Metacognitive monitoring and help-seeking decisions on mathematical equivalence problems
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.
KeywordsMetacognition 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 was obtained from participants’ parents and all children assented to participate.
Conflict of interest
The authors declare that they have no conflict of interest.
- 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
- 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. https://doi.org/10.5951/jresematheduc.46.1.0039.CrossRefGoogle Scholar
- Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth: Heinemann.Google Scholar
- 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. https://doi.org/10.3758/BF03207799.CrossRefGoogle Scholar
- 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
- 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
- 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. https://doi.org/10.1007/s11409-015-9139-1.CrossRefGoogle Scholar
- Järvelä, S. (2011). How does help seeking help? – New prospects in a variety of contexts. Learning and Instruction, 21(2), 297–299. https://doi.org/10.1016/j.learninstruc.2010.07.006.CrossRefGoogle Scholar
- 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
- Matthews, P. G., & Fuchs, L. S. (2018). Keys to the gate? Equal sign knowledge at second grade predicts fourth-grade algebra competence. Child Development. https://doi.org/10.1111/cdev.13144.
- 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. https://doi.org/10.1037/a0019105.CrossRefGoogle 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(5), 1620–1633. https://doi.org/10.1111/j.1467-8624.2011.01622.x.CrossRefGoogle Scholar
- 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. https://doi.org/10.1037/a0028997.CrossRefGoogle Scholar
- 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
- NGA Center & CCSSO. (2010). Common core state standards for mathematics. Washington, DC: Author. Retrieved August 23, 2019 from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf.
- 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
- 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. https://doi.org/10.1080/13546783.2010.509052.CrossRefGoogle Scholar
- 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. https://doi.org/10.1016/j.learninstruc.2016.12.002.CrossRefGoogle Scholar
- 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. https://doi.org/10.1016/j.learninstruc.2012.08.005.CrossRefGoogle Scholar