Difference between revisions of "Sequencing learning with multiple representations of rational numbers (Aleven, Rummel, & Rau)"
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Revision as of 13:24, 16 February 2011
Learning with Multiple Reprsentations in a Complex, Realworld Domain: Intelligent Tutoring Systems for Fractions
Vincent Aleven, Nikol Rummel, and Martina Rau
Summary Table
PIs  Vincent Aleven & Nikol Rummel 
Other Contributers  Graduate Students: Martina Rau (CMU HCII) 
2008 study  N = 132 6thgrade students 
2009 study  N = 388 5th and 6thgrade students 
2010 study  N = 690 4th and 5thgrade students 
Study Start Date  September 1st, 2008 
Study End Date  August 31st, 2012 
Total Number of Students to date  N = 1210 
Total Participant Hours  ~6000 
Data available in DataShop  Dataset: Fraction Study Spring 2009 (log data only)Dataset: Mathtutor Fractions MERs Spring 2009 (revised)

Abstract
We investigate a key issue in coordinative learning, namely, how learning with multiple graphical representations should bue used to effectively support students’ conceptual understanding of fractions. In a previous experiment (Rau, Aleven, & Rummel, 2009), we demonstrated that students benefit from learning with multiple graphical representations when compared to a single graphical representation, provided that they were prompted to relate the graphical representations to the symbolic representation of fractions (e.g., 1/2). In two consecutive studies, we investigated how multiple representations should be sequenced. Prior research on contextual interference has demonstrated that interleaving different types of learning tasks can foster a deep understanding of the underlying concepts. Do the same advantages apply to interleaving representations? In future studies, we plan to investigate ways to explicitly support students in relating the different graphical representations to one another. We focus on fractions as a challenging topic area for students in which multiple representations are often used and likely to support robust learning. This research will contribute to the literature on early mathematics learning, learning with multiple representations, and learning with intelligent tutoring systems. It will also add to the portfolio of studies in the PSLC’s coordinative learning cluster.
Background & Significance
A quintessential form of coordinative learning occurs when learners work with multiple external representations (MERs) of subject matter. Accumulating evidence points towards the promise of learning with MERs (Ainsworth, Bibby, & Wood, 2002; Larkin & Simon, 1987; Seufert, 2003), and also to the need for students to make sense out of the different representations by connecting and abstracting from them (Ainsworth, 1999).
This research focuses on a difficult area of early mathematics learning: fractions. Both teachers’ experiences and research in educational psychology show that students have difficulties with fraction arithmetic and with the various representations for fractions (e.g. Brinker, 1997; Callingham & Watson, 2004; Caney & Watson, 2003; Person et al., 2004; PittaPantazi, Gray & Christou, 2004). Coordinating between MERs is regarded as a key process for learning across areas of mathematics (Kilpatrick, Swafford, & Findell, 2001; NCTM, 2000), including fractions (e.g. Kieren, 1993; Moss & Case, 1999; Martinie & BayWilliams, 2003; Thompson & Saldanha, 2003).
A number of authors have argued, based on observational studies, that MERs can lead to deeper conceptual understanding of fractions (Corwin et al., 1990; Cramer et al., 1997a, 1997b; Steiner & Stoeckling, 1997). However, we know of no experimental studies that have investigated the advantages of instruction with multiple (graphical) fraction representations over instruction that focuses on a single representation, with one exception: an in vivo experiment, in which 132 6thgrade students used four versions of CTATbuilt tutors (Rau, Aleven, & Rummel, 2009). Students learning with MERs and prompted to selfexplain performed best on a posttest and delayed posttest assessing procedural and conceptual knowledge of fractions.
At this point, however, we do not know enough about the circumstances that may influence the effectiveness of learning with multiple representations of fractions, a criticism that has been leveraged against the existing body of research on learning with MERs more generally (Ainsworth, 2006; Goldman, 2003). Learning with multiple representations is challenging. An important prerequisite for benefiting from the multiplicity of multiple graphical representations is that students conceptually understand each one of them (Ainsworth, 2006).
When designing intelligent tutoring systems that use multiple graphical representations, designers must decide how to temporally sequence the GRs. How often should the curriculum alternate between multiple graphical representations? Practice schedules are likely to impact how students understand each GR. In particular, it may matter whether items with the same attributes (e.g., task types) are practiced in a “blocked” manner (e.g., A – A – B – B) or are interleaved with practice of other item types (e.g., A – B – A – B). Research on contextual interference shows that interleaving task types leads to better learning results than blocked practice [5, 6]. A common interpretation of this finding is that interleaved practice encourages deep processing [6]. Since students cannot hold all relevant knowledge components in working memory, they must reactivate taskspecific knowledge components as they come up again in the task sequence. When working with multiple graphical representations, a relevant question is: Which task attribute(s) should designers of ITS interleave: task types or multiple graphical representations? Or both? Another previous study of ours provides a partial answer [7]: we contrasted the effects of interleaving task types (while blocking multiple graphical representations) and interleaving multiple graphical representations(while blocking task types) in an ITS for fractions. Our results show an advantage for interleaving task types over interleaving multiple graphical representations. However, the question of whether students benefit most from blocked or interleaved multiple graphical representations when task types are interleaved remains open. The study presented in this paper addresses this question.
