Abstract
Our objective in this chapter is to present a framework that can be used as a guide for designers of virtual manipulatives and for researchers who study their effects on student learning in mathematics. Because a significant amount of research has been devoted to the effects of concrete manipulatives on student learning, the crux of the framework is based on the existing literature in this area. Specifically, the framework consists of three interrelated components that align with the research on students’ learning with external representations: the surface features of the representations themselves, the pedagogical contexts that support students’ meaning making, and the students’ perceptions and interpretations of the representations. Where applicable, we integrate the research on virtual manipulatives to support the validity of the framework itself and its applicability for researchers of virtual mathematics tools.
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Adrien, E., Duponsel, N., & Osana, H. P. (2015). Effects of explanations and presentation order of manipulatives and written symbols in second-grade addition instruction. Presentation at the annual meeting of the American Educational Research Association, Chicago, IL.
Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. Learning and Instruction, 16, 183–198.
Ambrose, R. C. (2002). Are we overemphasizing manipulatives in the primary grades to the detriment of girls? Teaching Children Mathematics, 9(1), 16–21. Retrieved from www.nctm.org/resources/nea/TCM2002-09-16a.pdf
Ball, D. L. (1992). Magical hopes: Manipulatives and the reform of math education. American Educator, 16(14–18), 46–47.
Bartlett, F. C. (1932). Remembering. Cambridge: Cambridge University Press.
Beishuizen, M. (1993). Mental strategies and materials or models for addition and subtraction up to 100 in Dutch second grades. Journal for Research in Mathematics Education, 24(4), 294–323. doi:10.2307/749464.
Bell, P., & Davis, E. A. (2000). Designing Mildred: Scaffolding students’ reflection and argumentation using a cognitive software guide. In B. Fishman, & S. O’Connor-Divelbiss (Eds.), Fourth International Conference of the Learning Sciences (pp. 142–149). Mahwah, NJ: Erlbaum.
Bransford, J. D., Brown, A. L., & Cocking, R. R. (2000). How people learn. Washington, DC: National Research Council.
Burns, B. A., & Hamm, E. M. (2011). A comparison of concrete and virtual manipulative use in third- and fourth-grade mathematics. School Science and Mathematics, 111(6), 256.
Callanan, M. A., Jipson, J. L., & Soennichsen, M. S. (2002). Maps, globes, and videos: Parent-child conversations about representational objects. In S. G. Paris (Ed.), Perspectives on children’s object-centered learning in museums (pp. 261–283). Mahwah, NJ: Erlbaum.
Carbonneau, K. J., Marley, S. C., & Selig, J. P. (2013). A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. Journal of Educational Psychology, 105, 380–400. doi:10.1037/a0031084.
Carpenter, K. K. (2013). Strategy instruction in early childhood math software: Detecting and teaching single-digit addition strategies (Doctoral Dissertation, Columbia University). Retrieved from http://academiccommons.columbia.edu/catalog/ac:160522
Carraher, D. W., & Schliemann, A. D. (2002). Early algebra and algebraic reasoning. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (Vol. II, pp. 669–705). Charlotte, NC: Information Age Publishing.
Chao, S.-J., Stigler, J. W., & Woodward, J. A. (2000). The effects of physical materials on Kindergartners’ learning of number concepts. Cognition and Instruction, 18(3), 285–316. doi:10.1207/S1532690XCI1803_1.
Clements, D. H. (1999). “Concrete” manipulatives, concrete ideas. Contemporary Issues in Early Childhood, 1(1), 45–60.
Corbiere, D. (2003). Digi-block companion to everyday mathematics: Grade 1. Digi-Block, Inc.
DeLoache, J. S. (1987). Rapid change in the symbolic functioning of very young children. Science, 238, 1556–1557. doi:10.1126/science.2446392.
DeLoache, J. S. (1989). Young children’s understanding of the correspondence between a scale model and a larger space. Cognitive Development, 4, 121–129. doi:10.1016/0885-2014(89)90012-9.
DeLoache, J. S. (1995). Early understanding and use of symbols: The model model. Current Directions in Psychological Science, 4(4), 109–113. doi:10.1111/1467-8721.ep10772408.
DeLoache, J. S. (2000). Dual representation and young children’s use of scale models. Child Development, 71, 329–338. doi:10.1111/1467-8624.00148.
DeLoache, J. S., & Sharon, T. (2005). Symbols and similarity: You can get too much of a good thing. Journal of Cognition and Development, 6(1), 33–49. doi:10.1207/s15327647jcd0601_3.
