Metacognition and Learning

, Volume 14, Issue 2, pp 167–187 | Cite as

Metacognitive monitoring and help-seeking decisions on mathematical equivalence problems

  • Lindsey J. NelsonEmail author
  • Emily R. Fyfe


Metacognition is central to children’s cognitive development. However, there is conflicting evidence about children’s ability to accurately monitor their performance and subsequently control their behavior. This is of particular interest for mathematics topics on which children exhibit persistent misconceptions—that is, when children’s knowledge of a topic is inaccurate, yet resistant to change. This study investigated elementary school children’s metacognitive regulation on mathematical equivalence problems (N = 52, ages 6.7–9.8 years), including their ability to accurately monitor their certainty and their ability to control their behavior by making strategic help-seeking decisions. Results revealed that children were exceedingly confident—even when their answers were incorrect—resulting in relatively low accurate monitoring scores. However, their help-seeking decisions were largely strategic—reflecting children’s tendency to not ask for help when feeling confident—resulting in relatively high control scores. Additionally, individual differences in accurate monitoring and in strategic control were positively correlated with children’s comprehensive knowledge of mathematical equivalence, and the correlation with accurate monitoring held up after controlling for baseline accuracy, certainty, and help-seeking. Collectively, these results suggest that children may face unique, but critically important, metacognitive challenges when solving mathematical equivalence problems.


Metacognition Monitoring Help-seeking Mathematical equivalence 



The authors thank Nicholas Vest for help with data collection as well as the teachers and students at the participating schools.


Support for this research was provided by the Eunice Kennedy Shriver National Institute of Child Health and Human Development of the National Institutes of Health under Award Number T32HD007475. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

Compliance with ethical standards

Informed consent

Informed consent was obtained from participants’ parents and all children assented to participate.

