Educational Psychology Review

, Volume 26, Issue 1, pp 9–25 | Cite as

Concreteness Fading in Mathematics and Science Instruction: a Systematic Review

  • Emily R. Fyfe
  • Nicole M. McNeil
  • Ji Y. Son
  • Robert L. Goldstone
Review Article


A longstanding debate concerns the use of concrete versus abstract instructional materials, particularly in domains such as mathematics and science. Although decades of research have focused on the advantages and disadvantages of concrete and abstract materials considered independently, we argue for an approach that moves beyond this dichotomy and combines their advantages. Specifically, we recommend beginning with concrete materials and then explicitly and gradually fading to the more abstract. Theoretical benefits of this “concreteness fading” technique for mathematics and science instruction include (1) helping learners interpret ambiguous or opaque abstract symbols in terms of well-understood concrete objects, (2) providing embodied perceptual and physical experiences that can ground abstract thinking, (3) enabling learners to build up a store of memorable images that can be used when abstract symbols lose meaning, and (4) guiding learners to strip away extraneous concrete properties and distill the generic, generalizable properties. In these ways, concreteness fading provides advantages that go beyond the sum of the benefits of concrete and abstract materials.


Concrete manipulatives Abstract symbols Learning and instruction 


  1. Ainsworth, S. (1999). The functions of multiple representations. Computers and Education, 33, 131–152.CrossRefGoogle Scholar
  2. Ainsworth, S. (2006). DeFT: a conceptual framework for considering learning with multiple representations. Learning and Instruction, 16, 183–198.CrossRefGoogle Scholar
  3. Ainsworth, S., Bibby, P. A., & Wood, D. (2002). Examining the effects of different multiple representational systems in learning primary mathematics. Journal of the Learning Sciences, 11(1), 25–61.CrossRefGoogle Scholar
  4. Alfieri, L., Brooks, P. J., Aldrich, N. J., & Tenenbaum, H. R. (2011). Does discovery-based instruction enhance learning? Journal of Educational Psychology, 103(1), 1–18.CrossRefGoogle Scholar
  5. Allsopp, D. H., Kyger, M., Ingram, R., & Lovin, L. (2006). MathVIDS2. Virginia Department of Education.
  6. Baranes, R., Perry, M., & Stigler, J. W. (1989). Activation of real-world knowledge in the solution of word problems. Cognition and Instruction, 6, 287–318.CrossRefGoogle Scholar
  7. Barsalou, L. W. (2003). Situated simulation in the human conceptual system. Language & Cognitive Processes, 18, 513–562.CrossRefGoogle Scholar
  8. Belenky, D., & 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, in press.Google Scholar
  9. Berkas, N., & Pattison, C. (2007). Manipulatives: more than a special education intervention. NCTM News Bulletin. Retrieved May 27, 2013, from NCTM Web site:
  10. Berthold, K., & Renkl, A. (2009). Instructional aids to support a conceptual understanding of multiple representations. Journal of Educational Psychology, 101(1), 70–87.CrossRefGoogle Scholar
  11. Braithwaite, D. W., & Goldstone, R. L. (2013). Integrating formal and grounded representations in combinatorics learning. Journal of Educational Psychology. doi: 10.1037/a0032095.Google Scholar
  12. Brown, M. C., McNeil, N. M., & Glenberg, A. M. (2009). Using concreteness in education: real problems, potential solutions. Child Development Perspectives, 3(3), 160–164.CrossRefGoogle Scholar
  13. Bruner, J. S. (1961). The art of discovery. Harvard Educational Review, 31, 21–32.Google Scholar
  14. Bruner, J. S. (1966). Toward a theory of instruction. Cambridge: Belknap.Google Scholar
  15. Bryan, C. A., Wang, T., Perry, B., Wong, N. Y., & Cai, J. (2007). Comparison and contrast: similarities and differences of teachers’ views of effective mathematics teaching and learning from four regions. ZDM Mathematics Education, 39, 329–340.CrossRefGoogle Scholar