The presented research investigates the effect of sequencing multiple graphical representations on students' learning of fractions.
Glossary
 Conceptual knowledge: knowledge about the rationale of a solution procedure
 Procedural knowledge: knowledge of the components of a correct procedure involving knowledge about stepbystep actions for solving problems
Research questions
 Contrasting interleaving topics versus multiple graphical representations
 Sequencing multiple graphical representations
Hypotheses
 We hypothesize that a mix of these two designs (i.e., an intermediate position on the continuum between highly infrequent and highly frequent switching between the representations) would be best as it allows learners to gain some experience with one representation before moving on to the next, but also facilitates making connections across representations as the (temporal) distance between representations is smaller than in the highly infrequently switching design.
 We hypothesize that gaining fluency with each of the representations is more important at the beginning of a tutoring session than towards the end. Therefore, we expect a sequence that transitions from infrequent to frequent switching between representations to be more effective than the extremes of the continuum between highly infrequent and highly infrequent switching between the representations.
Dependent variables
 Previously validated pretest, immediate posttest, and delayed posttest measuring student performance on:
 Reproduction of conceptual knowledge
 Reproduction of procedural knowledge
 Transfer of conceptual knowledge
 Transfer of procedural knowledge
 Log data collected during tutor use, used to assess:
 Learning curves
 Time on task
 Error rates
 Hint usage
 Latency of responses
Independent Variables
 Switch frequently – students will switch representations after every other tutor problem.
 Switch with moderate frequency – students will switch representations after every 5 tutor problems
 Switch infrequently – within each unit of the tutor curriculum, the representations are used in a “blocked” manner: students first work with one representation, then with the next, then with the third
 Switch with graduallyincreasing frequency – students start out within each unit by switching representations with moderate frequency (as in the second condition), but the frequency increases gradually until at the end of each unit it reaches the same frequency as the first condition
Findings
Data collection is still in progress.
Explanation
Data collection is still in progress.
Further Information
Connections
Annotated Bibliography
References
 Ainsworth, S. (1999). Designing effective multirepresentational learning environments (No. 58). Nottingham: ESRC Centre for Research in Development, Instruction & Training Department of Psychology.
 Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. Learning and Instruction, 16(3), 183198.
 Ainsworth, S. (in press). How do animations influence learning? In Robinson & Schraw (Eds.), Current Perspectives on Cognition, Learning, and Instruction: Recent Innovations in Educational Technology that Facilitate Student Learning.
 Ainsworth, S., Bibby, P., & Wood, D. (2002). Examining the effects of different multiple representational systems in learning primary mathematics. Journal of the Learning Sciences, 11(1), 2561.
 Bodemer, D., Ploetzner, R., Bruchmüller, K., & Häcker, S. (2005). Supporting learning with interactive multimedia through active integration of representations. Instructional Science, 33(1), 7395.
 Bodemer, D., & Faust, U. (2006). External and mental referencing of multiple representations. Computers in Human Behavior, 22(1), 2742.
 Brinker, L. (1997). Using Structured Representations To Solve Fraction Problems: A Discussion of Seven Students' Strategies.
 Callingham, R., & Watson, J. (2004). A Developmental Scale of Mental Computation with PartWhole Numbers. Mathematics Education Research Journal, 16(2), 6986.
 Caney, A., & Watson, J. M. (2003). Mental Computation Strategies for PartWhole Numbers. Paper presented at the International Educational Research Conference, Auckland, New Zealand.
 Corwin, R. B., Russell, S. J., & Tierney, C. C. (1990). Seeing fractions: A unit for the upper elementary grades. Sacramento, CA: California Dept. of Education. (ED 348 211).
 Cramer, K., Behr, M., Post, T., & Lesh, R. (1997a). Rational Number Project: Fraction Lessons for the Middle Grades: Level 1. Dubuque, IA: Kendall/Hunt Publishing.
 Cramer, K., Behr, M., Post, T., & Lesh, R. (1997b). Rational Number Project: Fraction Lessons for the Middle Grades: Level 2. Dubuque, IA: Kendall/Hunt Publishing.
 Goldman, S. R., Mertz, D. L., & Pellegrino, J. W. (1989). Individual differences in extended practice functions and solution strategies for basic addition facts. Journal of Educational Psychology, 81(4), 481496.