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Ackerman, R., & Thompson, V. A. (2015). Meta-reasoning. In A. Feeney & V. Thompson (Eds.), Reasoning as memory (pp. 164–182). New York, NY: Psychology Press.Google Scholar
  2. Baars, M., Van Gog, T., de Bruin, A., & Paas, F. (2014). Effects of problem solving after worked example study on primary school children's monitoring accuracy. Applied Cognitive Psychology, 28(3), 382–391. Scholar
  3. Baars, M., Van Gog, T., de Bruin, A., & Paas, F. (2018). Accuracy of primary school children’s immediate and delayed judgments of learning about problem-solving tasks. Studies in Educational Evaluation, 58, 51–59. Scholar
  4. Baten, E., Praet, M., & Desoete, A. (2017). The relevance and efficacy of metacognition for instructional design in the domain of mathematics. ZDM, 49, 613–623. Scholar
  5. Blanton, M., Stephens, A., Knuth, E., Gardiner, A. M., Lsler, I., & Kim, J. S. (2015). The development of children’s early algebraic thinking: the impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, 46, 39–87. Scholar
  6. Boekaerts, M. (1997). Self-regulated learning: a new concept embraced by researchers, policy makers, educators, teachers, and students. Learning and Instruction, 7, 161–186. Scholar
  7. Byrd, C. E., McNeil, N. M., Chesney, D. L., & Matthews, P. G. (2015). A specific misconception of the equal sign acts as a barrier to children’s learning of early algebra. Learning and Individual Differences, 38, 61–67. Scholar
  8. Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth: Heinemann.Google Scholar
  9. Coughlin, C., Hembacher, E., Lyons, K. E., & Ghetti, S. (2015). Introspection on uncertainty and judicious help-seeking during the preschool years. Developmental Science, 18, 957–971. Scholar
  10. De Clercq, A., Desoete, A., & Roeyers, H. (2000). EPA2000: a multilingual, programmable computer assessment of off-line metacognition in children with mathematical-learning disabilities. Behavior Research Methods, Instruments, & Computers, 32(2), 304–311. Scholar
  11. Desoete, A. (2008). Multi-method assessment of metacognitive skills in elementary school children: how you test is what you get. Metacognition and Learning, 3(3), 189–206. Scholar
  12. Desoete, A., Roeyers, H., & Buysse, A. (2001). Metacognition and mathematical problem solving in grade 3. Journal of Learning Disabilities, 34(5), 435–447. Scholar
  13. Desoete, A., Roeyers, H., & De Clercq, A. (2003). Can offline metacognition enhance mathematical problem solving? Journal of Educational Psychology, 95(1), 188–200. Scholar
  14. Desoete, A., Roeyers, H., & Huylebroeck, A. (2006). Metacognitive skills in Belgian third grade children (age 8 to 9) with and without mathematical learning disabilities. Metacognition and Learning, 1(2), 119–135. Scholar
  15. Destan, N., Hembacher, E., Ghetti, S., & Roebers, C. M. (2014). Early metacognitive abilities: the interplay of monitoring and control processes in 5-to 7-year-old children. Journal of Experimental Child Psychology, 126, 213–228. Scholar
  16. Destan, N., Spiess, M. A., de Bruin, A., van Loon, M., & Roebers, C. M. (2017). 6-and 8-year-olds’ performance evaluations: do they differ between self and unknown others? Metacognition and Learning, 12(3), 315–336. Scholar
  17. Dignath, C., & Büttner, G. (2008). Components of fostering self-regulated learning among students. a meta-analysis on intervention studies at primary and secondary school level. Metacognition and Learning, 3(3), 231–264. Scholar
  18. Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P., et al. (2007). School readiness and later achievement. Developmental Psychology, 43(6), 1428. Scholar
  19. Erickson, S., & Heit, E. (2015). Metacognition and confidence: comparing math to other academic subjects. Frontiers in Psychology, 6, 742. Scholar
  20. Flavell, J. H. (1976). Metacognitive aspects of problem solving. In L. B. Resnick (Ed.), The nature of intelligence (pp. 231–236). Hillsdale: Erlbaum.Google Scholar
  21. Fyfe, E. R., & Rittle-Johnson, B. (2016). Feedback both helps and hinders learning: The causal role of prior knowledge. Journal of Educational Psychology, 108(1), 82. Scholar
  22. Fyfe, E. R., Matthews, P. G., & Amsel, E. (2017a). College students’ knowledge of the equal sign and its relation to solving equations. In E. Galindo & J. Newton (Eds.), Proceedings of the 39th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 279–282). Indianapolis, IN: Hoosier Association of Mathematics Teacher Educators.Google Scholar
  23. Fyfe, E. R., Evans, J. L, Matz, L., Hunt, K., & Alibali, M. W. (2017b). Relations between patterning skill and differing aspects of early mathematics knowledge. Cognitive Development, 44, 1–11. Scholar
