, Volume 42, Issue 2, pp 149–161 | Cite as

Metacognition and mathematics education

  • Wolfgang SchneiderEmail author
  • Cordula Artelt
Original Article


The role of metacognition in mathematics education is analyzed based on theoretical and empirical work from the last four decades. Starting with an overview on different definitions, conceptualizations and models of metacognition in general, the role of metacognition in education, particularly in mathematics education, is discussed. The article emphasizes the importance of metacognition in mathematics education, summarizing empirical evidence on the relationships between various aspects of metacognition and mathematics performance. As a main result of correlational studies, it can be shown that the impact of declarative metacognition on mathematics performance is substantial (sharing about 15–20% of common variance). Moreover, numerous intervention studies have demonstrated that “normal” learners as well as those with especially low mathematics performance do benefit substantially from metacognitive instruction procedures.


Metacognition Mathematics achievement Training effects 


  1. Allardice, B. S., & Ginsburg, H. P. (1983). Children’s psychological difficulties in mathematics. In H. P. Ginsburg (Ed.), The development of mathematical thinking. New York: Academic Press.Google Scholar
  2. Artelt, C., Schiefele, U., & Schneider, W. (2001). Predictors of reading literacy. European Journal of Psychology of Education, 16, 363–383.CrossRefGoogle Scholar
  3. Best, D. L., & Ornstein, P. A. (1986). Children’s generation and communication of mnemonic organizational strategies. Developmental Psychology, 22, 845–853.CrossRefGoogle Scholar
  4. Braten, I., & Throndsen, I. S. (1998). Cognitive strategies in mathematics. Part II: Teaching a more advanced addition strategy to an eight-year-old girl with learning difficulties. Scandinavian Journal of Educational Research, 42, 151–171.CrossRefGoogle Scholar
  5. Brophy, J. (1986). Teaching and learning mathematics: Where research should be going. Journal for Research in Mathematics Education, 17, 323–346.CrossRefGoogle Scholar
  6. Brown, A. L. (1978). Knowing when, where, and how to remember: A problem of metacognition. In R. Glaser (Ed.), Advances in instructional psychology (pp. 77–165). Hillsdale, NJ: Erlbaum.Google Scholar
  7. Brown, A. L., Bransford, J. D., Ferrara, R. A., & Campione, J. C. (1983). Learning, remembering, and understanding. In J. H. Flavell & E. M. Markham (Eds.), Handbook of child psychology: Vol. 3. Cognitive development (pp. 77–166). New York: Wiley.Google Scholar
  8. Carr, M., Alexander, J., & Folds-Bennett, T. (1994). Metacognition and mathematics strategy use. Applied Cognitive Psychology, 8, 583–595.CrossRefGoogle Scholar
  9. Carr, M., & Jessup, D. L. (1995). Cognitive and metacognitive predictors of mathematics strategy use. Learning and Instruction, 7, 235–247.Google Scholar
  10. Carr, M., Kurtz, B. E., Schneider, W., Turner, L. A., & Borkowski, J. G. (1989). Strategy acquisition and transfer: Environmental influences on metacognitive development. Developmental Psychology, 25, 765–771.CrossRefGoogle Scholar
  11. Cavanaugh, J. C., & Perlmutter, M. (1982). Metamemory: A critical examination. Child Development, 53, 11–28.CrossRefGoogle Scholar
  12. Clarke, D. J., Waywood, A., & Stephens, M. (1993). Probing the structure of mathematical writing. Educational Studies in Mathematics, 25(3), 235–250.CrossRefGoogle Scholar
