Actions with manipulatives support second graders’ learning about place-value concepts
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Helena P. Osana
Department of Education, Faculty of Arts and Science, Concordia University, Montreal, QC H3G 1M8, Canada
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Joel R. Levin
Department of Educational Psychology, The University of Arizona, Tucson, AZ 85721, USA
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Anne Lafay
Université Grenoble Alpes, Université Savoie Mont Blanc, CNRS, LPNC, 38000 Grenoble, France
Abstract
The aim of the present study was to examine the impact of conceptual transparency of mathematics manipulatives used in instruction on the learning of place-value concepts in typically developing second graders (n = 88) and those at-risk for mathematics learning disabilities (n = 29). We randomly assigned the children of each classification to three instructional conditions that varied according to the extent to which place-value concepts were made visible in the objects’ perceptual features. In one condition, the ten and hundreds denominations were already grouped and the ones were visible in each denomination; in the second condition the denominations were already grouped but the ones were not visible; in the third condition, we provided children with individual beads that could be attached in groups of tens and hundreds. We assessed the accuracy of the children’s representations of two- and three-digit numerals using manipulatives and their place-value knowledge on symbolic tasks. Contrary to our expectations, we found that the manipulatives requiring children to construct their own denominations were related to gains in the accuracy of the physical representations, but not to gains on the symbolic measures. We speculate that the actions involved in constructing the denominations provided opportunities for children to encode the materials’ salient features in ways that led to the greatest benefits. We suggest that teachers ensure that students encode manipulatives used during instruction in meaningful ways.
Copyright (c) 2026 Helena P. Osana, Joel R. Levin, Anne Lafay
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
[1]Abrahamson, D., Nathan, M. J., Williams-Pierce, C., et al. (2020). The future of embodied design for mathematics teaching and learning. Frontiers in Education, 5, 147. https://doi.org/10.3389/feduc.2020.00147
[2]Alfieri, L., Nokes-Malach, T. J., & Schunn, C. D. (2013). Learning through case comparisons: A meta-analytic review. Educational Psychologist, 48(2), 87–113. https://doi.org/10.1080/00461520.2013.775712
[3]Berthold, K., Eysink, T. H., & Renkl, A. (2009). Assisting self-explanation prompts are more effective than open prompts when learning with multiple representations. Instructional Science, 37(4), 345–363. https://doi.org/10.1007/s11251-008-9051-z
[4]Bower, C. A., Mix, K. S., Yuan, L., et al. (2022). A network analysis of children’s emerging place-value concepts. Psychological Science, 33(7), 1112–1127. https://doi.org/10.1177/09567976211070242
[5]Bruner, J. S. (1966). Towards a Theory of Instruction. Cambridge, MA: Harvard University Press.
[6]Byrge, L., Smith, L. B., & Mix, K. S. (2014). Beginnings of place value: How preschoolers write three‐digit numbers. Child Development, 85(2), 437–443. https://doi.org/10.1111/cdev.12162
[7]Byrne, E. M., Jensen, H., Thomsen, B. S., et al. (2023). Educational interventions involving physical manipulatives for improving children's learning and development: A scoping review. Review of Education, 11(2), e3400. https://doi.org/10.1002/rev3.3400
[8]Carbonneau, K. J., Marley, S. C., & Selig, J. P. (2013). A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. Journal of Educational Psychology, 105(2), 380–400. https://doi.org/10.1037/a0031084
[9]Carpenter, T. P., Franke, M. L., Jacobs, V. R., et al. (1997). A longitudinal study of invention and understanding in children's multidigit addition and subtraction. Journal for Research in Mathematics Education, 29(1), 3–20. https://doi.org/10.5951/jresematheduc.29.1.0003
[10]Castro-Alonso, J. C., Ayres, P., Zhang, S., et al. (2024). Research avenues supporting embodied cognition in learning and instruction. Educational Psychology Review, 36(1), 10. https://doi.org/10.1007/s10648-024-09847-4
[11]Chase, K., & Abrahamson, D. (2013). Rethinking transparency: Constructing meaning in a physical and digital design for algebra. In: Proceedings of the 12th International Conference on Interaction Design and Children; 24–27 June 2013; New York, NY, USA. pp. 475–478. https://doi.org/10.1145/2485760.2485834
[12]Chen, Z. (1996). Children’s analogical problem solving: The effects of superficial, structural, and procedural similarity. Journal of Experimental Child Psychology, 62(3), 410–431. https://doi.org/10.1006/jecp.1996.0037
[13]Cheung, P., & Ansari, D. (2021). Cracking the code of place value: The relationship between place and value takes years to master. Developmental Psychology, 57(2), 227–240. https://doi.org/10.1037/dev0001145
[14]Craik, F. I. M., & Tulving, E. (1975). Depth of processing and the retention of words in episodic memory. Journal of Experimental Psychology: General, 104(3), 268–294. https://doi.org/10.1037/0096-3445.104.3.268
[15]Crooks, N. M., & Alibali, M. W. (2014). Defining and measuring conceptual knowledge in mathematics. Developmental Review, 34(4), 344–377. https://doi.org/10.1016/j.dr.2014.10.001
[16]De Vos, T. (1992). Tempo Test Rekenen (TTR). Arithmetic Number Fact Test. Nijmegen: Berkhout.
