Skip to main content
Log in

Not a One-Way Street: Bidirectional Relations Between Procedural and Conceptual Knowledge of Mathematics

Educational Psychology Review Aims and scope Submit manuscript

Abstract

There is a long-standing and ongoing debate about the relations between conceptual and procedural knowledge (i.e., knowledge of concepts and procedures). Although there is broad consensus that conceptual knowledge supports procedural knowledge, there is controversy over whether procedural knowledge supports conceptual knowledge and how instruction on the two types of knowledge should be sequenced. A review of the empirical evidence for mathematics learning indicates that procedural knowledge supports conceptual knowledge, as well as vice versa, and thus that the relations between the two types of knowledge are bidirectional. However, alternative orderings of instruction on concepts and procedures have rarely been compared, with limited empirical support for one ordering of instruction over another. We consider possible reasons for why mathematics education researchers often believe that a conceptual-to-procedural ordering of instruction is optimal and why so little research has evaluated this claim. Future empirical research on the effectiveness of different ways to sequence instruction on concepts and procedures is greatly needed.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Alcock, L., Ansari, D., Batchelor, S., Bisson, M.-J., De Smedt, B., Gilmore, C. K., … Weber, K. (2014). Challenges in mathematical cognition: A collaboratively-derived research agenda. Manuscript under review.

  • Ball, D. L., Ferrini-Mundy, J., Kilpatrick, J., Milgram, R. J., Schmid, W., & Schaar, R. (2005). Reaching for common group in K-12 mathematics education. Notices of the AMS, 52, 1055–1058.

    Google Scholar 

  • Baroody, A. J. (1992). The development of preschoolers’ counting skills and principles. In J. Bideaud, C. Meljac, & J. P. Fischer (Eds.), Pathway to Numbers: Children’s developing numerical abilities (pp. 99–126). Hillsdale: Erlbaum.

    Google Scholar 

  • Baroody, A. J. (2003). The development of adaptive expertise and flexibility: The integration of conceptual and procedural knowledge. In A. J. Baroody & A. Dowker (Eds.), The Development of Arithmetic Concepts and Skills: Constructing Adaptive Expertise (pp. 1–34). Mahwah: Erlbaum.

    Google Scholar 

  • Baroody, A. J., & Ginsburg, H. (1986). The relationship between initial meaningful and mechanical knowledge of arithmetic. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics. Hillsdale: Lawrence Erlbaum Associates, Inc.

    Google Scholar 

  • Baroody, A. J., Feil, Y., & Johnson, A. R. (2007). An alternative reconceptualization of procedural and conceptual knowledge. Journal for Research in Mathematics Education, 38, 115–131.

    Google Scholar 

  • Blöte, A. W., Van der Burg, E., & Klein, A. S. (2001). Students’ flexibility in solving two-digit addition and subtraction problems: instruction effects. Journal of Educational Psychology, 93, 627–638. doi:10.1037//0022-0663.93.3.627.

    Article  Google Scholar 

  • Byrnes, J. P., & Wasik, B. A. (1991). Role of conceptual knowledge in mathematical procedural learning. Developmental Psychology, 27, 777–786. doi:10.1037//0012-1649.27.5.777.

    Article  Google Scholar 

  • Canobi, K. H. (2009). Concept-procedure interactions in children’s addition and subtraction. Journal of Experimental Child Psychology, 102, 131–149. doi:10.1016/j.jecp.2008.07.008.

    Article  Google Scholar 

  • Cobb, P., Wood, T., Yackel, E., Nicholls, J., Wheatley, G., Trigatti, B., & Perlwitz, M. (1991). Assessment of a problem-centered second-grade mathematics project. Journal for Research in Mathematics Education, 22, 3–29. doi:10.2307/749551.

    Article  Google Scholar 

  • Cowan, R., Donlan, C., Shepherd, D.-L., Cole-Fletcher, R., Saxton, M., & Hurry, J. (2011). Basic calculation proficiency and mathematics achievement in elementary school children. Journal of Educational Psychology, 103, 786–803. doi:10.1037/a0024556.

    Article  Google Scholar 

  • Crooks, N. M., & Alibali, M. W. (2014). Defining and measuring conceptual knowledge in mathematics. Developmental Review, 34, 344–377. doi:10.1016/j.dr.2014.10.001.

  • Fuson, K. C. (1988). Children’s counting and concept of number. New York: Springer-Verlag.

    Book  Google Scholar 

  • Fuson, K. C., & Briars, D. J. (1990). Using a base-ten blocks learning/teaching approach for first- and second-grade place-value and multidigit addition and subtraction. Journal for Research in Mathematics Education, 21, 180–206. doi:10.2307/749373.

    Article  Google Scholar 

  • Gelman, R., & Williams, E. M. (1998). Enabling constraints for cognitive development and learning: domain specificity and epigenesis. In D. Kuhn & R. S. Siegler (Eds.), Handbook of child psychology: Cognition, perception, and language (5th ed., Vol. 2, pp. 575–630). New York: John Wiley & Sons, Inc.

