Abstract
In this article we review the research on the flexible or adaptive use of solution strategies in school mathematics, with a focus on the most recent work in the field. After a short introduction, we provide an overview of the various ways in which strategy flexibility has been conceptualized and investigated in the research literature. Then we review the research that has looked at the relationship between strategy flexibility and task proficiency, followed by studies that analyzed the association of strategy flexibility and other learner variables, including learners’ age, general mathematical ability, prior knowledge, executive functions, gender, and affect. Studies addressing the socio-cultural and educational embeddedness of strategy flexibility are reviewed next, and, finally, we discuss the intervention studies that have tried to stimulate learners’ strategy flexibility by means of various instructional approaches. While this review reveals that strategy flexibility is increasingly recognized as an important and valuable construct in research and practice of mathematics education, and that recently substantial progress has been made in our understanding of this construct, there are many aspects of it that are still not well-understood and that need further investigation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
In this cross-national study, Spanish, Finnish, and Swedish middle and high school students’ procedural flexibility was examined, with the goal of determining whether and how students’ equation-solving accuracy and flexibility varied by country. The results revealed substantial within-country as well as between-country variation in students’ reliance on standard versus situationally flexible strategies.
This author investigated Dutch fourth graders’ adaptive use of the direct subtraction versus subtraction-by-addition strategy to solve multidigit subtractions. Children solved multidigit subtraction problems in one choice and in two no-choice conditions, either with random assignment to mental computation, written computation, or free choice between the two. One third of the children adaptively switched their strategy according to the numerical characteristics of the problems, and the likelihood to do so was highest in the mandatory mental computation condition.
The authors investigated elementary school children's use of direct subtraction and subtraction-by-addition when mentally solving multi-digit subtractions. Flemish fourth- to sixth-grade children of varying mathematical achievement levels were offered subtractions using a choice/no-choice design. The findings yielded additional evidence for the frequent, efficient, and adaptive use of SBA, even in younger and mathematically lower achieving children.
This study analyzed the effects on students of a professional development program aimed at improving algebra instruction that paid ample attention to supporting trainees in comparing and discussing multiple strategies (CDMS) when teaching linear equation solving. CDMS was found to increase how often teachers actually engaged their students in comparison and discussion of multiple strategies, as well as these teachers’ students’ flexibility in handling linear equations.
These authors investigated the learning opportunities provided by textbooks regarding adaptive expertise in multi-digit addition and subtraction and found large discrepancies in the textbooks' quality in this respect. Furthermore, data of a large-scale three-year longitudinal study showed an effect of the textbook quality on third-graders' adaptive expertise.
References
Acevedo Nistal, A., Van Dooren, W., Clarebout, G., Elen, J., & Verschaffel, L. (2009). Conceptualising, investigating and stimulating representational flexibility in mathematical problem solving and learning: A critical review. ZDM-the International Journal on Mathematics Education, 41, 627–636. https://doi.org/10.1007/s11858-009-0189-1
Bakker, M., Torbeyns, J., Verschaffel, L., & De Smedt, B. (2022). The mathematical, motivational, and cognitive characteristics of high mathematics achievers in primary school. Journal of Educational Psychology, 114, 992–1004. https://doi.org/10.1037/edu0000678
Bakker, M., Torbeyns, J., Wijns, N., Verschaffel, L., & De Smedt, B. (2019). Gender equality in four- and five-year-old preschoolers’ early numerical competencies. Developmental Science, 22(1), e12718. https://doi.org/10.1111/desc.12718