DeLoache, J. S., Miller, K. F., & Rosengren, K. S. (1997). The credible shrinking room: Very young children’s performance with symbolic and nonsymbolic relations. Psychological Science, 8(4), 308–313.
DeLoache, J. S., Peralta de Mendoza, O. A., & Anderson, K. N. (1999). Multiple factors in early symbol use instructions, similarity, and age in understanding a symbol-referent relation. Cognitive Development, 14(2), 299–312. doi:10.1016/S0885-2014(99)00006-4.
Dienes, Z. P. (1963). An experimental study of mathematics learning. London, England: Hutchinson of London.
diSessa, A. A. (2004). Metarepresentation: Native competence and targets for instruction. Cognition and Instruction, 22, 293–331.
diSessa, A. A., & Sherin, B. L. (2000). Meta-representation: An introduction. Mathematical Behavior, 19, 385–398.
Dufour-Janvier, B., Bednarz, N., & Belanger, M. (1987). Pedagogical considerations concerning the problem of representation. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 109–122). Mahwah, NJ: Erlbaum.
Englard, L. (2010). Raise the bar on problem solving. Teaching Children Mathematics, 17(3), 156–163.
English, L. D. (2004). Mathematical and analogical reasoning. In L. English (Ed.), Mathematical and analogical reasoning of young learners (pp. 1–22). Mahwah, NJ: Erlbaum.
Fujimura, N. (2001). Facilitating children’s proportional reasoning: A model of reasoning processes and effects of intervention on strategy change. Journal of Educational Psychology, 93, 589–603. doi:10.1037/0022-0663.93.3.589.
Fyfe, E. R., McNeil, N. M., Son, J. Y., & Goldstone, R. L. (2014). Concreteness fading in mathematics and science instruction: A systematic review. Educational Psychology Review, 26, 9–25.
Gelman, S. A., Chesnick, R. J., & Waxman, S. R. (2005). Mother-child conversations about pictures and objects: Referring to categories and individuals. Child Development, 76(6), 1129–1143.
Gentner, D., & Colhoun, J. (2010). Analogical processes in human thinking and learning. In B. M. Glatzeder, V. Goel, & A. von Müller (Eds.), Toward a theory of thinking: Building blocks for a conceptual framework (pp. 35–48). Berlin, Germany: Springer.
Gentner, D., & Markman, A. B. (1997). Structure mapping in analogy and similarity. American Psychologist, 52, 45–56.
Gentner, D., Loewenstein, J., & Thompson, L. (2003). Learning and transfer: A general role for analogical encoding. Journal of Educational Psychology, 95, 393–408.
Gick, M. L., & Holyoak, K. J. (1983). Schema induction and analogical transfer. Cognitive Psychology, 15, 1–38.
Ginsburg, H. P., Jamalian, A., & Creighan, S. (2013). Cognitive guidelines for the design and evaluation of early mathematics software: The example of MathemAntics. In L. D. English, & J. T. Mulligan (Eds.), Reconceptualizing early mathematics learning (pp. 83–120). Springer, Netherlands. Retrieved from http://link.springer.com/chapter/10.1007/978-94-007-6440-8_6
Glenberg, A. M., Jaworski, B., Rischal, M., & Levin, J. R. (2007). What brains are for: Action, meaning, and reading comprehension. In D. McNamara (Ed.), Reading comprehension strategies: Theories, interventions, and technologies (pp. 221–240). Mahwah, NJ: Lawrence Erlbaum Publishers.
Goldin, G. A. (1998). Representational systems, learning, and problem solving in mathematics. The Journal of Mathematical Behavior, 17, 137–165. doi:10.1016/S0364-0213(99)80056-1.
Goldstone, R. L., & Day, S. B. (2012). Introduction to “New Conceptualizations of Transfer of Learning”. Educational Psychologist, 47(3), 149–152.
Goldstone, R. L., & Sakamoto, Y. (2003). The transfer of abstract principles governing complex adaptive systems. Cognitive Psychology, 46, 414–466. doi:10.1016/S0010-0285(02)00519-4.
Goldstone, R. L., & Son, J. Y. (2005). The transfer of scientific principles using concrete and idealized simulations. Journal of the Learning Sciences, 14, 69–110.
Goswami, U. (2004). Commentary: Analogical reasoning and mathematical development. In L. English (Ed.), Mathematical and analogical reasoning of young learners (pp. 169–186). Mahwah, NJ: Erlbaum.
Gravemeijer, K. (2002). Preamble: From models to modeling. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 7–22). The Netherlands: Kluwer Academic Publishers.
Gürbüz, R. (2010). The effect of activity-based instruction on conceptual development of seventh grade students in probability. International Journal of Mathematical Education in Science and Technology, 41, 743–767. doi:10.1080/00207391003675158.