    Google Scholar 

  • Gilmore, C. K., & Papadatou-Pastou, M. (2009). Patterns of individual differences in conceptual understanding and arithmetical skill: a meta-analysis. Mathematical Thinking and Learning, 11, 25–40. doi:10.1080/10986060802583923.

    Article  Google Scholar 

  • Goldin Meadow, S., Alibali, M. W., & Church, R. B. (1993). Transitions in concept acquisition: using the hand to read the mind. Psychological Review, 100, 279–297. doi:10.1037//0033-295X.100.2.279.

    Article  Google Scholar 

  • Grouws, D. A., & Cebulla, K. J. (2000). Improving student achievement in mathematics. Geneva: International Academy of Education.

    Google Scholar 

  • Halford, G. S. (1993). Children’s understanding: The development of mental models. Hillsdale: Erlbaum.

    Google Scholar 

  • Hallett, D., Nunes, T., & Bryant, P. (2010). Individual differences in conceptual and procedural knowledge when learning fractions. Journal of Educational Psychology, 102, 395–406. doi:10.1037/a0017486.

    Article  Google Scholar 

  • Hallett, D., Nunes, T., Bryant, P., & Thorpe, C. M. (2012). Individual differences in conceptual and procedural fraction understanding: the role of abilities and school experience. Journal of Experimental Child Psychology, 113, 469–486. doi:10.1016/j.jecp.2012.07.009.

    Article  Google Scholar 

  • Hecht, S. A., & Vagi, K. J. (2010). Sources of group and individual differences in emerging fraction skills. Journal of Educational Psychology, 102, 843–859. doi:10.1037/a0019824.

    Article  Google Scholar 

  • Hecht, S. A., & Vagi, K. J. (2012). Patterns of strengths and weaknesses in children’s knowledge about fractions. Journal of Experimental Child Psychology, 111, 212–229. doi:10.1016/j.jecp.2011.08.012.

    Article  Google Scholar 

  • Hiebert, J., & Grouws, D. (2007). Effective teaching for the development of skill and conceptual understanding of number: what is most effective? In J. Reed (Ed.), Research brief. Reston: National Council of Teachers of Mathematics.

    Google Scholar 

  • Hiebert, J., & LeFevre, P. (1986). Conceptual and procedural knowledge in mathematics: an introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Hillsdale: Lawrence Erlbaum Associates, Inc.

    Google Scholar 

  • Hiebert, J., & Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and subtraction. Cognition and Instruction, 14, 251–283. doi:10.1207/s1532690xci1403_1.

    Article  Google Scholar 

  • Kamii, C., & Dominick, A. (1997). To teach or not to teach algorithms. The Journal of Mathematical Behavior, 16, 51–61.

    Article  Google Scholar 

  • Kamii, C., & Dominick, A. (1998). The harmful effects of algorithms in grades 1-4. In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics. 1998 Yearbook (pp. 130–140). Reston: National Council of Teachers of Mathematics.

    Google Scholar 

  • Karmiloff-Smith, A. (1992). Beyond modularity: A developmental perspective on cognitive science. Cambridge: MIT Press.

    Google Scholar 

  • Kilpatrick, J., Swafford, J. O., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington DC: National Academy Press.

    Google Scholar 

  • Mack, N. K. (1990). Learning fractions with understanding: building on informal knowledge. Journal for Research in Mathematics Education, 21, 16–32. doi:10.2307/749454.

    Article  Google Scholar 

  • Matthews, P., & Rittle-Johnson, B. (2009). In pursuit of knowledge: comparing self-explanations, concepts, and procedures as pedagogical tools. Journal of Experimental Child Psychology, 104, 1–21. doi:10.1016/j.jecp.2008.08.004.

    Article  Google Scholar 

  • McNeil, N. M., Fyfe, E. R., Petersen, L. A., Dunwiddie, A. E., & Brletic-Shipley, H. (2011). Benefits of practicing 4 = 2 + 2: nontraditional problem formats facilitate children’s understanding of mathematical equivalence. Child Development, 82, 1620–1633.

    Article  Google Scholar 

  • McNeil, N. M., Chesney, D. L., Matthews, P. G., Fyfe, E. R., Petersen, L. A., Dunwiddie, A. E., & Wheeler, M. C. (2012). It pays to be organized: organizing arithmetic practice around equivalent values facilitates understanding of math equivalence. Journal of Educational Psychology. doi:10.1037/a0028997.

    Google Scholar 

  • McNeil, N. M., Fyfe, E. R., & Dunwiddie, A. E. (2014). Arithmetic practice can be modified to promote understanding of mathematical equivalence. Journal of Educational Psychology. doi:10.1037/a0037687.

