Skip to main content
SpringerLink
Log in

Concreteness Fading in Mathematics and Science Instruction: a Systematic Review

  • Review Article
  • Published:
Educational Psychology Review Aims and scope Submit manuscript

Abstract

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.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Ainsworth, S. (1999). The functions of multiple representations. Computers and Education, 33, 131–152.

    Article  Google Scholar 

  • Ainsworth, S. (2006). DeFT: a conceptual framework for considering learning with multiple representations. Learning and Instruction, 16, 183–198.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Allsopp, D. H., Kyger, M., Ingram, R., & Lovin, L. (2006). MathVIDS2. Virginia Department of Education. http://www.coedu.usf.edu/main/departments/sped/mathvids/index.html.

  • 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.

    Article  Google Scholar 

  • Barsalou, L. W. (2003). Situated simulation in the human conceptual system. Language & Cognitive Processes, 18, 513–562.

    Article  Google Scholar 

  • 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.

  • Berkas, N., & Pattison, C. (2007). Manipulatives: more than a special education intervention. NCTM News Bulletin. Retrieved May 27, 2013, from NCTM Web site: http://www.nctm.org/news/release_list.aspx?id=12698.

  • Berthold, K., & Renkl, A. (2009). Instructional aids to support a conceptual understanding of multiple representations. Journal of Educational Psychology, 101(1), 70–87.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • Bruner, J. S. (1961). The art of discovery. Harvard Educational Review, 31, 21–32.

    Google Scholar 

  • Bruner, J. S. (1966). Toward a theory of instruction. Cambridge: Belknap.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Carraher, T. N., & Schliemann, A. D. (1985). Computation routines prescribed by schools: help or hindrance? Journal for Research in Mathematics Education, 16, 37–44.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Christie, S., & Gentner, D. (2010). Where hypotheses come from: learning new relations by structural alignment. Journal of Cognition and Development, 11(3), 356–373.

    Article  Google Scholar 

  • 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 

  • Cronbach, L. J., & Snow, R. E. (1977). Aptitudes and instructional methods: a handbook for research on interactions. New York: Irvington.

    Google Scholar 

  • de Jong, T., Linn, M. C., & Zacharia, Z. C. (2013). Physical and virtual laboratories in science and engineering education. Science, 340, 305–308.

    Article  Google Scholar 

  • Devlin, K. (2011). What exactly is multiplication? [Web post for Mathematical Association of America]. Retrieved from http://www.maa.org/devlin/devlin_01_11.html.

  • Evans-Martin, F. F. (2005). The nervous system. New York: Chelsea House.

    Google Scholar 

  • Freudenthal, H. (1991). Revisiting mathematics education: China lectures. Dordrecht: Kluwer Academic.

    Google Scholar 

  • 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.

  • 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.

  • Gentner, D., & Medina, J. (1998). Similarity and the development of rules. Cognition, 65, 263–297.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Goldstone, R. L., & Sakamoto, Y. (2003). The transfer of abstract principles governing complex adaptive systems. Cognitive Psychology, 46, 414–466.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • Hofstadter, D., & Sander, E. (2013). Surfaces and essences: analogies as the fuel and fire of thinking. New York: Basic Books.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

  • 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.

    Article  Google Scholar 

  • Kalyuga, S. (2007). Expertise reversal effect and its implications for learner-entailed instruction. Educational Psychology Review, 19, 509–539.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Koedinger, K., & Anderson, J. (1998). Illustrating principled design: the early evolution of a cognitive tutor for algebra symbolization. Interactive Learning Environments, 5, 161–179.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Kotovsky, L., & Gentner, D. (1996). Comparison and categorization in the development of relational similarity. Child Development, 67(6), 2797–2822.

    Article  Google Scholar 

  • 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 

  • Lakoff, G., & Nunez, R. E. (2000). Where mathematics comes from: how the embodied mind brings mathematics into being. New York: Basic Books.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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 

  • Mann, R. L. (2004). Balancing act: the truth behind the equals sign. Teaching Children Mathematics, 11(2), 65–69.

