Educational Psychology Review

, Volume 27, Issue 4, pp 587–597 | Cite as

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

  • Bethany Rittle-JohnsonEmail author
  • Michael Schneider
  • Jon R. Star
Review Article


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.


Conceptual knowledge Procedural knowledge Mathematics learning 


  1. 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.Google Scholar
  2. 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
  3. 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
  4. 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
  5. 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
  6. 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
  7. 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.CrossRefGoogle Scholar
  8. 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.CrossRefGoogle Scholar
  9. 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.CrossRefGoogle Scholar
  10. 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.CrossRefGoogle Scholar
  11. 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.CrossRefGoogle Scholar
  12. 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.
  13. Fuson, K. C. (1988). Children’s counting and concept of number. New York: Springer-Verlag.CrossRefGoogle Scholar
  14. 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.CrossRefGoogle Scholar
  15. 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
  16. 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.CrossRefGoogle Scholar
  17. 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.CrossRefGoogle Scholar
  18. Grouws, D. A., & Cebulla, K. J. (2000). Improving student achievement in mathematics. Geneva: International Academy of Education.Google Scholar
  19. Halford, G. S. (1993). Children’s understanding: The development of mental models. Hillsdale: Erlbaum.Google Scholar
  20. 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.CrossRefGoogle Scholar
  21. 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.CrossRefGoogle Scholar
  22. 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.CrossRefGoogle Scholar
  23. 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.CrossRefGoogle Scholar
  24. 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
  25. 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
  26. 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.CrossRefGoogle Scholar
  27. Kamii, C., & Dominick, A. (1997). To teach or not to teach algorithms. The Journal of Mathematical Behavior, 16, 51–61.CrossRefGoogle Scholar
  28. 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
  29. Karmiloff-Smith, A. (1992). Beyond modularity: A developmental perspective on cognitive science. Cambridge: MIT Press.Google Scholar
  30. Kilpatrick, J., Swafford, J. O., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington DC: National Academy Press.Google Scholar
  31. 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.CrossRefGoogle Scholar
  32. 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.CrossRefGoogle Scholar
  33. 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.CrossRefGoogle Scholar
  34. 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
  35. 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
  36. 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.CrossRefGoogle Scholar
  37. 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
  38. NCTM. (1989). Curriculum and evaluation standards for school mathematics. Reston: NCTM.Google Scholar
  39. NCTM. (2000). Principles and standards for school mathematics. Reston: NCTM.Google Scholar
  40. NCTM. (2014). Principles to actions: Ensuring mathematical success for all. Reston: National Council of Teachers of Mathematics, Inc.Google Scholar
  41. Perry, M. (1991). Learning and transfer: instructional conditions and conceptual change. Cognitive Development, 6, 449–468. doi: 10.1016/0885-2014(91)90049-J.CrossRefGoogle Scholar
  42. 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.CrossRefGoogle Scholar
  43. 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.CrossRefGoogle Scholar
  44. Resnick, L. B., & Ford, W. W. (1981). The psychology of mathematics for instruction. Hillsdale: Lawrence Erlbaum Associates.Google Scholar
  45. 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.CrossRefGoogle Scholar
  46. 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.CrossRefGoogle Scholar
  47. 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.CrossRefGoogle Scholar
  48. 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.Google Scholar
  49. 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
  50. 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.CrossRefGoogle Scholar
  51. 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.CrossRefGoogle Scholar
  52. 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
  53. 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.CrossRefGoogle Scholar
  54. 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:
  55. Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36, 404–411. doi: 10.1037/a0024997.Google Scholar
  56. Star, J. R. (2007). Foregrounding procedural knowledge. Journal for Research in Mathematics Education, 38, 132–135.Google Scholar
  57. 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.CrossRefGoogle Scholar
  58. 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.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Bethany Rittle-Johnson
    • 1
    Email author
  • Michael Schneider
    • 2
  • Jon R. Star
    • 3
  1. 1.Department of Psychology and Human Development, Peabody CollegeVanderbilt UniversityNashvilleUSA
  2. 2.Department of Educational PsychologyUniversity of TrierTrierGermany
  3. 3.Graduate School of EducationHarvard UniversityCambridgeUSA

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