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IES Grant

Title: The Role of External Representations in Learning and Transfer of Mathematical Knowledge
Center: NCER Year: 2007
Principal Investigator: Sloutsky, Vladimir Awardee: Ohio State University
Program: Cognition and Student Learning      [Program Details]
Award Period: 4 years Award Amount: $1,760,669
Goal: Development and Innovation Award Number: R305B070407
Description:

Purpose: Abstract mathematical concepts pose challenges for learners, especially when learners are young. One way that classroom teachers address this challenge is to introduce abstract concepts through concrete representations or instantiations. For example, manipulatives such as differently colored blocks representing ones, tens, and hundreds could aid the learning of arithmetic. However, extraneous information in concrete representations may distract the learner from the relevant mathematical structure and, as a result, hinder transfer. At the same time, generic representations, such as traditional mathematical symbols, can be learned by children and can, in turn, allow for transfer. The purpose of this project is to study how learning mathematical concepts using concrete learning aids affects both learning and transfer of the learned knowledge to new situations. This type of research has the potential to support mathematics learning, and may have important implications for the teaching of mathematics.

Project Activities: The researchers will conduct a series of nine experimental studies with kindergarten through sixth grade students examining learning and transfer of fractions using instructional methods that vary in their degree of concreteness. The researchers predict that greater concreteness will hinder transfer of skills across problems as compared to more abstract, generic instantiations. The researchers will also examine why concrete learning aids hinder transfer and how these negative effects can be alleviated.

Products: Products from this study will include a set of principles describing how the use of concrete learning aids can support initial mastery of a mathematical concept and subsequent transfer of this knowledge to new situations. Peer reviewed publications will also be produced.

Structured Abstract

Setting: The research will take place in 13 elementary and middle schools in Ohio.

Sample: Participants will include kindergarten through sixth grade students. The sample size will be 15–20 participants per experimental condition.

Intervention: The researchers will develop learning situations that vary along two dimensions of concreteness: perceptual (number of dimensions of variations in the stimuli constituting an instantiation) and conceptual (the amount of prior knowledge associated with a particular instantiation). Researchers will teach children about fractions and examine effects of concreteness on transfer of this knowledge to novel instantiations and novel tasks involving fractions. They will also examine the durability of learning by giving a delayed test after several weeks. Next, they will propose ways of optimizing both learning and transfer and testing these ways of learning empirically. The researchers will then examine the generalizability of these findings for other mathematical concepts in possible follow-up experiments. Finally, the researchers will make instructional recommendations and design ways of testing effects of these recommendations on learning and transfer of mathematical knowledge.

Research Design and Methods: The researchers plan to complete a series of nine experiments, all of which follow a similar pattern. Factors to be manipulated include a set of five instantiations: the standard number line (one-dimensional diagram with minimal amount of extraneous information); discrete-generic (represent fractions as a subset of a set of shaded shapes), continuous-generic (represent fractions as proportions of geometric shapes); and discrete-concrete and continuous-concrete, pictures of cupcakes and pizzas which have additional dimensions of variation such as colors of the inside surface. The effects of concreteness on the following will be examined: basic fractional knowledge (e.g., proportion), the acquisition of standard mathematical notation, learning fractions as real numbers, the acquisition of higher order fractional knowledge (e.g., operations with fractions), and solving word problems. The hypothesis is that learning will be more successful with continuous than with discrete instantiations; generic instantiations will result in greater transfer than concrete ones; and the number line will elicit the best transfer performance for tasks involving higher order operations. For example, in one experiment, training will consist of teaching participants how to label proportions with standard notation. Following training, participants will be presented with 12 learning problems. Then, participants will receive 24 learning problems with multiple-choice answers. Questions will involve either discrete entities such as trees or toys, or continuous entities such as pies or liquid. In addition the questions will be of one or two types: fraction to whole number, or whole number to fraction. There will be six questions for each of the four resulting categories. While the test will be presented via computer, participants may use paper and pencil instead.

Control Condition: Students in the control condition will be presented with standard classroom instruction in fractions.

