Skip to main content
SpringerLink
Log in

Supporting Meaningful Use of Manipulatives in Kindergarten: The Role of Dual Representation in Early Mathematics

  • Chapter
  • First Online:
Mathematical Learning and Cognition in Early Childhood

  • 1695 Accesses

Abstract

Concrete objects (otherwise known as “manipulatives”) are used often in elementary mathematics classrooms. Teaching effectively with manipulatives is in part contingent on the extent to which children acquire dual representation (DeLoache, 1995), the notion that the manipulatives are objects in their own right as well as objects that “stand for” target concepts. We explored the instructional conditions that support the development of children’s dual representation of manipulatives and the moderating effects of prior numeracy knowledge. The participants were 12 kindergarten students from two classrooms. Explicit instruction on using base ten blocks as representations of quantity was delivered over six lessons. The students’ understanding of the blocks was assessed after each lesson and an application task measuring performance in a novel context was administered at the end of the unit. Meaningful use of the blocks increased steadily from lesson to lesson, with explicit instruction most beneficial for the students with lower prior knowledge. Students with higher prior knowledge were better able to transfer their learning and demonstrated superior performance on the application task.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Alfieri, L., Brooks, P. J., Aldrich, N. J., & Tenenbaum, H. R. (2011). Does discovery-based instruction enhance learning? Journal of Educational Psychology, 103, 1–18.

    Article  Google Scholar 

  • Alfieri, L., Nokes-Malach, T. J., & Schunn, C. D. (2014). Learning through case comparisons: A meta-analytic review. Educational Psychologist, 48(2), 87–113.

    Article  Google Scholar 

  • Ambrose, R. C. (2002). Are we overemphasizing manipulatives in the primary grades to the detriment of girls? Teaching Children Mathematics, 9, 16–21.

    Google Scholar 

  • Ball, D. L. (1992). Magical hopes: Manipulatives and the reform of mathematics education. American Educator, 16(2), 14–18, 46–47.

    Google Scholar 

  • Barnett, S. M., & Ceci, S. J. (2002). When and where do we apply what we learn? A taxonomy for far transfer. Psychological Bulletin, 128(4), 612–637.

    Article  PubMed  Google Scholar 

  • Baroody, A. J. (2017). The use of concrete experiences in early childhood mathematics instruction. In J. Sarama, D. H. Clements, C. Germeroth, & C. Day-Hess (Eds.), Advances in child development and behavior: The development of early childhood mathematics education (pp. 43–87). Cambridge, MA: Academic Press.

    Chapter  Google Scholar 

  • Belenky, D. M., & Schalk, L. (2014). The effects of idealized and grounded materials on learning, transfer, and interest: An organizing framework for categorizing external knowledge representations. Educational Psychology Review, 26, 27–50.

    Article  Google Scholar 

  • Booth, J. L., & Siegler, R. S. (2008). Numerical magnitude representations influence arithmetic learning. Child Development, 79, 1016–1031.

    Article  PubMed  Google Scholar 

  • Brown, M., McNeil, N., & Glenberg, A. (2009). Using concreteness in education: Real problems, potential solutions. Child Development Perspectives, 3, 160–164.

    Article  Google Scholar 

  • Bruner, J. (1964). The course of cognitive growth. American Psychologist, 19(1), 1–15.

    Article  Google Scholar 

  • Byrd, C. E., McNeil, N. M., Chesney, D. L., & Matthews, P. G. (2015). A specific misconception of the equal sign acts as a barrier to children's learning of early algebra. Learning and Individual Differences, 38, 61–67.

    Article  Google Scholar 

  • Carbonneau, K. J., & Marley, S. C. (2015). Instructional guidance and realism of manipulatives influence preschool children's mathematics learning. Journal of Experimental Education, 83(4), 495–513.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (2014). Children’s mathematics: Cognitively guided instruction (2nd ed.). Portsmouth, NH: Heinemann.

    Google Scholar 

  • Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Empson, S. B. (1998). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education, 29, 3–20.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Clarke, T., Ayres, P., & Sweller, J. (2005). The impact of sequencing and prior knowledge on learning mathematics through spreadsheet applications. Educational Technology Research and Development, 53(3), 15–24.

    Article  Google Scholar 

  • Cook, M. P. (2006). Visual representations in science education: The influence of prior knowledge and cognitive load theory on instructional design principles. Science Education, 90(6), 1073–1091.

    Article  Google Scholar 

  • DeLoache, J. S. (1987). Rapid change in the symbolic functioning of very young children. Science, 238, 1556–1557.