  13. Coffman, J. L., Ornstein, P. A., McCall, L. E., & Curran, P. J. (2008). Linking teachers’ memory-relevant language and the development of children’s memory skills. Developmental Psychology, 44, 1640–1654.CrossRefGoogle Scholar
  14. Cohors-Fresenborg, E., & Kaune, C. (2001). Mechanisms of the taking effect of metacognition in understanding processes in mathematics teaching. In G. Törner, R. Bruder, N. Neill, A. Peter-Koop, & B. Wollring (Eds.), Developments in mathematics education in German-speaking countries, selected papers from the annual conference on didactics of mathematics, Ludwigsburg (pp. 29–38). Hildesheim: Franzbecker.Google Scholar
  15. Cornoldi, C., Lucangeli, D., Caponi, B., Falco, G., Focchiatti, R., & Todeschini, M. (1995). Matematica e Metacognizione. Trento: Erickson.Google Scholar
  16. Desoete, A., Roeyers, H., & De Clercq, A. (2001). EPA2000: Een instrument om metacognitieve en rekenvaardigheden te meten. [EPA2000: An instrument for measuring metacognitive and arithmetic skills]. Kind en Adolescent, 22(2), 85–94.CrossRefGoogle Scholar
  17. Desoete, A., Roeyers, H., & De Clercq, A. (2003). Can offline metacognition enhance mathematical problem solving? Journal of Educational Psychology, 95(1), 188–200.CrossRefGoogle Scholar
  18. Desoete, A., & Veenman, M. (Eds.). (2006a). Metacognition in mathematics education. Haupauge, NY: Nova Science.Google Scholar
  19. Desoete, A., & Veenman, M. (2006b). Metacognitions in mathematics: Critical issues on nature, theory, assessment and treatment. In A. Desoete & M. Veenman (Eds.), Metacognition in mathematics education (pp. 1–10). Haupauge, NY: Nova Science.Google Scholar
  20. Flavell, J. H. (1971). First discussant’s comments: What is memory development the development of? Human Development, 14, 272–278.CrossRefGoogle Scholar
  21. Flavell, J. H. (1979). Metacognition and cognitive monitoring. A new area of cognitive-developmental inquiry. American Psychologist, 34, 906–911.CrossRefGoogle Scholar
  22. Flavell, J. H., Miller, P. H., & Miller, S. A. (2002). Cognitive development (4th ed.). Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
  23. Flavell, J. H., & Wellman, H. M. (1977). Metamemory. In R. Kail & J. Hagen (Eds.), Perspectives on the development of memory and cognition (pp. 3–33). Hillsdale, NJ: Erlbaum.Google Scholar
  24. Garofalo, J., & Lester, F. K. (1985). Metacognition, cognitive monitoring, and mathematical performance. Journal of Research in Mathematics Education, 16, 163–176.CrossRefGoogle Scholar
  25. Ghatala, E. S., Levin, J. R., Pressley, M., & Goodwin, D. (1986). A componential analysis of effects of derived and supplied strategy–utility information on children’s strategy selections. Journal of Experimental Child Psychology, 41, 76–92.CrossRefGoogle Scholar
  26. Goswami, U. (2008). Cognitive development—The learning brain (2nd ed.). Hove, UK: Psychology Press.Google Scholar
  27. Graesser, A. C., McNamara, D. S., & VanLehn, K. (2005). Scaffolding deep comprehension strategies through Point & Query, AutoTutor, and iSTART. Educational Psychologist, 40(4), 225–234.CrossRefGoogle Scholar
  28. Holland Joyner, M. H., & Kurtz-Costes, B. (1997). Metamemory development. In N. Cowan (Ed.), The development of memory in childhood (pp. 275–300). Hove, UK: Psychology Press.Google Scholar
  29. Kaune, C. (2006). Reflection and metacognition in mathematics education—tools for the improvement of teaching quality. Zentralblatt für Didaktik der Mathematik, 38(4), 350–360.CrossRefGoogle Scholar
  30. Kramarski, B. (2008). Promoting teachers’ algebraic reasoning and self-regulation with metacognitive guidance. Metacognition and Learning, 3(2), 83–99.CrossRefGoogle Scholar