Key Measures: Researchers will use multiple choice testing to measure learning and transfer in basic fractional knowledge (e.g., proportion), the acquisition of standard mathematical notation, learning fractions as real numbers, the acquisition of higher order fractional knowledge (e.g., operations with fractions), and solving word problems.

Data Analytic Strategy: Research methods will differ slightly across experiments. Overall, researchers will conduct sample t-tests and analysis of variance on composite learning and transfer scores, and differences will be examined across conditions. Follow-up analyses with multiple factors will also be included in the design. Post-hoc Tukey comparisons will examine whether there are any differences across conditions and Bonferroni adjusted scores will determine if conditions differ from chance. Time will be equated or accounted for across conditions. Analysis of Covariance will be used with learning and transfer scores as covariates.

Publications

Book chapter

Kaminski, J.A., and Sloutsky, V.M. (2011). Representation and Transfer of Abstract Mathematical Concepts. In V. F. Reyna (Ed.), The Adolescent Brain: Learning, Reasoning, and Decision Making (pp. 67–93). Washington, DC: APA.

Sloutsky, V.M., and Fisher, A.V. (2011). The Development of Categorization. In B.H. Ross (Ed.), The Psychology of Learning and Motivation (pp. 142–166). New York: Academic Press.

Journal article, monograph, or newsletter

Hupp, J., Sloutsky, V.M., and Culicover, P.W. (2009). Evidence for a Domain-General Mechanism Underlying the Suffixation Preference in Language. Language and Cognitive Processes, 24(6): 876–909.

Kaminski, J.A., and Sloutsky, V.M. (2013). Extraneous Perceptual Information Can Interfere With Children's Acquisition of Mathematical Knowledge. Journal of Educational Psychology, 105(2): 351–363.

Kaminski, J.A., Sloutsky, V.M., and Heckler, A.F. (2008). Response to J. Mourrat, L. Cultrona, and S. Reed. Science, 322: 1633.

Kaminski, J.A., Sloutsky, V.M., and Heckler, A.F. (2008). Response to McCallum. Science Online, 320: 454–455.

Kaminski, J.A., Sloutsky, V.M., and Heckler, A.F. (2009). Concrete Instantiations of Mathematics: A Double-Edged Sword. Journal for Research in Mathematics Education, 40(2): 90–93.

Kaminski, J.A., Sloutsky, V.M., and Heckler, A.F. (2009). Transfer of Mathematical Knowledge: The Portability of Generic Instantiations. Child Development Perspectives, 3(3): 151–155.

Kaminski, J.A., Sloutsky, V.M., and Heckler, A.F. (2013). The Cost of Concreteness: The Effect of Nonessential Information on Analogical Transfer. Journal of Experimental Psychology: Applied, 19(1): 14–29.

Kaminski, K.A., Sloutsky, V.M., and Heckler, A.F. (2008). The Advantage of Abstract Examples in Learning Math. Science, 320(5875): 454–455.

Robinson, C. W., and Sloutsky, V.M. (2010). Effects of Multimodal Presentation and Stimulus Familiarity on Auditory and Visual Processing. Journal of Experimental Child Psychology, 107(3): 351–358.

Robinson, C.W., and Sloutsky, V.M. (2008). Effects of Auditory Input in Individuation Tasks. Developmental Science, 11(6): 86–881.

Robinson, C.W., and Sloutsky, V.M. (2010). Development of Cross-Modal Processing. Wiley Interdisciplinary Reviews: Cognitive Science, 1(1): 135–141.

Robinson, C.W., and Sloutsky, V.M. (2013). When Audition Dominates Vision: Evidence From Cross-Modal Statistical Learning. Experimental Psychology, 60(2): 113–121.

Sloutsky, V.M. (2008). Analogy is to Priming as Relations are to Transformations. Behavioral and Brain Sciences, 31(4): 396–397.

Sloutsky, V.M. (2009). Theories About "Theories": Where is the Explanation? Comment on Waxman and Gelman. Trends in Cognitive Sciences, 13(8): 331–332.