    Article  PubMed  Google Scholar 

  • DeLoache, J. S. (1995). Early understanding and use of symbols: The model model. Current Directions in Psychological Science, 4(4), 109–113.

    Article  Google Scholar 

  • DeLoache, J. S. (2000). Dual representation and young children’s use of scale models. Child Development, 71, 329–338.

    Article  PubMed  Google Scholar 

  • DeLoache, J. S. (2004). Becoming symbol-minded. Trends in Cognitive Sciences, 8(2), 66–70.

    Article  PubMed  Google Scholar 

  • Dienes, Z. P. (1964). Building up mathematics (2nd ed.). London, England: Hutchinson of London.

    Google Scholar 

  • Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P., … Japeli, C. (2007). School readiness and later achievement. Developmental Psychology, 43, 1428–1446.

    Article  PubMed  Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • Fosnot, C. T., & Dolk, M. (2001). Young mathematicians at work: Constructing number sense, addition, and subtraction. Westport, CT: Heinemann.

    Google Scholar 

  • Fyfe, E. R., DeCaro, M. S., & Rittle-Johnson, B. (2014a). An alternative time for telling: When conceptual instruction prior to problem solving improves mathematical knowledge. British Journal of Educational Psychology, 84(3), 502–519.

    Article  PubMed  Google Scholar 

  • Fyfe, E. R., McNeil, N. M., Son, J. Y., & Goldstone, R. L. (2014b). Concreteness fading in mathematics and science instruction: A systematic review. Educational Psychology Review, 26, 9–25.

    Article  Google Scholar 

  • Fyfe, E. R., Rittle-Johnson, B., & DeCaro, M. S. (2012). The effects of feedback during exploratory mathematics problem solving: Prior knowledge matters. Journal of Educational Psychology, 104(4), 1094–1108.

    Article  Google Scholar 

  • Goldin, G. A. (1998). Representational systems, learning, and problem solving in mathematics. The Journal of Mathematical Behavior, 17, 137–165.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • Gravemeijer, K., Doorman, M., & Drijvers, P. (2010). Symbolizing and the development of meaning in computer-supported algebra education. In L. Verschaffel, E. De Corte, T. de Jong, & J. Elen (Eds.) (2010). Use of representations in reasoning and problem solving (pp. 191–208). New York, NY: Routledge.

    Google Scholar 

  • 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.

    Google Scholar 

  • Hushman, C. J., & Marley, S. C. (2015). Guided instruction improves elementary student learning and self-efficacy in science. The Journal of Educational Research, 108(5), 371–381.

    Article  Google Scholar 

  • Janvier, C. E. (Ed.). (1987). Problems of representation in the teaching and learning of mathematics. Mahwah, NJ: Erlbaum.

    Google Scholar 

  • Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41(2), 75–86.

    Article  Google Scholar 

  • Lee, C. Y., & Chen, M. J. (2014). The impacts of virtual manipulatives and prior knowledge on geometry learning performance in junior high school. Journal of Educational Computing Research, 50(2), 179–201.

    Article  Google Scholar 

  • Lobato, J. (2012). The actor-oriented transfer perspective and its contributions to educational research and practice. Educational Psychologist, 47, 232–247.

    Article  Google Scholar 

  • Marley, S. C., & Carbonneau, K. J. (2014a). Theoretical perspectives and empirical evidence relevant to classroom instruction with manipulatives. Educational Psychology Review, 26, 1–7.

    Article  Google Scholar 

  • Marley, S. C., & Carbonneau, K. J. (2014b). Future directions for theory and research with instructional manipulatives [Special issue]. Educational Psychology Review, 26, 91–100.

    Google Scholar 

  • Martí, E., Scheuer, N., & de la Cruz, M. (2013). Symbolic use of quantitative representations in young children. In B. M. Brizuela & B. E. Gravel (Eds.), Show me what you know: Exploring student representations across STEM disciplines (pp. 7–21). New York, NY: Teachers College Press.

    Google Scholar 

  • 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.

    Article  PubMed  Google Scholar 

  • Marzolf, D. P., & DeLoache, J. S. (1994). Transfer in young children’s understanding of spatial representations. Child Development, 65(1), 1–15.

    Article  PubMed  Google Scholar 

  • McGuire, P., & Kinzie, M. B. (2013). Analysis of place value instruction and development in pre-kindergarten mathematics. Early Childhood Education Journal, 41(5), 355–364.

    Article  Google Scholar 

  • McNeil, N. M., & Jarvin, L. (2007). When theories don’t add up: Disentangling the manipulatives debate. Theory Into Practice, 46, 309–316.