  16. Butler, F. M., Miller, S. P., Crehan, K., Babbitt, B., & Pierce, T. (2003). Fraction instruction for students with mathematics disabilities: comparing two teaching sequences. Learning Disabilities Research & Practice, 18(2), 99–111.CrossRefGoogle Scholar
  17. Carraher, T. N., & Schliemann, A. D. (1985). Computation routines prescribed by schools: help or hindrance? Journal for Research in Mathematics Education, 16, 37–44.CrossRefGoogle Scholar
  18. Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1985). Mathematics in the streets and in the schools. British Journal of Developmental Psychology, 3, 21–29.CrossRefGoogle Scholar
  19. Carroll, W. M., & Issacs, A. C. (2003). Achievement of students using the university of Chicago school mathematics project’s everyday mathematics. In S. Senk & D. Thompson (Eds.), Standards-based school mathematics curricula (pp. 9–22). Mahwah: Erlbaum.Google Scholar
  20. Chandler, P., & Sweller, J. (1992). The split-attention effect as a factor in the design of instruction. British Journal of Educational Psychology, 62(2), 233–246.CrossRefGoogle Scholar
  21. Chi, M. T. H., Feltovich, P. J., & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices. Cognitive Science, 5, 121–152.CrossRefGoogle Scholar
  22. Christie, S., & Gentner, D. (2010). Where hypotheses come from: learning new relations by structural alignment. Journal of Cognition and Development, 11(3), 356–373.CrossRefGoogle Scholar
  23. Clements, D. H., & Sarama, J. (2007). Effects of a preschool mathematics curriculum: summative research on the building blocks project. Journal for Research in Mathematics Education, 38(2), 136–163.Google Scholar
  24. Cronbach, L. J., & Snow, R. E. (1977). Aptitudes and instructional methods: a handbook for research on interactions. New York: Irvington.Google Scholar
  25. de Jong, T., Linn, M. C., & Zacharia, Z. C. (2013). Physical and virtual laboratories in science and engineering education. Science, 340, 305–308.CrossRefGoogle Scholar
  26. Devlin, K. (2011). What exactly is multiplication? [Web post for Mathematical Association of America]. Retrieved from
  27. Evans-Martin, F. F. (2005). The nervous system. New York: Chelsea House.Google Scholar
  28. Freudenthal, H. (1991). Revisiting mathematics education: China lectures. Dordrecht: Kluwer Academic.Google Scholar
  29. Fyfe, E. R., & McNeil, N. M. (2009). Benefits of “concreteness fading” for children with low knowledge of mathematical equivalence. Poster presented at the Cognitive Development Society, San Antonio, TX.Google Scholar
  30. Fyfe, E. R., & McNeil, N. M. (2013). The benefits of “concreteness fading” generalize across task, age, and prior knowledge. In K. Mix (chair), Learning from concrete models. Symposium presented at the Society for Research in Child Development, Seattle, WA.Google Scholar
  31. Gentner, D., & Medina, J. (1998). Similarity and the development of rules. Cognition, 65, 263–297.CrossRefGoogle Scholar
  32. Gick, M. L., & Holyoak, K. J. (1983). Schema induction and analogical transfer. Cognitive Psychology, 15, 1–38. doi: 10.1016/0010-0285(83)90002-6.CrossRefGoogle Scholar
  33. Glenberg, A. M., Gutierrez, T., Levin, J. R., Japuntich, S., & Kaschak, M. P. (2004). Activity and imagined activity can enhance young children’s reading comprehension. Journal of Educational Psychology, 96, 424–436. doi: 10.1037/0022-0663.96.3.424.CrossRefGoogle Scholar
  34. Goldstone, R. L., & Sakamoto, Y. (2003). The transfer of abstract principles governing complex adaptive systems. Cognitive Psychology, 46, 414–466.CrossRefGoogle Scholar
  35. Goldstone, R. L., & Son, J. Y. (2005). The transfer of scientific principles using concrete and idealized simulations. The Journal of the Learning Sciences, 14, 69–110.CrossRefGoogle Scholar