Berk, D., Taber, S. B., Gorowara, C. C., & Poetzl, C. (2009). Developing prospective elementary teachers’ flexibility in the domain of proportional reasoning. Mathematical Thinking and Learning, 11, 113–135. https://doi.org/10.1080/10986060903022714
Bjorklund, D. F., & Rosenblum, K. E. (2002). Context effects in children’s selection and use of simple arithmetic strategies. Journal of Cognition and Development, 3, 225–242. https://doi.org/10.1207/S15327647JCD0302_5
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. https://doi.org/10.1037/0022-0663.93.3.627
Bødtker Sunde, P. (2019). Strategies in single-digit addition: patterns and perspectives. [Doctoral dissertation, Aarhus University, Denmark]. Retrieved from: https://edu.au.dk/fileadmin/edu/phdafhandlinger/Pernille_B_Sunde_Thesis_2019.pdf. Accessed 20 Nov 2022
Bull, R., & Lee, K. (2014). Executive functioning and mathematics achievement. Child Development Perspectives, 8, 36–41. https://doi.org/10.1111/cdep.12059
Bye, J. K., Harsch, R. M., & Varma, S. (2022). Decoding fact fluency and strategy flexibility in solving one-step algebra problems: An individual differences analysis. Journal of Numerical Cognition, 8, 281–294. https://doi.org/10.5964/jnc.7093
Carr, M., & Jessup, D. L. (1997). Gender differences in first-grade mathematics strategy use: Social and metacognitive influences. Journal of Educational Psychology, 89, 318–328. https://doi.org/10.1037/0022-0663.89.2.318
De Smedt, B., Torbeyns, J., Stassens, N., Ghesquière, P., & Verschaffel, L. (2010). Frequency, efficiency and flexibility of indirect addition in two learning environments. Learning and Instruction, 20, 205–215. https://doi.org/10.1016/j.learninstruc.2009.02.020
del Olmo-Muñoz, J., González-Calero, J. A., Diago, P. D., Arnau, D., & Arevalillo-Herráez, M. (2022). Using intra-task flexibility on an intelligent tutoring system to promote arithmetic problem-solving proficiency. British Journal of Educational Technology, 53, 1976–1992. https://doi.org/10.1111/bjet.13228. in press.
Dowker, A., Sarkar, A., & Looi, C. Y. (2016). Mathematics anxiety: What have we learned in 60 years? Frontiers in Psychology, 7, 508. https://doi.org/10.3389/fpsyg.2016.00508
Durkin, K., Rittle-Johnson, B., Star, J. R., & Loehr, A. (2021). Comparing and discussing multiple strategies: An approach to improving algebra instruction. The Journal of Experimental Education. https://doi.org/10.1080/00220973.2021.1903377
Elia, I., den Heuvel-Panhuizen, M., & Kolovou, A. (2009). Exploring strategy use and strategy flexibility in nonroutine problem solving by primary school high achievers in mathematics. ZDM-the International Journal on Mathematics Education, 41, 605–618. https://doi.org/10.1007/s11858-009-0184-6
Ellis, S. (1997). Strategy choice in sociocultural context. Developmental Review, 17, 490–524. https://doi.org/10.1006/DREV.1997.0444
Fazio, L. K., DeWolf, M., & Siegler, R. S. (2016). Strategy use and strategy choice in fraction magnitude comparison. Journal of Experimental Psychology: Learning Memory and Cognition, 42, 1–16. https://doi.org/10.1037/xlm0000153
Fryer, R. G., & Levitt, S. D. (2010). An empirical analysis of the gender gap in mathematics. American Economic Journal: Applied Economics, 2, 210–240. https://doi.org/10.1257/app.2.2.210
Hatano, G. (2003). Foreword. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills (pp. xi–xii). Lawrence Erlbaum Associates.
Heinze, A., Arend, J., Gruessing, M., & Lipowsky, F. (2018). Instructional approaches to foster third graders’ adaptive use of strategies: An experimental study on the effects of two learning environments on multi-digit addition and subtraction. Instructional Science, 46, 869–891. https://doi.org/10.1007/s11251-018-9457-1
Heinze, A., Marschick, F., & Lipowsky, F. (2009). Addition and subtraction of three-digit numbers: Adaptive strategy use and the influence of instruction in German third grade. Mathematics Education, 41, 591–604. https://doi.org/10.1007/s11858-009-0205-5
Heirdsfield, A. M., Cooper, T., Mulligan, J., & Irons, C. (1999). Children's mental multiplication and division strategies. In O. Zaslavsky (Ed.), Proceedings of 23rd Conference of the International Group for the Psychology of Mathematics Education (pp. 89–96). Technion, Haifa.