  31. Kramarski, B., & Mevarech, Z. R. (2003). Enhancing mathematical reasoning in the classroom: The effects of cooperative learning and meta-cognitive training. American Educational Research Journal, 40, 281–310.CrossRefGoogle Scholar
  32. Kreutzer, M. A., Leonard, C., & Flavell, J. H. (1975). An interview study of children’s knowledge about memory. Monographs of the Society for Research in Child Development, 40(serial no. 159).Google Scholar
  33. Kron-Sperl, V., Schneider, W., & Hasselhorn, M. (2008). The development and effectiveness of memory strategies in kindergarten and elementary school: Findings from the Würzburg and Göttingen longitudinal studies. Cognitive Development, 23, 79–104.CrossRefGoogle Scholar
  34. Kuhn, D. (1999). Metacognitive development. In L. Balter & C. S. Tamis-LeMonda (Eds.), Child Psychology: A Handbook of Contemporary Issues (pp. 259–286). Philadelphia, PA: Psychology Press.Google Scholar
  35. Kuhn, D. (2000). Theory of mind, metacognition, and reasoning: A life-span perspective. In P. Mitchell & K. J. Riggs (Eds.), Children’s reasoning and the mind (pp. 301–326). Hove, UK: Psychology Press.Google Scholar
  36. Lester, F. K. (1982). Building bridges between psychological and mathematics education research on problem solving. In F. K. Lester & J. Garofalo (Eds.), Mathematical problem solving (pp. 55–85). Philadelphia: The Franklin Institute Press.Google Scholar
  37. Lockl, K., & Schneider, W. (2002). Developmental trends in children’s feeling-of-knowing judgements. International Journal of Behavioral Development, 26, 327–333.CrossRefGoogle Scholar
  38. Lucangeli, D., & Cornoldi, C. (1997). Mathematics and metacognition: What is the nature of the relationship? Mathematical Cognition, 3, 121–139.CrossRefGoogle Scholar
  39. Mevarech, Z. R., & Kramarski, B. (1997). IMPROVE: A multidimensional method for teaching mathematics in heterogeneous classrooms. American Educational Research Journal, 34, 365–394.Google Scholar
  40. Mevarech, Z. R., & Kramarski, B. (2003). The effects of metacognitive training versus worked-out examples on students’ mathematical reasoning. British Journal of Educational Psychology, 73(4), 449–471.CrossRefGoogle Scholar
  41. Mevarech, Z. R., Tabuk, A., & Sinai, O. (2006). Meta-cognitive instruction in mathematics classrooms: Effects on the solution of different kinds of problems. In A. Desoete & M. Veenman (Eds.), Metacognition in mathematics education (pp. 73–81). Haupauge, NY: Nova Science.Google Scholar
  42. Moely, B. E., Santulli, K. A., & Obach, M. S. (1995). Strategy instruction, metacognition, and motivation in the elementary school classroom. In F. E. Weinert & W. Schneider (Eds.), Memory performance and competencies: Issues in growth and development (pp. 301–321). Mahwah, NJ: Erlbaum.Google Scholar
  43. National Institute of Child Health and Human Development (2000). Report of the National Reading Panel. Teaching children to read: an evidence-based assessment of the scientific research literature on reading and its implications for reading instruction: Reports of the subgroups (NIH Publication No. 00-4754). Washington, DC: U.S. Government Printing Office.Google Scholar
  44. Nelson, T. O., & Narens, L. (1994). Why investigate metacognition? In J. Metcalfe & A. P. Shimamura (Eds.), Metacognition. Knowing about knowing (pp. 1–25). Cambridge, MA: MIT Press.Google Scholar
  45. Nunes, T., Schliemann, A. D., & Carraher, D. W. (1993). Street mathematics and school mathematics. Cambridge, UK: Cambridge University Press.Google Scholar