  36. Graham, S. A., Namy, L. L., Gentner, D., & Meagher, K. (2010). The role of comparison in preschoolers’ novel object categorization. Journal of Experimental Child Psychology, 107(3), 280–290.CrossRefGoogle Scholar
  37. Gravemeijer, K. (2002). Preamble: from models to modeling. In K. Gravemeijer, R. Lehrer, B. Oers, & L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 7–22). Dordrecht: Kluwer.CrossRefGoogle Scholar
  38. Hofstadter, D., & Sander, E. (2013). Surfaces and essences: analogies as the fuel and fire of thinking. New York: Basic Books.Google Scholar
  39. Homer, B. D., & Plass, J. L. (2009). Expertise reversal for iconic representations in science visualizations. Instructional Science, 38(3), 259–276. doi: 10.1007/s11251-009-9108-7.CrossRefGoogle Scholar
  40. Jaakkola, T., Nurmi, S., & Veermans, K. (2009). Comparing the effectiveness of semi-concrete and concreteness fading computer-simulations to support inquiry learning. Paper presented at the EARLI conference.Google Scholar
  41. Johnson, A. M., Reisslein, J., & Reisslein, M. (2014). Representation sequencing in computer-based engineering education. Computers & Education, 72, 249–261. doi: 10.1016/j.compedu.2013.11.010.CrossRefGoogle Scholar
  42. Kalyuga, S. (2007). Expertise reversal effect and its implications for learner-entailed instruction. Educational Psychology Review, 19, 509–539.CrossRefGoogle Scholar
  43. Kaminski, J. A., & Sloutsky, V. M. (2009). The effect of concreteness on children’s ability to detect common proportion. In N. Taatgen & H. van Rijn (Eds.), Proceedings of the conference of the cognitive science society (pp. 335–340). Mahwah: Erlbaum.Google Scholar
  44. Kaminski, J. A., Sloutsky, V. M., & Heckler, A. F. (2008). The advantage of abstract examples in learning math. Science, 320, 454–455. doi: 10.1126/science.1154659.CrossRefGoogle Scholar
  45. Kaminski, J. A., Sloutsky, V. M., & Heckler, A. F. (2009). Transfer of mathematical knowledge: the portability of generic instantiations. Child Development Perspectives, 3, 151–155.CrossRefGoogle Scholar
  46. Koedinger, K., & Anderson, J. (1998). Illustrating principled design: the early evolution of a cognitive tutor for algebra symbolization. Interactive Learning Environments, 5, 161–179.CrossRefGoogle Scholar
  47. Koedinger, K. R., & Nathan, M. J. (2004). The real story behind story problems: effects of representations on quantitative reasoning. Journal of the Learning Sciences, 13, 129–164.CrossRefGoogle Scholar
  48. Kotovsky, L., & Gentner, D. (1996). Comparison and categorization in the development of relational similarity. Child Development, 67(6), 2797–2822.CrossRefGoogle Scholar
  49. Kurtz, K. J., Boukrina, O., & Gentner, D. (2013). Comparison promotes learning and transfer of relational categories. Journal of Experimental Psychology: Learning, Memory, and Cognition, 39(4), 1303–1310. doi: 10.1037/a0031847.Google Scholar
  50. Lakoff, G., & Nunez, R. E. (2000). Where mathematics comes from: how the embodied mind brings mathematics into being. New York: Basic Books.Google Scholar
  51. Leeper, R. (1935). A study of a neglected portion of the field of learning—the development of sensory organization. The Pedagogical Seminary and Journal of Genetic Psychology, 46, 41–75.CrossRefGoogle Scholar
  52. Lehrer, R., & Schauble, L. (2002). Symbolic communication in mathematics and science: co-constituting inscription and thought. In E. D. Amsel & J. P. Byrnes (Eds.), Language, literacy, and cognitive development. The development and consequences of symbolic communication (p. 167e192). Mahwah: Lawrence Erlbaum Associates.Google Scholar
  53. Lesh, R. (1979). Mathematical learning disabilities: considerations for identification, diagnosis, remediation. In R. Lesh, D. Mierkiewicz, & M. G. Kantowski (Eds.), Applied mathematical problem solving (p. 111e180). Columbus: ERIC.Google Scholar
  54. Mann, R. L. (2004). Balancing act: the truth behind the equals sign. Teaching Children Mathematics, 11(2), 65–69.Google Scholar