Heirdsfield, A. M., & Cooper, T. J. (2002). Flexibility and inflexibility in accurate mental addition and subtraction: Two case studies. Journal of Mathematical Behavior, 21, 57–74. https://doi.org/10.1016/S0732-3123(02)00103-7
Hickendorff, M. (2018). Dutch sixth graders’ use of shortcut strategies in solving multidigit arithmetic problems. European Journal of Psychology of Education, 33, 577–594. https://doi.org/10.1007/s10212-017-0357-6
Hickendorff, M. (2020). Fourth graders’ adaptive strategy use in solving multidigit subtraction problems. Learning and Instruction, 67, 101311. https://doi.org/10.1016/j.learninstruc.2020.101311
Hickendorff, M., McMullen, J., & Verschaffel, L. (2022). Mathematical flexibility: Theoretical, methodological, and educational considerations. Journal of Numerical Cognition, 8, 326–334. https://doi.org/10.5964/jnc.10085
Hodzik, S., & Lemaire, P. (2011). Inhibition and shifting capacities mediate adults’ age-related differences in strategy selection and repertoire. Acta Psychologica, 137, 335–344. https://doi.org/10.1016/j.actpsy.2011.04.002
Hutchison, J. E., Lyons, I. M., & Ansari, D. (2018). More similar than different: Gender differences in children’s basic numerical skills are the exception not the rule. Child Development, 90, e66–e79. https://doi.org/10.1111/cdev.13044
Imbo, I., & Vandierendonck, A. (2007a). The development of strategy use in elementary school children: Working memory and individual differences. Journal of Experimental Child Psychology, 96, 284–309. https://doi.org/10.1016/j.jecp.2006.09.001
Imbo, I., & Vandierendonck, A. (2007b). The role of phonological and executive working memory resources in simple arithmetic strategies. European Journal of Cognitive Psychology, 19, 910–933. https://doi.org/10.1080/09541440601051571
Jiang, C., Hwang, S., & Cai, J. (2014). Chinese and Singaporean sixth-grade students’ strategies for solving problems about speed. Educational Studies in Mathematics, 87, 27–50. https://doi.org/10.1007/s10649-014-9559-x
Kabinet Vlaams minister van Onderwijs. (2017). Dalende trend resultaten wiskunde basisonderwijs vraagt om verder onderzoek [Decreases in children’s mathematical performance in primary school requires further investigation]. Retrieved from: https://www.hildecrevits.be/nieuws/dalende-trend-resultaten-wiskunde-basisonderwijs-vraagt-om-verder-onderzoek/. Accessed 20 Nov 2022
Keleş, T., & Yazgan, Y. (2021). Gifted eighth, ninth, tenth and eleventh graders’ strategic flexibility in non-routine problem solving. The Journal of Educational Research, 114, 332–345. https://doi.org/10.1080/00220671.2021.1937913
Klein, A. S., Beishuizen, M., & Treffers, A. (1998). The empty number line in Dutch second grades: Realistic versus gradual program design. Journal for Research in Mathematics Education, 29, 443–464. https://doi.org/10.2307/749861
Korten, L. (2017). The fostering of flexible mental calculation in an inclusive mathematics classroom during Mutual Learning. In T. Dooley & G. Gueudet (Eds.), Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (CERME 10) (pp. 362–2370). Institute of Education and ERME, Dublin City University. https://hal.archives-ouvertes.fr/CERME10/public/CERME10_Complete.pdf. Accessed 20 Nov 2022
Krutetskii, V. A. (1976). The psychology of mathematical ability in schoolchildren. University of Chicago Press.
Lemaire, P., & Siegler, R. S. (1995). Four aspects of strategic change: Contributions to children’s learning of multiplication. Journal of Experimental Psychology: General, 124, 83–97. https://doi.org/10.1037/0096-3445.124.1.83
Lindberg, S. M., Hyde, J. S., Petersen, J. L., & Linn, M. C. (2010). New trends in gender and mathematics performance: A meta-analysis. Psychological Bulletin, 136, 1123–1135. https://doi.org/10.1037/a0021276
Liu, R., Wang, J., Star, J. R., Zhen, R., Jiang, R., & Fu, X. (2018). Turning potential flexibility into flexible performance: Moderating effect of self-efficacy and use of flexible cognition. Frontiers in Psychology, 9, 646. https://doi.org/10.3389/fpsyg.2018.00646
Luwel, K., Onghena, P., Torbeyns, J., Schillemans, V., & Verschaffel, L. (2009). Strengths and weaknesses of the choice/no-choice method in research on strategy use. European Psychologist, 14, 351–362. https://doi.org/10.1027/1016-9040.14.4.351
Luwel, K., Verschaffel, L., Onghena, P., & De Corte, E. (2001). Strategic aspects of children’s numerosity judgement. European Journal of Psychology of Education, 16, 233–255. http://www.jstor.org/stable/23421419. Accessed 20 Nov 2022
Machi, L. A., & McEvoy, B. T. (2016). The literature review: six steps to success (3rd ed.). Berlin: Corwin, a SAGE Company.