    Article  Google Scholar 

  • Moyer, P. S. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics. Educational Studies in Mathematics, 47, 175–197.

    Article  Google Scholar 

  • Okamoto, Y., & Case, R. (1996). Number knowledge test. Monographs of the Society for Research in Child Development, 61(1–2), 27–58.

    Google Scholar 

  • Osana, H. P., Adrien, E., & Duponsel, N. (2017). Effects of instructional guidance and sequencing of manipulatives and written symbols on second graders’ numeration knowledge. Education Sciences, 7(2), 52–74.

    Article  Google Scholar 

  • 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 (AERA), San Francisco, CA.

    Google Scholar 

  • Osana, H. P., Przednowek, K., Cooperman, A., & Adrien, E. (2018). Encoding effects on first-graders’ use of manipulatives. Journal of Experimental Education, 2, 154–172.

    Article  Google Scholar 

  • Pagar, D. (2013). The effects of a grouping by tens manipulative on children’s strategy use, base ten understanding and mathematical knowledge. Doctoral dissertation. Available from ProQuest Dissertations and Theses database. (UMI No. 3560909).

    Google Scholar 

  • Peled, I., Meron, R., & Rota, S. (2007). Using a multiplicative approach to construct decimal structure. In Woo, J. H., Lew, H. C., Park, K. S. & Seo, D. Y. (Eds.). Proceedings of the 31st Conference of
the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 65–72). Seoul, Korea: PME.

    Google Scholar 

  • Petersen, L. A., & McNeil, N. M. (2013). Effects of perceptually rich manipulatives on preschoolers’ counting performance: Established knowledge counts. Child Development, 84, 1020–1033.

    Article  PubMed  Google Scholar 

  • Ramani, G. B., & Siegler, R. S. (2011). Reducing the gap in numerical knowledge between low- and middle-income preschoolers. Journal of Applied Developmental Psychology, 32, 146–159.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • Richland, L. E., Stigler, J. W., & Holyoak, K. J. (2012). Teaching the conceptual structure of mathematics. Educational Psychologist, 47(3), 189–203.

    Article  Google Scholar 

  • Rittle-Johnson, B., Star, J. R., & Durkin, K. (2009). The importance of prior knowledge when comparing examples: Influences on conceptual and procedural knowledge of equation solving. Journal of Educational Psychology, 101, 836–852.

    Article  Google Scholar 

  • Sarama, J., & Clements, D. H. (2009). ‘Concrete’ computer manipulatives in mathematics education. Child Development Perspectives, 3(3), 145–150.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • Uttal, D. H., Schreiber, J. C., & DeLoache, J. S. (1995). Waiting to use a symbol: The effects of delay on children’s use of models. Child Development, 66, 1875–1889.

    Article  PubMed  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Uttal, D. H., & Yuan, L. (2014). Using symbols: Developmental perspectives. Wiley Interdisciplinary Reviews: Cognitive Science, 5(3), 295–304.

    PubMed  Google Scholar 

  • Vendetti, M. S., Matlen, B. J., Richland, L. E., & Bunge, S. A. (2015). Analogical reasoning in the classroom: Insights from cognitive science. Mind, Brain, and Education, 9(2), 100–106.

    Article  Google Scholar 

  • Verschaffel, L., De Corte, E., de Jong, T., & Elen, J. (Eds.). (2010). Use of representations in reasoning and problem solving. New York, NY: Routledge.

    Google Scholar 

  • Wearne, D., & Hiebert, J. (1988). A cognitive approach to meaningful mathematics instruction: Testing a local theory using decimal numbers. Journal for Research in Mathematics Education, 19(5), 371–384.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Education, Concordia University, Montreal, QC, Canada

    Helena P. Osana & Nicole Pitsolantis

Authors
  1. Helena P. Osana
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicole Pitsolantis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Helena P. Osana .

Editor information

Editors and Affiliations

  1. Department of Psychology, University of Regina, Regina, SK, Canada

    Katherine M. Robinson

  2. Department of Education, Concordia University, Montreal, QC, Canada

    Helena P. Osana

  3. Department of Mathematics, Wilfred Laurier University, Waterloo, ON, Canada

    Donna Kotsopoulos

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Osana, H.P., Pitsolantis, N. (2019). Supporting Meaningful Use of Manipulatives in Kindergarten: The Role of Dual Representation in Early Mathematics. In: Robinson, K., Osana, H., Kotsopoulos, D. (eds) Mathematical Learning and Cognition in Early Childhood. Springer, Cham. https://doi.org/10.1007/978-3-030-12895-1_7

Download citation

Publish with us

Policies and ethics

Search

Navigation