the feasibility of implementing the system as well as the promise of the intervention's potential success.

Key Measures: Key measures include researcher-developed assessments of perceptual learning of algebraic structures, problem-solving ability (procedural fluency), and conceptual understanding.

Data Analytic Strategy: In the pilot study, the pre-test, post-test, and retention test will be analyzed using an analysis of variance to statistically evaluate improvement.


Journal article, monograph, or newsletter

Goldstone, R.L., Landy, D., and Brunel, L. (2011). Improving Perception to Make Distant Connections Closer. Frontiers in Perception Science, 2(385): 1–10.

Landy, D., Allen, C., and Zednik, C. (2014). A Perceptual Account of Symbolic Reasoning. Frontiers in Psychology, 5: 275.

Landy, D., Brookes, D., and Smout, R. (2014). Abstract Numeric Relations and the Visual Structure of Algebra. Journal of Experimental Psychology: Learning, Memory, and Cognition, 40(5): 1404–1418.

Landy, D., Silbert, N. and Goldin, A. (2013). Estimating Large Numbers. Cognitive Science, 37(5): 775–799.


Landy, D., Brookes, D., and Smout, R. (2012). Modeling Abstract Numeric Relations Using Concrete Notations. In Proceedings of the 33rd Annual Conference of the Cognitive Science Society (pp. 102–107). Boston: Cognitive Science Society.

Landy, D., Charlesworth, A., and Ottmar, E. (2014). Cutting in Line: Discontinuities in the use of Large Numbers in Adults. In Proceedings of the 36th Annual Conference of the Cognitive Science Society (pp. 815–820). Quebec: Cognitive Science Society.

Ottmar, E., Landy, D., and Goldstone, R.L. (2012). Teaching the Perceptual Structure of Algebraic Expressions: Preliminary Findings From the Pushing Symbols Intervention. In Proceedings of the 34th Annual Conference of the Cognitive Science Society (pp. 2156–2161). Austin, TX: Cognitive Science Society.

Sears, K., Landy, D., and Lesky, J. (2012). Interactions Between Actions and Apparent Distance. In Proceedings of the 34th Annual Conference of the Cognitive Science Society (pp. 2300–2305). Austin: Cognitive Science Society.