Hiebert, J. (1992). Mathematical, cognitive, and instructional analyses of decimal fractions. In G. Leinhardt, R. Putnam, & R. Hattrup (Eds.), Analysis of arithmetic for mathematics teaching (pp. 283–322). Hillsdale, NJ: Erlbaum.
Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371–404). Reston, VA: National Council of Teachers of Mathematics.
Hiebert, J., & Wearne, D. (1992). Links between teaching and learning place value with understanding in first grade. Journal for Research in Mathematics Education, 23, 98–122. doi:10.2307/749496.
Hiebert, J., Carpenter, T. R., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., et al. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.
Hughes, M. (1986). Children and number: Difficulties in learning mathematics. Oxford, England: Basil Blackwell.
Kamii, C., Lewis, B. A., & Kirkland, L. (2001). Manipulatives: when are they useful? Journal of Mathematical Behavior, 20, 21–31. doi:10.1016/S0732-3123(01)00059-1.
Kaminski, J. A., Sloutsky, V. M., & Heckler, A. (2009). Transfer of mathematical knowledge: The portability of generic instantiations. Child Development Perspectives, 3(3), 151–155. doi:10.1111/j.1750-8606.2009.00096.x.
Kim, R., & Albert, L. R. (2014). The history of base-ten-blocks: Why and who made base-ten-blocks? Mediterranean Journal of Social Sciences, 5, 356–365.
Ladel, S., & Kortenkamp, U. (2013). An activity-theoretic approach to multi-touch tools in early maths learning. The International Journal for Technology in Mathematics Education, 20(1). Retrieved from http://www.tech.plym.ac.uk/research/mathematics_education/field%20of%20work/ijtme/volume_20/number_one.html⋕one
Lee, C.-Y., & Yuan, Y. (2010). Gender differences in the relationship between Taiwanese adolescents’ mathematics attitudes and their perceptions toward virtual manipulatives. International Journal of Science and Mathematics Education, 8, 937–950.
Markman, A. B. (1999). Knowledge representation. Mahwah, NJ: Erlbaum.
Martin, T. (2009). A theory of physically distributed learning: How external environments and internal states interact in mathematics learning. Child Development Perspectives, 3(3), 140–144.
Martin, T., & Schwartz, D. L. (2005). Physically distributed learning: Adapting and reinterpreting physical environments in the development of fraction concepts. Cognitive Science, 29, 587–625.
Marzolf, D. P., & DeLoache, J. S. (1994). Transfer in young children’s understanding of spatial representations. Child Development, 65(1), 1–15. doi:10.2307/1131361.
Mayer, R. E. (2011). Instruction based on visualizations. In R. E. Mayer & P. A. Alexander (Eds.), Handbook of research on learning and instruction (pp. 427–445). New York, NY: Routledge.
McNeil, N. M., Uttal, D. H., Jarvin, L., & Sternberg, R. J. (2009). Should you show me the money? Concrete objects both hurt and help performance on mathematics problems. Learning and Instruction, 19(2), 171–184.
Moyer, P. S. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics. Educational Studies in Mathematics, 47, 175–197.
Moyer, P. S., Bolyard, J. J., & Spikell, M. A. (2002). What are virtual manipulatives? Teaching Children Mathematics, 8, 372–377.
Moyer-Packenham, P. S., & Westenskow, A. (2013). Effects of virtual manipulatives on student achievement and mathematics learning. International Journal of Virtual and Personal Learning Environments, 4(3).
Moyer-Packenham, P., Baker, J., Westenskow, A., Anderson, K., Shumway, J., Rodzon, K., et al. (2013). A study comparing virtual manipulatives with other instructional treatments in third- and fourth-grade classrooms. Journal of Education, 193(2), 25–39.
Moyer-Packenham, P. S., Shumway, J. F., Bullock, E., Tucker, S. I., Anderson-Pence, K. L., & Westenskow, A. (2015). Young children’s learning performance and efficiency when using virtual manipulative mathematics iPad apps. Journal of Computers in Mathematics and Science Teaching, 34(1), 41–69.
Namukasa, I. K., Stanley, D., & Tuchtie, M. (2009). Virtual manipulative materials in secondary mathematics: A theoretical discussion. The Journal of Computers in Mathematics and Science Teaching, 28(3), 277.
Ng, S. F., & Lee, K. (2009). Model method: A visual tool to support algebra word problem solving at the primary level. In K. Y. Wong, P. Y. Lee, B. Kaur, P. Y. Foong, & S. F. Ng (Eds.), Mathematics education: The Singapore journey (pp. 169–203). Singapore: World Scientific.