 Kaput, J.J. (1989). Linking representations in the symbolic systems of algebra. In S. Wagner & C. Kieran (Eds.), Research agenda for mathematics education: Research issues in the learning and teaching of algebra (pp.167194). Reston, VA: National Council of Teachers of Mathematics.
 Kieren, T. E. (1993). Rational and fractional numbers: From quotient fields to recursive understanding. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 4984). Hillsdale, NJ: Erlbaum.
 Kilpatrick, Jeremy; Swafford, Jane; Findell, Bradford (Eds.); Mathematics Learning Study Committee, National Research Council (2001). Conclusions and recommendations. In Adding It Up: Helping Children Learn Mathematics (pp. 407432). Washington, D.C.: The National Academies Press.
 Kozma, R. B., Russell, J., Jones, T., Marx, N., & Davis, J. (1996). The use of multiple, linked representations to facilitate science understanding. In S. Vosniadou, E. De Corte, R. Glaser & H. Mandl (Eds.), International perspectives on the design of technologysupported learning environments. (pp. 4160). Hillsdale, NJ, England: Lawrence Erlbaum Associates, Inc.
 Larkin, J. H., & Simon, H. A. (1987). Why a diagram is (sometimes) worth ten thousand words. Cognitive Science: A Multidisciplinary Journal, 11(1), 65100.
 Lemaire, P., & Siegler, R. S. (1995). Four aspects of strategic change: Contributions to children's learning of multiplication. Journal of Experimental Psychology: General, 124(1), 8397.
 Lowe, R. K. (1999). Extrating information from an animation during complex visual learning. European Journal of Psychology of Education, 14(2), 225244.
 Martin, T., & Schwartz, D. L. (2005). Physically Distributed Learning: Adapting and Reinterpreting Physical Environments in the Development of Fraction Concepts. Cognitive Science: A Multidisciplinary Journal, 29(4), 587625.
 Martinie, S.L., & BayWilliams, J.M. (2003) Investigating students’ conceptual understanding of decimal fractions using multiple representations. Mathematics Teaching in the Middle School, 8(5), 244248.
 Millsaps, G. M., & Reed, M. K. (1998). Curricula for Teaching about Fractions. ERIC Digest.
 Moss, J., & Case, R. (1999). Developing children's understanding of the rational numbers: A new model and an experimental curriculum. Journal for Research in Mathematics Education, 30(2), 122147.
 Moss, J. (2005). Pipes, Tubes, and Beakers: New approaches to teaching the rationalnumber system. In J. Brantsford & S. Donovan (Eds.), How people learn: A targeted report for teachers (pp. 309349): National Academy Press.
 Ni, Y. (2001). Semantic domains of rational numbers and the acquisition of fraction equivalence. Contemporary Educational Psychology, 26(3), 400417.
 NCTM. (2000). Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.
 Paik, J. H. (2005). Fraction concepts: A complex system of mappings. ProQuest Information & Learning, US.
 Person, A. C., Berenson, S. B., & Greenspon, P. J. (2004). The Role of Number in Proportional Reasoning: A Prospective Teacher's Understanding. International Group for the Psychology of Mathematics Education.
 PittaPantazi, D., Gray, E. M., & Christou, C. (2004). Elementary School Students' Mental Representation of Fractions. International Group for the Psychology of Mathematics Education.
 Plötzner, R., Bodemer, D., & Neudert, S. (2008). Successful and less successful use of dynamic visualizations in instructional texts. In R. K. Lowe & W. Schnotz (Eds.), Learning with Animation. Research implications for design. New York: Cambridge University Press.
 Seufert, T. (2003). Supporting coherence formation in learning from multiple representations. Learning and Instruction, 13(2), 227237.
 Spiro, R. J. & Jehng, J. C. (1990). Cognitive flexibility and hypertext: Theory and technology for the nonlinear and multidimensional traversal of complex subject matter. In D. Nix & R. Spiro (Eds.), Cognition, education, and multimedia: Exploring ideas in high technology (pp. 163205). Hillsdale, NJ: Lawrence Erlbaum Associates.
 Steiner, G. F., & Stoecklin, M. (1997). Fraction calculation: A didactic approach to constructing mathematical networks. Learning and Instruction, 7(3), 211233.
 Thompson, P.W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 95–113). Reston, VA: National Council of Teachers of Mathematics.
 Uttal, D. H., Scudder, K. V., & DeLoache, J. S. (1997). Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics. Journal of Applied Developmental Psychology, 18(1), 3754.
 Witherspoon, M. L. (1993). Fractions: In Search of Meaning. Arithmetic Teacher, 40(8), 482485.
 Yang, D.C., & Reys, R. E. (2001). One fraction problem, many solution paths. Mathematics Teaching in the Middle School, 7(3), 164166.