[17]Dennis, M. S., Sharp, E., Chovanes, J., et al. (2016). A meta–analysis of empirical research on teaching students with mathematics learning difficulties. Learning Disabilities Research & Practice, 31(3), 156–168. https://doi.org/10.1111/ldrp.12107
[18]Dinsmore, D. L., & Alexander, P. A. (2012). A critical discussion of deep and surface processing: What it means, how it is measured, the role of context, and model specification. Educational Psychology Review, 24(4), 499–567. https://doi.org/10.1007/s10648-012-9198-7
[19]Donovan, A. M., & Alibali, M. W. (2021). Toys or math tools: Do children’s views of manipulatives affect their learning? Journal of Cognition and Development, 22(2), 281–304. https://doi.org/10.1080/15248372.2021.1890602
[20]Donovan, A. M., & Alibali, M. W. (2022). Manipulatives and mathematics learning: The roles of perceptual and interactive features. In: Macrine, S. L., Fugate, J. M. B. (editors). Movement Matters: How Embodied Cognition Informs Teaching and Learning. MIT Press. pp. 147–161. https://doi.org/10.7551/mitpress/13593.003.0017
[21]Donovan, A. M., & Fyfe, E. R. (2022). Connecting concrete objects and abstract symbols promotes children’s place value knowledge. Educational Psychology, 42(8), 1008–1026. https://doi.org/10.1080/01443410.2022.2077915
[22]English, L. D. (2004). Mathematical and Analogical Reasoning in Early Childhood. In: English, L. (editor). Mathematical and Analogical Reasoning of Young Learners. Routledge. pp. 1–22.
[23]Fiorella, L. (2021). The embodiment principle in multimedia learning. In: Mayer, R. E., Fiorella, L. (editors). The Cambridge Handbook of Multimedia Learning, 3rd ed. Cambridge University Press. pp. 286–295. https://doi.org/10.1017/9781108894333.030
[24]Fiorella, L. (2023). Making sense of generative learning. Educational Psychology Review, 35(2), 50. https://doi.org/10.1007/s10648-023-09769-7
[25]Fuson, K. C. (1998). Pedagogical, mathematical, and real-world conceptual-support nets: A model for building children’s multidigit domain knowledge. Mathematical Cognition, 4(2), 147–186. https://doi.org/10.1080/135467998387370
[26]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(3), 180–206. https://doi.org/10.5951/jresematheduc.21.3.0180
[27]Fyfe, E. R., Alibali, M. W., & Nathan, M. J. (2017). The promise and pitfalls of making connections in mathematics. In: Proceedings of the 39th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education; 5–8 October 2017; Indianapolis, IN USA. pp. 717–724.
[28]Fyfe, E. R., Borriello, G. A., & Merrick, M. (2023). A developmental perspective on feedback: How corrective feedback influences children’s literacy, mathematics, and problem solving. Educational Psychologist, 58(3), 130–145. https://doi.org/10.1080/00461520.2022.2108426
[29]Fyfe, E. R., DeCaro, M. S., & Rittle‐Johnson, B. (2014). An alternative time for telling: When conceptual instruction prior to problem solving improves mathematical knowledge. British Journal of Educational Psychology, 84(3), 502–519. https://doi.org/10.1111/bjep.12035
[30]Fyfe, E. R., & Nathan, M. J. (2019). Making “concreteness fading” more concrete as a theory of instruction for promoting transfer. Educational Review, 71(4), 403–422. https://doi.org/10.1080/00131911.2018.1424116
[31]Glenberg, A. M., Witt, J. K., & Metcalfe, J. (2013). From the revolution to embodiment: 25 years of cognitive psychology. Perspectives on Psychological Science, 8(5), 573–585. https://doi.org/10.1177/1745691613498098
[32]Godwin, K. E., Seltman, H., Almeda, M., et al. (2021). The elusive relationship between time on-task and learning: Not simply an issue of measurement. Educational Psychology, 41(4), 502–519. https://doi.org/10.1080/01443410.2021.1894324
[33]Götze, D., & Baiker, A. (2021). Language-responsive support for multiplicative thinking as unitizing: Results of an intervention study in the second grade. ZDM – Mathematics Education, 53(2), 263–275. https://doi.org/10.1007/s11858-020-01206-1
[34]Ho, C. S.-H., & Cheng, F. S.-F. (1997). Training in place-value concepts improves children’s addition skills. Contemporary Educational Psychology, 22(4), 495–506. https://doi.org/10.1006/ceps.1997.0947
[35]Jensen, S., Gasteiger, H., & Bruns, J. (2024). Place value and regrouping as helpful constructs to diagnose difficulties in understanding the place value system. Journal für Mathematik-Didaktik, 45(2), 11. https://doi.org/10.1007/s13138-024-00234-8
[36]Kiru, E. W., Doabler, C. T., Sorrells, A. M., et al. (2018). A synthesis of technology-mediated mathematics interventions for students with or at risk for mathematics learning disabilities. Journal of Special Education Technology, 33(2), 111–123. https://doi.org/10.1177/0162643417745835
[37]Kotovsky, L., & Gentner, D. (1996). Comparison and categorization in the development of relational similarity. Child Development, 67(6), 2797–2822. https://doi.org/10.1111/j.1467-8624.1996.tb01889.x
[38]Ladel, S., & Kortenkamp, U. (2016). Development of a flexible understanding of place value. In: Meaney, T., Helenius, O., Johansson, M. L., et al. (editors). Mathematics Education in the Early Years. Springer. pp. 289–307.