  55. Markman, A. B., & Gentner, D. (1993). Structural alignment during similarity comparisons. Cognitive Psychology, 25, 431–467.CrossRefGoogle Scholar
  56. Mayer, R. E. (2004). Should there be a three-strikes rule against pure discovery learning? The case for guided methods of instruction. American Psychologist, 59(1), 14–19.CrossRefGoogle Scholar
  57. McClelland, J. L., Fiez, J. A., & McCandliss, B. D. (2002). Teaching the /r/–/l/ discrimination to Japanese adults: behavioral and neural aspects. Physiology & Behavior, 77, 657–662.CrossRefGoogle Scholar
  58. McNeil, N. M., & Alibali, M. W. (2005). Why won’t you change your mind? Knowledge of operational patterns hinders learning and performance on equations. Child Development, 76(4), 883–899. doi: 10.1111/j.1467-8624.2005.00884.x.CrossRefGoogle Scholar
  59. McNeil, N. M., & Fyfe, E. R. (2012). “Concreteness fading” promotes transfer of mathematical knowledge. Learning and Instruction, 22, 440–448.CrossRefGoogle Scholar
  60. Medin, D. L., Goldstone, R. L., & Gentner, D. (1993). Respects for similarity. Psychological Review, 100, 254–278. doi: 10.1037/0033-295X.100.2.254.CrossRefGoogle Scholar
  61. Nathan, M. J. (2012). Rethinking formalisms in formal education. Educational Psychologist, 47(2), 125–148.CrossRefGoogle Scholar
  62. Petersen, L. A., & McNeil, N. M. (2013). Effects of perceptually rich manipulatives on preschoolers’ counting performance: established knowledge counts. Child Development, 84(3), 1020–1033. doi: 10.1111/cdev.12028.CrossRefGoogle Scholar
  63. Peterson, S. K., Mercer, C. D., & O’Shea, L. (1988). Teaching learning disabled students place value using the concrete to abstract sequence. Learning Disabilities Research, 4, 52–56.Google Scholar
  64. Piaget, J. (1970). Science of education and the psychology of the child. New York: Orion.Google Scholar
  65. Piaget, J. (1973). To understand is to invent. New York: Grossman.Google Scholar
  66. Quine, W. V. (1977). Natural kinds. In S. P. Schwartz (Ed.), Naming, necessity, and natural kinds. Ithaca: Cornell University Press.Google Scholar
  67. Renkl, A., Atkinson, R., Maier, U., & Staley, R. (2002). From example study to problem solving: smooth transitions help learning. Journal of Experimental Education, 70, 293–315.CrossRefGoogle Scholar
  68. Romberg, T. A., & Shafer, M. C. (2004). Purpose, plans, goals, and conduct of the study (Monograph 1). Madison: University of Wisconsin—Madison.Google Scholar
  69. Romberg, T. A., Shafer, M. C., Webb, D. C., & Folgert, L. (2005). The impact of MiC on student achievement (Monograph 5). Madison: University of Wisconsin—Madison.Google Scholar
  70. Ross, B. H. (1987). This is like that: the use of earlier problems and the separation of similarity effects. Journal of Experimental Psych: Learning, Memory, and Cognition, 13, 629–639.Google Scholar
  71. Rutherford, T., Kibrick, M., Burchinal, M., Richland, L., Conley, A., Osborne, K., et al., (2010). Spatial temporal mathematics at scale: an innovative and fully developed paradigm to boost math achievement among all learners. Paper presented at AERA, Denver CO.Google Scholar
  72. Scheiter, K., Gerjets, P., & Schuh, J. (2010). The acquisition of problem-solving skills in mathematics: how animations can aid understanding of structural problem features and solution procedures. Instructional Science, 38, 487–502. doi: 10.1007/s11251-009-9114-9.CrossRefGoogle Scholar
  73. Schliemann, A. D., & Carraher, D. W. (2002). The evolution of mathematical reasoning: everyday versus idealized understandings. Developmental Review, 22, 242–266.CrossRefGoogle Scholar
  74. Schwartz, D. L., Chase, C. C., Oppezzo, M. A., & Chin, D. B. (2011). Practicing versus inventing with contrasting cases: the effects of telling first on learning an transfer. Journal of Educational Psychology, 103(4), 759–775. doi: 10.1037/a0025140.CrossRefGoogle Scholar