    Google Scholar 

  • Markman, A. B., & Gentner, D. (1993). Structural alignment during similarity comparisons. Cognitive Psychology, 25, 431–467.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • McNeil, N. M., & Fyfe, E. R. (2012). “Concreteness fading” promotes transfer of mathematical knowledge. Learning and Instruction, 22, 440–448.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Nathan, M. J. (2012). Rethinking formalisms in formal education. Educational Psychologist, 47(2), 125–148.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • Piaget, J. (1970). Science of education and the psychology of the child. New York: Orion.

    Google Scholar 

  • Piaget, J. (1973). To understand is to invent. New York: Grossman.

    Google Scholar 

  • Quine, W. V. (1977). Natural kinds. In S. P. Schwartz (Ed.), Naming, necessity, and natural kinds. Ithaca: Cornell University Press.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Romberg, T. A., & Shafer, M. C. (2004). Purpose, plans, goals, and conduct of the study (Monograph 1). Madison: University of Wisconsin—Madison.

    Google Scholar 

  • 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 

  • 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 

  • 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.

  • 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.

    Article  Google Scholar 

  • Schliemann, A. D., & Carraher, D. W. (2002). The evolution of mathematical reasoning: everyday versus idealized understandings. Developmental Review, 22, 242–266.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Son, J. Y., & Goldstone, R. L. (2009). Contextualization in perspective. Cognition and Instruction, 27, 51–89.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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 

  • 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.

    Article  Google Scholar 

  • Terrace, H. S. (1963). Errorless transfer of a discrimination across two continua. Journal of the Experimental Analysis of Behavior, 6, 223–232.

    Article  Google Scholar 

  • Thomas, J., & Thomas, D. (2011). Singapore math: about us. Tualatin: Singapore Math. Retrieved November 27, 2013, from: http://www.singaporemath.com/aboutus.asp.

    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. doi:10.1016/S0193-3973(97)90013-7.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

  • 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.

    Article  Google Scholar 

  • What Works Clearinghouse (2007). Real math building blocks [intervention report]. Retrieved http://ies.ed.gov/ncee/wwc/pdf/intervention_reports/WWC_Building_Blocks_072307.pdf.

  • What Works Clearinghouse (2008). Mathematics in context [intervention report]. Retrieved: http://ies.ed.gov/ncee/wwc/pdf/intervention_reports/wwc_mathinc_082608.pdf.

  • What Works Clearinghouse (2009). Singapore math [intervention report]. Retrieved: http://ies.ed.gov/ncee/wwc/pdf/intervention_reports/wwc_singaporemath_042809.pdf.

  • What Works Clearinghouse (2010). Everyday mathematics [intervention report]. Retrieved: http://ies.ed.gov/ncee/wwc/pdf/intervention_reports/wwc_everyday_math_091410.pdf.

  • 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.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Psychology and Human Development, Vanderbilt University, 230 Appleton Place, Peabody #552, Nashville, TN, 37203, USA

    Emily R. Fyfe

  2. Department of Psychology, University of Notre Dame, Notre Dame, IN, USA

    Nicole M. McNeil

  3. Psychology Department, California State University, Los Angeles, Los Angeles, CA, USA

    Ji Y. Son

  4. Department of Psychological and Brain Sciences, Indiana University, Bloomington, IN, USA

    Robert L. Goldstone

Authors
  1. Emily R. Fyfe
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicole M. McNeil
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ji Y. Son
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Robert L. Goldstone
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emily R. Fyfe.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fyfe, E.R., McNeil, N.M., Son, J.Y. et al. Concreteness Fading in Mathematics and Science Instruction: a Systematic Review. Educ Psychol Rev 26, 9–25 (2014). https://doi.org/10.1007/s10648-014-9249-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10648-014-9249-3

Keywords

Search

Navigation