  24. García, T., Rodríguez, C., González-Castro, P., González-Pienda, J. A., & Torrance, M. (2016). Elementary students’ metacognitive processes and post-performance calibration on mathematical problem-solving tasks. Metacognition and Learning, 11(2), 139–170. Scholar
  25. Garrett, A. J., Mazzocco, M. M., & Baker, L. (2006). Development of the metacognitive skills of prediction and evaluation in children with or without math disability. Learning Disabilities Research & Practice, 21(2), 77–88. Scholar
  26. Gascoine, L., Higgins, S., & Wall, K. (2017). The assessment of metacognition in children aged 4–16 years: a systematic review. Review of Education, 5(1), 3–57. Scholar
  27. Järvelä, S. (2011). How does help seeking help? – New prospects in a variety of contexts. Learning and Instruction, 21(2), 297–299. Scholar
  28. Karabenick, S. A., & Berger, J. (2013). Chapter 8: Help seeking as a self-regulated learning strategy. In H. Bembenutty, T. J. Cleary, & A. Kitsantas (Eds.), Applications of self-regulated learning across diverse disciplines. A tribute to Barry J. Zimmerman (pp. 237–261). Charlotte: IAP.Google Scholar
  29. Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12, 317–326. Scholar
  30. Koriat, A., Nussinson, R., & Ackerman, R. (2014). Judgments of learning depend on how learners interpret study effort. Journal of Experimental Psychology: Learning, Memory, and Cognition, 40(6), 1624. Scholar
  31. Kramarski, B., & Mevarech, Z. R. (2003). Enhancing mathematical reasoning in the classroom: The effects of cooperative learning and metacognitive training. American Educational Research Journal, 40(1), 281–310. Scholar
  32. Lipko, A. R., Dunlosky, J., & Merriman, W. E. (2009). Persistent overconfidence despite practice: The role of task experience in preschoolers’ recall predictions. Journal of Experimental Child Psychology, 103(2), 152–166. Scholar
  33. Lockl, K., & Schneider, W. (2003). Metacognitive monitoring and self-control processes for children’s allocation of study time. Zeitschrift fur Padagogische Psychologie, 17, 173–182. .CrossRefGoogle Scholar
  34. Lucangeli, D., & Cornoldi, C. (1997). Mathematics and metacognition: what is the nature of the relationship? Mathematical Cognition, 3(2), 121–139. Scholar
  35. Lyons, K. E., & Ghetti, S. (2013). I don't want to pick! Introspection on uncertainty supports early strategic behavior. Child Development, 84(2), 726–736. Scholar
  36. Marulis, L. M., Palincsar, A. S., Berhenke, A. L., & Whitebread, D. (2016). Assessing metacognitive knowledge in 3–5 year olds: the development of a metacognitive knowledge interview (McKI). Metacognition and Learning, 11(3), 339–368. Scholar
  37. Matthews, P. G., & Fuchs, L. S. (2018). Keys to the gate? Equal sign knowledge at second grade predicts fourth-grade algebra competence. Child Development.
  38. McNeil, N. M. (2014). A change-resistance account of children’s difficulties understanding mathematical equivalence. Child Development Perspectives, 8, 42–47. Scholar
  39. 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. Scholar
  40. McNeil, N. M., Weinberg, A., Hattikudur, S., Stephens, A. C., Asquith, P., Knuth, E. J., & Alibali, M. W. (2010). A is for apple: Mnemonic symbols hinder the interpretation of algebraic expressions. Journal of Educational Psychology, 102(3), 625–634. Scholar
  41. 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(5), 1620–1633. Scholar
  42. McNeil, N. M., Chesney, D. L., Matthews, P. G., Fyfe, E. R., Petersen, L. A., & Dunwiddie, A. E. (2012). It pays to be organized: organizing arithmetic practice around equivalent values facilitates understanding of math equivalence. Journal of Educational Psychology, 104(4), 1109–1121. Scholar
  43. McNeil, N. M., Fyfe, E. R., & Dunwiddie, A. E. (2015). Arithmetic practice can be modified to promote understanding of mathematical equivalence. Journal of Educational Psychology, 107(2), 423–436. Scholar
  44. McNeil, N. M., Hornburg, C. B., Devlin, B. L., Carrazza, C., & McKeever, M. O. (2017). Consequences of individual differences in children’s formal understanding of mathematical equivalence. Child Development, 90, 940–956. Scholar
  45. Metcalfe, J. (2009). Metacognitive judgments and control of study. Current Directions in Psychological Science, 18(3), 159–163. Scholar
  46. Mevarech, Z., & Fridkin, S. (2006). The effects of IMPROVE on mathematical knowledge, mathematical reasoning and meta-cognition. Metacognition and Learning, 1(1), 85–97. Scholar
  47. 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. Scholar
  48. Nelson, T. O., & Narens, L. (1990). Metamemory: A theoretical framework and new findings. In G. H. Bower (Ed.), The psychology of learning and instruction: Advances in research and theory (Vol. 26, pp. 125–141). New York: Academic Press.Google Scholar