    Google Scholar 

  • Muldoon, K. P., Lewis, C., & Berridge, D. (2007). Predictors of early numeracy: is there a place for mistakes when learning about number? British Journal of Developmental Psychology, 25, 543–558. doi:10.1348/026151007x174501.

    Article  Google Scholar 

  • National Mathematics Advisory Panel. (2008). Foundations of success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.

    Google Scholar 

  • NCTM. (1989). Curriculum and evaluation standards for school mathematics. Reston: NCTM.

    Google Scholar 

  • NCTM. (2000). Principles and standards for school mathematics. Reston: NCTM.

    Google Scholar 

  • NCTM. (2014). Principles to actions: Ensuring mathematical success for all. Reston: National Council of Teachers of Mathematics, Inc.

    Google Scholar 

  • Perry, M. (1991). Learning and transfer: instructional conditions and conceptual change. Cognitive Development, 6, 449–468. doi:10.1016/0885-2014(91)90049-J.

    Article  Google Scholar 

  • Pesek, D. D., & Kirshner, D. (2000). Interference of instrumental instruction in subsequent relation learning. Journal for Research in Mathematics Education, 31, 524–540. doi:10.2307/749885.

    Article  Google Scholar 

  • Renkl, A., Stark, R., Gruber, H., & Mandl, H. (1998). Learning from worked-out examples: the effects of example variability and elicited self-explanations. Contemporary Educational Psychology, 23, 90–108. doi:10.1006/ceps.1997.0959.

    Article  Google Scholar 

  • Resnick, L. B., & Ford, W. W. (1981). The psychology of mathematics for instruction. Hillsdale: Lawrence Erlbaum Associates.

    Google Scholar 

  • Rittle-Johnson, B. (2006). Promoting transfer: effects of self-explanation and direct instruction. Child Development, 77, 1–15. doi:10.1111/j.1467-8624.2006.00852.x.

    Article  Google Scholar 

  • 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. doi:10.1037//0022-0663.91.1.175.

    Article  Google Scholar 

  • Rittle-Johnson, B., & Koedinger, K. R. (2009). Iterating between lessons concepts and procedures can improve mathematics knowledge. British Journal of Educational Psychology, 79, 483–500. doi:10.1348/000709908X398106.

    Article  Google Scholar 

  • Rittle-Johnson, B., & Schneider, M. (2015). Developing conceptual and procedural knowledge of mathematics. In R. C. Kadosh & A. Dowker (Eds.), Oxford handbook of numerical cognition (pp. 1102–1118). Oxford: Oxford University Press.

  • Rittle-Johnson, B., & Siegler, R. S. (1998). The relation between conceptual and procedural knowledge in learning mathematics: A review. In C. Donlan (Ed.), The development of mathematical skills (pp. 75–110). London: Psychology Press.

    Google Scholar 

  • Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: an iterative process. Journal of Educational Psychology, 93, 346–362. doi:10.1037//0022-0663.93.2.346.

    Article  Google Scholar 

  • Schneider, M., & Stern, E. (2010). The developmental relations between conceptual and procdural knowledge: a multimethod approach. Developmental Psychology, 46, 178–192. doi:10.1037/a0016701.

    Article  Google Scholar 

  • Schneider, M., Rittle-Johnson, B., & Star, J. R. (2011). Relations between conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Developmental Psychology. doi:10.1037/a0024997.

    Google Scholar 

  • Siegler, R. S., & Stern, E. (1998). Conscious and unconscious strategy discoveries: a microgenetic analysis. Journal of Experimental Psychology: General, 127, 377–397. doi:10.1037/0096-3445.127.4.377.

    Article  Google Scholar 

  • Sowder, J. T. (1998). What are the “math wars” in California all about? Reasons and perspectives. Retrieved June 1, 2005, from the Mathematically Sane Web site:http://mathematicallysane.com/analysis/mathwars.

  • Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36, 404–411. doi:10.1037/a0024997.

    Google Scholar 

  • Star, J. R. (2007). Foregrounding procedural knowledge. Journal for Research in Mathematics Education, 38, 132–135.

    Google Scholar 

  • Star, J. R., & Stylianides, G. J. (2013). Procedural and conceptual knowledge: exploring the gap between knowledge type and knowledge quality. Canadian Journal of Science, Mathematics, and Technology Education, 13, 169–181. doi:10.1080/14926156.2013.784828.

    Article  Google Scholar 

  • Vukovic, R. K., Fuchs, L. S., Geary, D. C., Jordan, N. C., Gersten, R., & Siegler, R. S. (2014). Sources of individual differences in children’s understanding of fractions. Child Development, 85, 1461–1476. doi:10.1111/cdev.12218.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Bethany Rittle-Johnson.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rittle-Johnson, B., Schneider, M. & Star, J.R. Not a One-Way Street: Bidirectional Relations Between Procedural and Conceptual Knowledge of Mathematics. Educ Psychol Rev 27, 587–597 (2015). https://doi.org/10.1007/s10648-015-9302-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10648-015-9302-x

Keywords

Navigation