Nührenbörger, M., & Steinbring, H. (2008). Manipulatives as tools in mathematics teacher education. In D. Tirosh & T. Wood (Eds.), International handbook of mathematics teacher education: Vol. 2: Tools and processes in mathematics teacher education (pp. 157–181). Rotterdam: Sense.
Osana, H. P., & Pitsolantis, N. (2013). Addressing the struggle to link form and understanding in fractions instruction. British Journal of Educational Psychology, 83, 29–56.
Osana, H. P., & Pitsolantis, N. (2015). Supporting Kindergarten children’s dual representation: Meaningful use of mathematics manipulatives. Paper presented at the American Educational Research Association (AERA), Washington, DC.
Osana, H. P., Przednowek, K., Cooperman, A., & Adrien, E. (2013). Making the most of math manipulatives: Play is not the answer. Presented at the annual meeting of the American Educational Research Association, San Francisco, CA.
Osana, H. P., Przednowek, K., Adrien, E., & Cooperman, A. (2014, May). Assessing dual representation in mathematics. In C. Sowinski (Chair), Math development: Measures and methodologies. Symposium Conducted at the Development 2014: A Canadian Conference on Developmental Psychology, Ottawa, Canada.
Ozel, S., Ozel, Z. E. Y., & Cifuentes, L. D. (2014). Effectiveness of an online manipulative tool and students’ technology acceptances. International Journal of Educational Studies in Mathematics, 1(1), 1–15.
Palmer, S. E. (1978). Fundamental aspects of cognitive representation. In E. Rosch & B. B. Lloyd (Eds.), Cognition and categorization (pp. 259–302). Erlbaum.
Parker, T. H., & Baldridge, S. J. (2008). Elementary mathematics for teachers. Okemos, USA: Sefton-Ash Publishing.
Perkins, D. N., & Salomon, G. (2012). Knowledge to go: A motivational and dispositional view of transfer. Educational Psychologist, 47(3), 248–258.
Petersen, L. A., & McNeil, N. M. (2013). Effects of perceptually rich manipulatives on preschoolers’ counting performance: Established knowledge counts. Child Development, 84, 1020–1033.
Pimm, D. (1995). Symbols and meanings in school mathematics. New York, NY: Routledge.
Przednowek, K., Osana, H. P., Cooperman, A., & Adrien, E. (2013). Introducing manipulatives: To play or not to play. In M. V. Martinez & A. C. Superfine (Eds.), Proceedings of the Thirty-Fifth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (p. 329). Chicago, IL: University of Illinois at Chicago.
Puchner, L., Taylor, A., O’Donnell, B., & Fick, K. (2008). Teacher learning and mathematics manipulatives: A collective case study about teacher use of manipulatives in elementary and middle school mathematics lessons. School Science and Mathematics, 108(7), 313–325.
Quintana, C., Reiser, B. J., Davis, E. A., Krajcik, J., Fretz, E., & Duncan, R. G. (2004). A scaffolding design framework for software to support science inquiry. Journal of the Learning Sciences, 13, 337–386.
Raphael, D., & Wahlstrom, M. (1989). The influence of instructional aids on mathematics achievement. Journal for Research in Mathematics Education, 20(2), 173–190.
Resnick, L. B., & Omanson, S. F. (1987). Learning to understand arithmetic. In R. Glaser (Ed.), Advances in instructional psychology (Vol. 3, pp. 41–95). Hillsdale, NJ: Erlbaum.
Reys, R. E., Lindquist, M. M., Lambdin, D. V., & Smith, N. L. (2014). Helping children learn mathematics (11th ed.). New York: Wiley.
Richland, L. E. (2011). Analogy and classroom mathematics learning. In N. L. Stein & S. R. Raudenbush (Eds.), Developmental cognitive science goes to school (pp. 203–218). New York, NY: Routledge.
Richland, L. E., Morrison, R. G., & Holyoak, K. J. (2006). Children’s development of analogical reasoning: Insights from scene analogy problems. Journal of Experimental Child Psychology, 94, 249–273.
Richland, L. E., Stigler, J. W., & Holyoak, K. J. (2012). Teaching the conceptual structure of mathematics. Educational Psychologist, 47(3), 189–203.
Rick, J. (2012). Proportion: a tablet app for collaborative learning. In Proceedings of the 11th International Conference on Interaction Design and Children (pp. 316–319). New York, NY, USA: ACM. doi:10.1145/2307096.2307155
Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology, 91, 175–189.