Sloutsky, V.M. (2010). From Perceptual Categories to Concepts: What Develops?. Cognitive Science, 34(7): 1244–1286.

Sloutsky, V.M. (2010). Mechanisms of Cognitive Development: Domain-General Learning or Domain-Specific Constraints?. Cognitive Science, 34(7): 1125–1130.

Sloutsky, V.M., and Fisher, A.V. (2011). Linguistic Labels: Conceptual Markers or Object Features?. Journal of Experimental Child Psychology, 111(1): 65–86.

Proceeding

Best, C.A., Robinson, C.W., and Sloutsky, V.M. (2010). The Effect Of Labels On Visual Attention: An Eye Tracking Study. In S. Ohlsson and R. Catrambone (Eds.), Proceedings Of The 32nd Annual Conference Of The Cognitive Science Society (pp. 1846–1851). Mahwah, NJ: Erlbaum.

Best, C.A., Robinson, C.W., and Sloutsky, V.M. (2011). The Effect of Labels on Categorization: Is Attention to Relevant Features a Good Index of Infants' Category Learning?. In Proceedings of the 33rd Annual Conference of the Cognitive Science Society (pp. 2751–2755). Mahwah, NJ: Erlbaum.

Best, C.A., Robinson, C.W., and Sloutsky, V.M. (2011). The Effect of Labels on Children's Category Learning. In Proceedings of the 33rd Annual Conference of the Cognitive Science Society (pp. 3332–3336). Mahwah, NJ: Erlbaum.

Deng. W., and Sloutsky, V.M. (2010). The Role of Linguistic Labels in Categorization. In S. Ohlsson and R. Catrambone (Eds.), Proceedings of the 32nd Annual Conference of the Cognitive Science Society (pp. 230–235). Mahwah, NJ: Erlbaum.

Kaminski, J.A., and Sloutsky, V.M. (2012). Children's Acquisition of Fraction Knowledge From Concrete Versus Generic Instantiations. In Proceedings of the 34th Annual Cognitive Science Society (pp. 1750–1755). Austin, TX: Cognitive Science Society.

Kaminski, J.A., and Sloutsky, V.M. (2009). The Effect of Concreteness on Children's Ability to Detect Common Proportion. In N. Taatgen, H. Van Rijn, L. Schomaker, and J. Nerbonne (Eds.), Proceedings of the 31st Annual Cognitive Science Society (pp. 335–339). Amsterdam: Cognitive Science Society.

Kaminski, J.A., and Sloutsky, V.M. (2010). Concreteness and Relational Matching in Preschoolers. In S. Ohlsson and R. Catrambone (Eds.), Proceedings of the 32nd Annual Meeting of the Cognitive Science Society (pp. 1828–1833). Portland, OR: Cognitive Science Society.

Osth, A., Dennis, S., and Sloutsky, V.M. (2010). Context and Category Information in Children and Adults. In S. Ohlsson and R. Catrambone (Eds.),  Proceedings of the 32nd Annual Conference of the Cognitive Science Society (pp. 842–847). Mahwah, NJ: Erlbaum.

Robinson, C.W., and Sloutsky, V.M. (2010). Attention and Cross-Modal Processing: Evidence From Heart Rate Analyses. In S. Ohlsson and R. Catrambone (Eds.), Proceedings of the 32nd Annual Conference of the Cognitive Science Society (pp. 2639–2643). Mahwah, NJ: Erlbaum.

Yao, X., and Sloutsky, V.M. (2010). Selective Attention and Development of Categorization: An Eye Tracking Study. In S. Ohlsson and R. Catrambone (Eds.),  Proceedings of the 32nd Annual Conference of the Cognitive Science Society (pp. 1980–1985). Mahwah, NJ: Erlbaum.

Yim, H., Best, C.B., and Sloutsky, V.M. (2011). Cost of Attention as an Indicator of Category Learning. In Proceedings of the 33rd Annual Conference of the Cognitive Science Society (pp. 1763–1769). Mahwah, NJ: Erlbaum.


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