Project Activities
The researchers followed an iterative development process using experiments to design a software package, From Here to There! (FH2T), intended to help students acquire algebraic notations. Using lab-based studies, the researchers first explore 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. They used these results to develop prototypes of the system and to test the prototypes in various stages of completeness.
Structured Abstract
Setting
The project took place in Indiana and Virginia.
Sample
The sample population included 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.
The intervention developed was From Here 2 There (FH2T), a self-paced interactive application that introduces students to mathematical content through discovery-based puzzles. Rather than simply applying procedures by rewriting different expressions, this technology allows students to physically and dynamically interact with algebraic expression elements, providing a potentially powerful source of perceptual-motor experiences.
Research design and methods
The research team followed an iterative process in which they generated materials and software and test prototypes of the system in various stages of completeness. The data collected in each round of collection was used for design improvements. First, the researchers tested the prototypes for usability and technical soundness with college undergraduates in remedial mathematics courses. Second, they developed 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, they conducted 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 studied the role of perception-action grounding in acquiring symbolic reasoning skills. They then systematically tested alternative versions of the FH2T system and explored the impact of several factors (namely, perceptual scaffolding, visual hints to structure, and sequencing instruction to introduce or remove perceptual supports) on student math learning. During the final stage in development, the researchers pilot tested FH2T in a small-town high school to assess its feasibility and its promise of improving mathematical reasoning.
Key measures
Key measures included 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 were analyzed using an analysis of variance to statistically evaluate improvement.
Key outcomes
- A study with elementary students found that they showed larger learning gains within FH2T when using the gamified version as compared to the nongamified version, that completing more items within FH2T led to better performance on a post-test, and that students with initially lower knowledge and performance improved more significantly when they completed more items in FH2T and spent more time engaging with those items (Hulse et al., 2019).
- The researchers found evidence that the ordering of categories (e.g., one category per block or categories mixed within a block) that the learners practice with leads to different types of encoding that may lead students to focus on different properties of math problems. (Carvalho & Goldstone, 2017).
People and institutions involved
IES program contact(s)
Project contributors
Products and publications
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ERIC Citations: Find available citations in ERIC for this award here
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Braithwaite, D. W., Goldstone, R. L., van der Maas, H. L., & Landy, D. H. (2016). Non-formal mechanisms in mathematical cognitive development: The case of arithmetic. Cognition, 149, 40-55.
Carvalho, P. F., & Goldstone, R. L. (2017). The sequence of study changes what information is attended to, encoded, and remembered during category learning. Journal of Experimental Psychology: Learning, Memory, and Cognition, 43(11), 1699.
Goldstone, R. L., de Leeuw, J. R., & Landy, D. H. (2015). Fitting perception in and to cognition. Cognition, 135, 24-29.
Goldstone, R. L., Weitnauer, E., Ottmar, E. R., Marghetis, T., & Landy, D. H. (2016). Modeling Mathematical Reasoning as Trained Perception-Action Procedures. Design recommendations for intelligent tutoring systems, 4, 213-223.
Goldstone, R.L., Landy, D., and Brunel, L. (2011). Improving Perception to Make Distant Connections Closer. Frontiers in Perception Science, 2(385): 1–10.
Guay, B., Chandler, C., Erkulwater, J., & Landy, D. (2016). Testing the effectiveness of a number-based classroom exercise. PS: Political Science & Politics, 49(2), 327-332.
Hulse, T., Daigle, M., Manzo, D., Braith, L., Harrison, A., & Ottmar, E. (2019). From here to there! Elementary: a game-based approach to developing number sense and early algebraic understanding. Educational Technology Research and Development, 67, 423-441.
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. (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., 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., Charlesworth, A., & Ottmar, E. (2017). Categories of large numbers in line estimation. Cognitive science, 41(2), 326-353.
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.
Landy, D., Silbert, N. and Goldin, A. (2013). Estimating Large Numbers. Cognitive Science, 37(5): 775–799.
Ottmar, E., & Landy, D. (2017). Concreteness fading of algebraic instruction: Effects on learning. Journal of the Learning Sciences, 26(1), 51-78.
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.
Ottmar, E., Landy, D., Weitnauer, E., & Goldstone, R. (2015). Graspable mathematics: Using perceptual learning technology to discover algebraic notation. In Integrating touch-enabled and mobile devices into contemporary mathematics education (pp. 24-48). IGI Global.
Ottmar, E.R., Landy, D., Goldstone, R., & Weitnauer, E. (2015). Getting From Here to There!: Testing the Effectiveness of an Interactive Mathematics Intervention Embedding Perceptual Learning. Proceedings of the 37th Annual Conference of the Cognitive Science Society. Pasadena, California: Cognitive Science Society.
Sawrey, K., Chan, J. Y. C., Ottmar, E., & Hulse, T. (2019). Experiencing Equivalence with Graspable Math: Results from a Middle-School Study. In Proceedings of the 41st Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA).
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.
Weitnauer, E., Landy, D., Goldstone, R. L., & Ritter, H. (2015). A Computational Model for Learning Structured Concepts from Physical Scenes. Proceedings of the 37th Annual Conference of the Cognitive Science Society. Pasadena, California: Cognitive Science Society.
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