  75. Schwonke, R., Berthold, K., & Renkl, A. (2009). How multiple external representations are used and how they can be made more useful. Applied Cognitive Psychology, 23(9), 1227–1243.CrossRefGoogle Scholar
  76. 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.CrossRefGoogle Scholar
  77. Sloutsky, V. M., Kaminski, J. A., & Heckler, A. F. (2005). The advantage of simple symbols for learning and transfer. Psychological Bulletin and Review, 12, 508–513.CrossRefGoogle Scholar
  78. Son, J. Y., & Goldstone, R. L. (2009). Contextualization in perspective. Cognition and Instruction, 27, 51–89.CrossRefGoogle Scholar
  79. Son, J. Y., Smith, L. B., & Goldstone, R. L. (2008). Simplicity and generalization: short-cutting abstraction in children’s object categorizations. Cognition, 108(3), 626–638.CrossRefGoogle Scholar
  80. Son, J. Y., Smith, L. B., & Goldstone, R. L. (2011). Connecting instances to promote children’s relational reasoning. Journal of Experimental Child Psychology, 108(2), 260–277.CrossRefGoogle Scholar
  81. Son, J. Y., Smith, L. B., Goldstone, R. G., & Leslie, M. (2012). The importance of being interpreted: grounded words and children’s relational reasoning. Frontiers in Developmental Psychology, 3, 45.Google Scholar
  82. Stigler, J. W., Givvin, K. B., & Thompson, B. (2010). What community college developmental mathematics students understand about mathematics. The MathAMATYC Educator, 10(3), 4–16.Google Scholar
  83. Tapola, A., Veermans, M., & Niemivirta, M. (2013). Predictors and outcomes of situational interest during a science learning task. Instructional Science, 41, 1047–1064. doi: 10.1007/s11251-013-9273-6.CrossRefGoogle Scholar
  84. Terrace, H. S. (1963). Errorless transfer of a discrimination across two continua. Journal of the Experimental Analysis of Behavior, 6, 223–232.CrossRefGoogle Scholar
  85. Thomas, J., & Thomas, D. (2011). Singapore math: about us. Tualatin: Singapore Math. Retrieved November 27, 2013, from: Scholar
  86. 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. doi: 10.1016/S0193-3973(97)90013-7.CrossRefGoogle Scholar
  87. Uttal, D. H., O’Doherty, K., Newland, R., Hand, L. L., & DeLoache, J. S. (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.CrossRefGoogle Scholar
  88. Wang-Iverson, P., Myers, P., & Lim, E. (2010). Beyond Singapore’s mathematics textbooks: focused and flexible supports for teaching and learning. American Educator, 28–38.Google Scholar
  89. Wecker, C., & Fischer, F. (2011). From guided to self-regulated performance of domain-general skills: the role of peer monitoring during the fading of instructional scripts. Learning and Instruction, 21, 746–756. doi: 10.1016/j.learninstruc.2011.05.001.CrossRefGoogle Scholar
  90. What Works Clearinghouse (2007). Real math building blocks [intervention report]. Retrieved
  91. What Works Clearinghouse (2008). Mathematics in context [intervention report]. Retrieved:
  92. What Works Clearinghouse (2009). Singapore math [intervention report]. Retrieved:
  93. What Works Clearinghouse (2010). Everyday mathematics [intervention report]. Retrieved:
  94. Zacharia, Z. C. (2007). Comparing and combining real and virtual experimentation: an effort to enhance students’ conceptual understanding of electric circuits. Journal of Computer Assisted Learning, 23, 120–132. doi: 10.111/j.1365-2729.2006.00215.x.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Emily R. Fyfe
    • 1
  • Nicole M. McNeil
    • 2
  • Ji Y. Son
    • 3
  • Robert L. Goldstone
    • 4
  1. 1.Department of Psychology and Human DevelopmentVanderbilt UniversityNashvilleUSA
  2. 2.Department of PsychologyUniversity of Notre DameNotre DameUSA
  3. 3.Psychology DepartmentCalifornia State University, Los AngelesLos AngelesUSA
  4. 4.Department of Psychological and Brain SciencesIndiana UniversityBloomingtonUSA

Personalised recommendations