  46. OECD (2004). Learning for tomorrow’s world. First results from PISA 2003. Paris: OECD.Google Scholar
  47. Palincsar, A. S. (1986). The role of dialogue in providing scaffolded instruction. Educational Psychologist, 21, 73–98.CrossRefGoogle Scholar
  48. Palincsar, A. S., & Brown, A. L. (1984). Reciprocal teaching of comprehension-fostering and comprehension-monitoring activities. Cognition and Instruction, 1, 117–175.CrossRefGoogle Scholar
  49. Paris, S. G., & Oka, E. R. (1986). Children’s reading strategies, metacognition, and motivation. Developmental Review, 6, 25–56.CrossRefGoogle Scholar
  50. Pintrich, P., Wolters, C. A., & Baxter, G. P. (2000). Assessing metacognition and self-regulated learning. In G. Schraw & J. C. Impara (Eds.), Issues in the measurement of metacognition (pp. 43–97). Lincoln, NE: Buros Institute of Mental Measurements, University of Nebraska Press.Google Scholar
  51. Polya, G. (1957). How to solve it (2nd ed.). New York: Doubleday.Google Scholar
  52. Polya, G. (1973). Induction and analogy in mathematics. Princeton, NJ: Princeton University Press.Google Scholar
  53. Pressley, M. (1986). The relevance of the Good Strategy User model to the teaching of mathematics. Educational Psychologist, 21, 139–161.CrossRefGoogle Scholar
  54. Pressley, M. (1995). What is intellectual development about in the 1990s? In F. E. Weinert & W. Schneider (Eds.), Memory performance and competencies: Issues in growth and development (pp. 1–25). Hillsdale, NJ: Erlbaum.Google Scholar
  55. Pressley, M. (2002). Comprehension strategies instruction. In C. C. Block & M. Pressley (Eds.), Comprehension instruction: Research-based best practices (pp. 11–27). New York: Guilford Press.Google Scholar
  56. Pressley, M., Borkowski, J. G., & O’Sullivan, J. T. (1985). Children’s metamemory and the teaching of memory strategies. In D. L. Forrest-Pressley, G. E. MacKinnon, & T. G. Waller (Eds.), Metacognition, cognition, and human performance (Vol. 1, pp. 111–153). Orlando, FL: Academic Press.Google Scholar
  57. Pressley, M., Borkowski, J. G., & Schneider, W. (1987). Cognitive strategies: Good strategy user’s coordinate metacognition and knowledge. In R. Vasta & G. Whitehurst (Eds.), Annals of Child Development (Vol. 5, pp. 89–129). New York: JAI.Google Scholar
  58. Pressley, M., Borkowski, J. G., & Schneider, W. (1989). Good information processing: What it is and what education can do to promote it. International Journal of Educational Research, 13, 857–867.CrossRefGoogle Scholar
  59. Renkl, A. (1996). Träges Wissen: Wenn Erlerntes nicht genutzt wird. Psychologische Rundschau, 47, 78–92.Google Scholar
  60. Russell, R. L., & Ginsburg, H. P. (1981). Cognitive analysis of children’s mathematical difficulties. Rochester, NY: University of Rochester.Google Scholar
  61. Schlagmüller, M., & Schneider, W. (2007). WLST-12. Würzburger Lesestrategie Wissenstest für die Klassen 7 bis 12. Göttingen: Hogrefe.Google Scholar
  62. Schneider, W. (1999). The development of metamemory in children. In D. Gopher & A. Koriat (Eds.), Attention and performance XVII: Cognitive regulation of performance: Interaction of theory and application (pp. 487–513). Cambridge, MA: MIT Press.Google Scholar
  63. Schneider, W. (2010). Memory development in childhood and adolescence. In U. Goswami (Ed.), The Blackwell handbook of cognitive development. London, UK: Blackwell (in press).Google Scholar
  64. Schneider, W., Körkel, J., & Weinert, F. E. (1987). The effects of intelligence, self-concept, and attributional style on metamemory and memory behavior. International Journal of Behavioral Development, 10, 281–299.Google Scholar