  49. Nelson-Le Gall, S. (1985). Help-seeking behavior in learning. American Educational Research Association, 12, 55–90. Scholar
  50. Newman, R. S. (2000). Social influences on the development of children's adaptive help seeking: the role of parents, teachers, and peers. Developmental Review, 20(3), 350–404. Scholar
  51. NGA Center & CCSSO. (2010). Common core state standards for mathematics. Washington, DC: Author. Retrieved August 23, 2019 from
  52. Özsoy, G., & Ataman, A. (2009). The effect of metacognitive strategy training on mathematical problem solving achievement. International Electronic Journal of Elementary Education, 1(2), 67–82.Google Scholar
  53. Pennequin, V., Sorel, O., Nanty, I., & Fontaine, R. (2010). Metacognition and low achievement in mathematics: the effect of training in the use of metacognitive skills to solve mathematical word problems. Thinking & Reasoning, 16(3), 198–220. Scholar
  54. Powell, S. R. (2012). Equations and the equal sign in elementary mathematics textbooks. The Elementary School Journal, 112, 627–648. Scholar
  55. Powell, S. R., & Fuchs, L. S. (2010). Contribution of equal-sign instruction beyond word-problem tutoring for third-grade students with mathematics difficulty. Journal of Educational Psychology, 102, 381–394. Scholar
  56. Raaijmakers, S. F., Baars, M., Schaap, L., Paas, F., & van Gog, T. (2017). Effects of performance feedback valence on perceptions of invested mental effort. Learning and Instruction, 51, 36–46. Scholar
  57. Raaijmakers, S. F., Baars, M., Paas, F., van Merriënboer, J. J., & van Gog, T. (2019). Effects of self-assessment feedback on self-assessment and task-selection accuracy. Metacognition and Learning, 26, 1–22. Scholar
  58. Rinne, L. F., & Mazzocco, M. M. (2014). Knowing right from wrong in mental arithmetic judgments: calibration of confidence predicts the development of accuracy. PLoS One, 9(7). Scholar
  59. Rittle-Johnson, B. (2006). Promoting transfer: effects of self-explanation and direct instruction. Child Development, 77, 1–15. Scholar
  60. Rittle-Johnson, B., Matthews, P. G., Taylor, R. S., & McEldoon, K. L. (2011). Assessing knowledge of mathematical equivalence: a construct-modeling approach. Journal of Educational Psychology, 103(1), 85–104. Scholar
  61. Roderer, T., & Roebers, C. M. (2010). Explicit and implicit confidence judgments and developmental differences in metamemory: an eye-tracking approach. Metacognition and Learning, 5(3), 229–250. Scholar
  62. Roebers, C. M. (2017). Executive function and metacognition: towards a unifying framework of cognitive self-regulation. Developmental Review, 45, 31–51. Scholar
  63. Roebers, C. M., & Spiess, M. (2017). The development of metacognitive monitoring and control in second graders: a short-term longitudinal study. Journal of Cognition and Development, 18(1), 110–128. Scholar
  64. Roebers, C. M., von der Linden, N., & Howie, P. (2007). Favourable and unfavourable conditions for children's confidence judgments. British Journal of Developmental Psychology, 25(1), 109–134. Scholar
  65. Schneider, W. (1998). Performance prediction in young children: effects of skill, metacognition and wishful thinking. Developmental Science, 1(2), 291–297. Scholar
  66. Schneider, W., & Artelt, C. (2010). Metacognition and mathematics education. ZDM, 42(2), 149–161. Scholar
  67. Shin, H., Bjorklund, D. F., & Beck, E. F. (2007). The adaptive nature of children's overestimation in a strategic memory task. Cognitive Development, 22(2), 197–212. Scholar
  68. Thiede, K. W., Anderson, M., & Therriault, D. (2003). Accuracy of metacognitive monitoring affects learning of texts. Journal of Educational Psychology, 95(1), 66–73. Scholar
  69. van Loon, M. H., de Bruin, A. B., van Gog, T., & van Merriënboer, J. J. (2013). Activation of inaccurate prior knowledge affects primary-school students’ metacognitive judgments and calibration. Learning and Instruction, 24, 15–25. Scholar
  70. Vo, V. A., Li, R., Kornell, N., Pouget, A., & Cantlon, J. F. (2014). Young children bet on their numerical skills: metacognition in the numerical domain. Psychological Science, 25(9), 1712–1721. Scholar
  71. Wang, M. C., Haertel, G. D., & Walberg, H. J. (1990). What influences learning? A content analysis of review literature. The Journal of Educational Research, 84(1), 30–43. Scholar
  72. Whitebread, D., & Coltman, P. (2010). Aspects of pedagogy supporting metacognition and self- regulation in mathematical learning of young children: evidence from an observational study. ZDM, 42(2), 163–178. Scholar

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Authors and Affiliations

  1. 1.Department of Psychological and Brain SciencesIndiana UniversityBloomingtonUSA

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