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, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
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.
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.
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.
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.
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.
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//0022-0618.104.22.1687.
Byrnes, J. P., & Wasik, B. A. (1991). Role of conceptual knowledge in mathematical procedural learning. Developmental Psychology, 27, 777–786. doi:10.1037//0012-1622.214.171.1247.
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.
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.
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.
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.
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.
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.
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.
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.
Grouws, D. A., & Cebulla, K. J. (2000). Improving student achievement in mathematics. Geneva: International Academy of Education.
Halford, G. S. (1993). Children’s understanding: The development of mental models. Hillsdale: Erlbaum.
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.
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.
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.
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.
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.
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.
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.
Kamii, C., & Dominick, A. (1997). To teach or not to teach algorithms. The Journal of Mathematical Behavior, 16, 51–61.
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.
Karmiloff-Smith, A. (1992). Beyond modularity: A developmental perspective on cognitive science. Cambridge: MIT Press.
Kilpatrick, J., Swafford, J. O., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington DC: National Academy Press.
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.
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.
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.
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.
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.
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.
National Mathematics Advisory Panel. (2008). Foundations of success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.
NCTM. (1989). Curriculum and evaluation standards for school mathematics. Reston: NCTM.
NCTM. (2000). Principles and standards for school mathematics. Reston: NCTM.
NCTM. (2014). Principles to actions: Ensuring mathematical success for all. Reston: National Council of Teachers of Mathematics, Inc.
Perry, M. (1991). Learning and transfer: instructional conditions and conceptual change. Cognitive Development, 6, 449–468. doi:10.1016/0885-2014(91)90049-J.
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.
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.
Resnick, L. B., & Ford, W. W. (1981). The psychology of mathematics for instruction. Hillsdale: Lawrence Erlbaum Associates.
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.
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//0022-06126.96.36.199.
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.
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.
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//0022-06188.8.131.526.
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.
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.
Siegler, R. S., & Stern, E. (1998). Conscious and unconscious strategy discoveries: a microgenetic analysis. Journal of Experimental Psychology: General, 127, 377–397. doi:10.1037/0096-34184.108.40.2067.
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.
Star, J. R. (2007). Foregrounding procedural knowledge. Journal for Research in Mathematics Education, 38, 132–135.
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.
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.
About this article
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
- Conceptual knowledge
- Procedural knowledge
- Mathematics learning