  65. Schneider, W., & Lockl, K. (2002). The development of metacognitive knowledge in children and adolescents. In T. J. Perfect & B. L. Schwartz (Eds.), Applied metacognition (pp. 224–257). Cambridge, UK: Cambridge University Press.Google Scholar
  66. Schneider, W., & Lockl, K. (2008). Procedural metacognition in children: Evidence for developmental trends. In J. Dunlosky & B. Bjork (Eds.), A handbook of memory and metacognition. Mahwah, NY: Erlbaum.Google Scholar
  67. Schneider, W., & Pressley, M. (1997). Memory development between 2 and 20. Hillsdale, NJ: Erlbaum.Google Scholar
  68. Schoenfeld, A. H. (1983). Episodes and executive decisions in mathematical problem solving. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 345–395). New York: Academic Press.Google Scholar
  69. Schoenfeld, A. H. (1987). What’s all that fuss about metacognition? In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 189–215). Hillsdale, NJ: Erlbaum.Google Scholar
  70. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). New York: Macmillan.Google Scholar
  71. Schraw, G. (1994). The effect of metacognitive knowledge on local and global monitoring. Contemporary Educational Psychology, 19, 143–154.CrossRefGoogle Scholar
  72. Silver, E. A. (1982). Knowledge organization and mathematical problem solving. In F. K. Lester & J. Garofalo (Eds.), Mathematical problem solving (pp. 15–25). Philadelphia: Franklin Institute Press.Google Scholar
  73. Sjuts, J. (2002). Metacognition in mathematics lessons. In H.-G. Weigand et al. (Eds.): Developments in mathematics education in German-speaking countries. Selected papers from the annual conference on didactics of mathematics, Bern, 1999 (pp. 76–87). Hildesheim: Verlag Franzbecker.Google Scholar
  74. Stillman, G. A., & Galbraith, P. L. (1998). Applying mathematics with real world connections: Metacognitive characteristics of secondary students. Educational Studies in Mathematics, 36(2), 157–195.CrossRefGoogle Scholar
  75. Teong, S. K. (2003). The effect of metacognitive training on mathematical word-problem solving. Journal of Computer Assisted Learning, 19(1), 46–55.CrossRefGoogle Scholar
  76. Van Luit, J. E. H., & Kroesbergen, E. H. (2006). Teaching metacognitive skills to students with mathematical disabilities. In A. Desoete & M. Veenman (Eds.), Metacognition in mathematics education (pp. 177–190). Haupauge, NY: Nova Science.Google Scholar
  77. Veenman, M. V. J. (2006). The role of intellectual and metacognitive skills in math problem solving. In A. Desoete & M. Veenman (Eds.), Metacognition in mathematics education (pp. 35–50). Haupauge, NY: Nova Science.Google Scholar
  78. Verschaffel, L. (1999). Realistic mathematical modelling and problem solving in the upper elementary school: Analysis and improvement. In J. H. M. Hamers, J. E. H. Van Luit, & B. Csapo (Eds.), Teaching and learning thinking skills. Contexts of learning (pp. 215–240). Lisse: Swets & Zeitlinger.Google Scholar
  79. Wang, M. C., Haertel, G. D., & Walberg, H. J. (1993). Toward a knowledge base for school learning. Review of Educational Research, 63, 249–294.Google Scholar
  80. Zimmerman, B. J., & Tsikalas, K. E. (2005). Can computer-based learning environments (CBLEs) be used as self-regulatory tools to enhance learning? Educational Psychologist, 40(4), 267–271.CrossRefGoogle Scholar

Copyright information

© FIZ Karlsruhe 2010

Authors and Affiliations

  1. 1.Department of PsychologyUniversity of WürzburgWürzburgGermany
  2. 2.Department of Educational ResearchUniversity of BambergBambergGermany

Personalised recommendations