Rittle-Johnson, B., & Koedinger, K. (2009). Iterating between lessons on concepts and procedures can improve mathematics knowledge. British Journal of Educational Psychology, 79, 1–21.
Rogoff, B. (1990). Apprenticeship in thinking: Cognitive development in social context. New York, NY: Oxford University Press.
Sarama, J., & Clements, D. H. (2002). Building Blocks for young children’s mathematical development. Journal of Educational Computing Research, 27, 93–110.
Sarama, J., & Clements, D. H. (2009). Building blocks and cognitive building blocks: Playing to know the world mathematically. American Journal of Play, 1, 313–337.
Sedig, K., & Liang, H.-N. (2006). Interactivity of visual mathematical representations: Factors affecting learning and cognitive processes. Journal of Interactive Learning Research, 17(2), 179–212.
Sedig, K., Klawe, M., & Westrom, M. (2001). Role of interface manipulation style and scaffolding on cognition and concept learning in Learnware. ACM Transactions on Computer-Human Interaction, 8, 34–59.
Segal, A., Tversky, B., & Black, J. (2014). Conceptually congruent actions can promote thought. Journal of Applied Research in Memory and Cognition. doi:10.1016/j.jarmac.2014.06.004
Sherman, J., & Bisanz, J. (2009). Equivalence in symbolic and non-symbolic contexts: Benefits of solving problems with manipulatives. Journal of Educational Psychology, 101, 88–100. doi:10.1037/a0013156
Singley, K., & Anderson, J. R. (1989). The transfer of cognitive skill. Cambridge, MA: Harvard University Press.
Stigler, J. W. (1984). “Mental abacus”: The effects of abacus training on Chinese children’s mental calculation. Cognitive Psychology, 16, 145–176.
Suh, J., & Moyer, P. S. (2007). Developing students’ representational fluency using virtual and physical algebra balances. The Journal of Computers in Mathematics and Science Teaching, 26(2), 155–173.
Suydam, M. N. (1986). Manipulative materials and achievement. Arithmetic Teacher, 33(6), 10–32.
Sweller, J., van Merrienboer, J. J. G., & Paas, F. G. W. C. (1998). Cognitive architecture and instructional design. Educational Psychology Review, 10, 251–296.
Thompson, P. W. (1994). Concrete materials and teaching for mathematical understanding. Arithmetic Teacher, 41(9), 556–558.
Toth, E. E., Suthers, D. D., & Lesgold, A. M. (2002). “Mapping to know”: The effects of representational guidance and reflective assessment on scientific inquiry. Science Education, 86, 264–286.
Tucker, S. I., Moyer-Packenham, P. S., Boyer-Thurgood, J. M., Anderson, K. L., Shumway, J. F., Westenskow, A., et al. (2014). Literature supporting an investigation of the nexus of mathematics, strategy, and technology in second-graders’ interactions with iPad-based virtual manipulatives. In Proceedings of the 12th Annual Hawaii International Conference on Education (HICE) (pp. 2338–2346). Honolulu, Hawaii. doi:10.13140/2.1.3392.4169
Uttal, D. H. (2003). On the relation between play and symbolic thought: The case of mathematics manipulatives. In O. Saracho & B. Spodek (Eds.), Contemporary perspectives in early childhood (pp. 97–114). Charlotte, NC: Information Age Press.
Uttal, D. H., & O’Doherty, K. (2008). Comprehending and learning from visual representations: A developmental approach. In J. Gilbert, M. Reiner, & M. Nakhleh (Eds.), Visualization: Theory and practice in science education (pp. 53–72). New York, NY: Springer.
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, 37–54.
Uttal, D. H., Liu, L. L., & DeLoache, J. S. (2006). Concreteness and symbolic development. In L. Balter & C. S. Tamis-LeMonda (Eds.), Child psychology: A handbook of contemporary issues (2nd ed., pp. 167–184). Philadelphia, PA: Psychology Press.
Uttal, D. H., O’Doherty, K., Newland, R., Hand, L. L., & DeLoache, J. (2009). Dual representation and the linking of concrete and symbolic representations. Child Development Perspectives, 3(3), 156–159. doi:10.1111/j.1750-8606.2009.00097.x
Wilson, M. (2002). Six views of embodied cognition. Psychonomic Bulletin and Review, 9, 625–636.
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Osana, H.P., Duponsel, N. (2016). Manipulatives, Diagrams, and Mathematics: A Framework for Future Research on Virtual Manipulatives. In: Moyer-Packenham, P. (eds) International Perspectives on Teaching and Learning Mathematics with Virtual Manipulatives. Mathematics Education in the Digital Era, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-32718-1_5
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