Maciejewski, W. (2020). Between confidence and procedural flexibility in calculus. International Journal of Mathematical Education in Science and Technology, 53, 1733–1750. https://doi.org/10.1080/0020739X.2020.1840639
Maciejewski, W., & Star, J. (2016). Developing flexible procedural knowledge in undergraduate calculus. Research in Mathematics Education, 18, 299–316. https://doi.org/10.1080/14794802.2016.1148626
McMullen, J., Brezovszky, B., Hannula-Sormunen, M. M., Veermans, K., Rodríguez-Aflecht, G., Pongsakdi, N., & Lehtinen, E. (2017). Adaptive number knowledge and its relation to arithmetic and pre-algebra knowledge. Learning and Instruction, 49, 178–187. https://doi.org/10.1016/j.learninstruc.2017.02.001
Mercier, E. M., & Higgins, S. E. (2013). Collaborative learning with multi-touch technology: Developing adaptive expertise. Learning and Instruction, 25, 13–23. https://doi.org/10.1016/j.learninstruc.2012.10.004
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. NCTM.
National Council of Teachers of Mathematics. (2014). Procedural fluency in mathematics: A position of the National Council of Teachers of Mathematics. NCTM.
Nemeth, L., Werker, K., Arend, J., Vogel, S., & Lipowsky, F. (2019). Interleaved learning in elementary school mathematics: Effects on the flexible and adaptive use of subtraction strategies. Frontiers in Psychology, 10, 2296. https://doi.org/10.3389/fpsyg.2019.02296
Newton, K. J., Lange, K., & Booth, J. L. (2020). Mathematical flexibility: Aspects of a continuum and the role of prior knowledge. The Journal of Experimental Education, 88, 503–515. https://doi.org/10.1080/00220973.2019.1586629
Newton, K. J., Star, J. R., & Lynch, K. (2010). Understanding the development of flexibility in struggling algebra students. Mathematical Thinking and Learning, 12, 282–305. https://doi.org/10.1080/10986065.2010.482150
Nunes, T., Carraher, D. W., & Schliemann, A. D. (1985). Mathematics in the streets and in schools. British Journal of Developmental Psychology, 3, 21–29. https://doi.org/10.1111/j.2044-835x.1985.tb00951.x
Rechtsteiner, C., & Rathgeb-Schnierer, E. (2017). ”Zahlenblickschulung” as approach to develop flexibility in mental calculation in all students. Journal of Mathematics Education, 10, 1–16. https://doi.org/10.26711/007577152790001
Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99, 561–574. https://doi.org/10.1037/0022-0663.99.3.561
Rittle-Johnson, B., Star, J. R., & Durkin, K. (2012). Developing procedural flexibility: Are novices prepared to learn from comparing procedures? British Journal of Educational Psychology, 82, 436–455. https://doi.org/10.1111/j.2044-8279.2011.02037.x
Schneider, M., Rittle-Johnson, B., & Star, J. R. (2011). Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Developmental Psychology, 47, 1525–1538. https://doi.org/10.1037/a0024997
Schukajlow, S., Krug, A., & Rakoczy, K. (2015). Effects of prompting multiple solutions for modelling problems on students’ performance. Educational Studies in Mathematics, 89, 393–417. https://doi.org/10.1007/s10649-015-9608-0
Selter, C. (2001). Addition and subtraction of three-digit numbers: German elementary children’s success, methods and strategies. Educational Studies in Mathematics, 47, 145–173. https://doi.org/10.1023/A:1014521221809
Selter, C. (2009). Creativity, flexibility, adaptivity, and strategy use in mathematics. ZDM - the International Journal on Mathematics Education, 41, 619–625. https://doi.org/10.1007/s11858-009-0203-7
Shaw, S. T., Pogossian, A. A., & Ramirez, G. (2020). The mathematical flexibility of college students: The role of cognitive and affective factors. British Journal of Educational Psychology, 90, 981–996. https://doi.org/10.1111/bjep.12340
Sievert, H., van den Ham, A. K., Niedermeyer, I., & Heinze, A. (2019). Effects of mathematics textbooks on the development of primary school children’s adaptive expertise in arithmetic. Learning and Individual Differences, 74, 101716. https://doi.org/10.1016/j.lindif.2019.02.006
Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. Zentralblatt Für Didaktik Der Mathematik, 29, 75–80. https://doi.org/10.1007/s11858-997-0003-x
Smith III, J. P. (1995). Competent reasoning with rational numbers. Cognition and Instruction, 13, 3–50. https://doi.org/10.1207/s1532690xci1301_1
Star, J. R., & Newton, K. J. (2009). The nature and development of experts’ strategy flexibility for solving equations. Zentralblatt Für Didaktik Der Mathematik, 41, 557–567. https://doi.org/10.1007/s11858-009-0185-5
Star, J. R., Newton, K., Pollack, C., Kokka, K., Rittle-Johnson, B., & Durkin, K. (2015). Student, teacher, and instructional characteristics related to students’ gains in flexibility. Contemporary Educational Psychology, 41, 198–208. https://doi.org/10.1016/j.cedpsych.2015.03.001
Star, J. R., & Rittle-Johnson, B. (2008). Flexibility in problem solving: The case of equation solving. Learning and Instruction, 18, 565–579. https://doi.org/10.1016/J.LEARNINSTRUC.2007.09.018
Star, J. R., & Rittle-Johnson, B. (2009). It pays to compare: An experimental study on computational estimation. Journal of Experimental Child Psychology, 102, 408–426. https://doi.org/10.1016/j.jecp.2008.11.004
Star, J. R., & Seifert, C. (2006). The development of flexibility in equation solving. Contemporary Educational Psychology, 31, 280–300. https://doi.org/10.1016/j.cedpsych.2005.08.001
Star, J. R., Tuomela, D., Joglar-Prieto, N., Hästö, P., Palkki, R., Abánades, M. Á., Peljare, J., Jiang, R. H., Lijia, L., & Liu, R. D. (2022). Exploring students’ procedural flexibility in three countries. International Journal of STEM Education, 9, 1–18. https://doi.org/10.1186/s40594-021-00322-y
Threlfall, J. (2009). Strategies and flexibility in mental calculation. ZDM-the International Journal on Mathematics Education, 41, 541–555. https://doi.org/10.1007/s11858-009-0195-3
Torbeyns, J., Hickendorff, M., & Verschaffel, L. (2017). The use of number-based versus digit-based strategies on multi-digit subtractions: 9–12-year-olds’ strategy use profiles and task performances. Learning and Individual Differences, 58, 64–74. https://doi.org/10.1016/j.lindif.2017.07.004
Torbeyns, J., Peters, G., De Smedt, B., Ghesquière, P., & Verschaffel, L. (2018). Subtraction by addition strategy use in children of varying mathematical achievement level: A choice/no-choice study. Journal of Numerical Cognition, 4, 215–234. https://doi.org/10.5964/jnc.v4i1.77
Torbeyns, J., De Smedt, B., Ghesquière, P., & Verschaffel, L. (2009). Acquisition and use of shortcut strategies by traditionally schooled children. Educational Studies in Mathematics, 71, 1–17. https://doi.org/10.1007/s10649-008-9155-z
Van Der Auwera, S., Torbeyns, J., De Smedt, B., Verguts, G., & Verschaffel, L. (2022). The remarkably frequent, efficient, and adaptive use of the subtraction by addition strategy: A choice/no-choice study in fourth- to sixth-graders with varying mathematical achievement levels. Learning and Individual Differences. https://doi.org/10.1016/j.lindif.2021.102107
Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24, 335–359. https://doi.org/10.1007/BF03174765
Xu, L., Liu, R. D., Star, J. R., Wang, J., Liu, Y., & Zhen, R. (2017). Measures of potential flexibility and practical flexibility in equation solving. Frontiers in Psychology, 8, 1–13. https://doi.org/10.3389/fpsyg.2017.01368
Zhou, H., Aheto, D. L., Gao, Q., & Chen, W. (2021). Mathematical calculation ability of primary school children: A comparative study between Ghana and China. Journal of Psychology in Africa, 31, 286–291. https://doi.org/10.1080/14330237.2021.1928924
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Verschaffel, L. Strategy flexibility in mathematics. ZDM Mathematics Education 56, 115–126 (2024). https://doi.org/10.1007/s11858-023-01491-6
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11858-023-01491-6