[39]Lafay, A., Osana, H. P., & Levin, J. R. (2023). Does conceptual transparency in manipulatives afford place-value understanding in children at risk for mathematics learning disabilities? Learning Disability Quarterly, 46(2), 92–105. https://doi.org/10.1177/07319487221124088
[40]Lambert, K., & Moeller, K. (2019). Place-value computation in children with mathematics difficulties. Journal of Experimental Child Psychology, 178, 214–225. https://doi.org/10.1016/j.jecp.2018.09.008
[41]Landerl, K., & Kölle, C. (2009). Typical and atypical development of basic numerical skills in elementary school. Journal of Experimental Child Psychology, 103(4), 546–565. https://doi.org/10.1016/j.jecp.2008.12.006
[42]Laski, E. V., Jor’dan, J. R., Daoust, C., et al. (2015). What makes mathematics manipulatives effective? Lessons from cognitive science and Montessori education. SAGE Open, 5(2), 1–8. https://doi.org/10.1177/2158244015589588
[43]Laski, E. V., & Siegler, R. S. (2014). Learning from number board games: You learn what you encode. Developmental Psychology, 50(3), 853–864. https://doi.org/10.1037/a0034321
[44]Laski, E. V., Vasilyeva, M., & Schiffman, J. (2016). Longitudinal comparison of place-value and arithmetic knowledge in Montessori and non-Montessori students. Journal of Montessori Research, 2(1), 1–15.
[45]Levin, J. R., Serlin, R. C., & Seaman, M. A. (1994). A controlled, powerful multiple-comparison strategy for several situations. Psychological Bulletin, 115(1), 153–159. https://doi.org/10.1037/0033-2909.115.1.153
[46]Marley, S. C., & Carbonneau, K. J. (2015). How psychological research with instructional manipulatives can inform classroom learning. Scholarship of Teaching and Learning in Psychology, 1(4), 412–424. https://doi.org/10.1037/stl0000047
[47]Martin, T., & Schwartz, D. L. (2005). Physically distributed learning: Adapting and reinterpreting physical environments in the development of fraction concepts. Cognitive Science, 29(4), 587–625. https://doi.org/10.1207/s15516709cog0000_15
[48]McNeil, N. M., Uttal, D. H., Jarvin, L., et al. (2009). Should you show me the money? Concrete objects both hurt and help performance on mathematics problems. Learning and Instruction, 19(2), 171–184. https://doi.org/10.1016/j.learninstruc.2008.03.005
[49]Mix, K. S., Bower, C. A., Hancock, G. R., et al. (2022). The development of place value concepts: Approximation before principles. Child Development, 93(3), 778–793. https://doi.org/10.1111/cdev.13724
[50]Mix, K. S., Bower, C. A., Yuan, L., et al. (2023). Predictive relations between early place value understanding and multidigit calculation: approximate versus syntactic measures. Educational Psychology, 43(7), 795–813. https://doi.org/10.1080/01443410.2023.2254528
[51]Mix, K. S., Smith, L. B., & Crespo, S. (2019). Leveraging relational learning mechanisms to improve place value instruction. In: Norton, A., Alibali, M. W. (editors). Constructing Number: Merging Perspectives from Psychology and Mathematics Education. Springer. pp. 87–121. https://doi.org/10.1007/978-3-030-00491-0_5
[52]Moura, R., Wood, G., Pinheiro-Chagas, P., et al. (2013). Transcoding abilities in typical and atypical mathematics achievers: The role of working memory and procedural and lexical competencies. Journal of Experimental Child Psychology, 116(3), 707–727. https://doi.org/10.1016/j.jecp.2013.07.008
[53]Nathan, M. J. (2021). Foundations of Embodied Learning: A Paradigm for Education. New York, NY: Routledge. https://doi.org/10.4324/9780429329098
[54]Nathan, M. J. (2024). Advances in grounded and embodied mathematical reasoning: Theory, technology, and research methods. In: Shapiro, L., Spaulding, S. (editors). The Routledge Handbook of Embodied Cognition, 2nd ed. Routledge. pp. 260–278. https://doi.org/10.4324/9781003322511-28
[55]Nathan, M. J., & Alibali, M. W. (2021). An embodied theory of transfer of mathematical learning. In: Hohensee, C., Lobato, J. (editors). Transfer of learning: Progressive Perspectives for Mathematics Education and Related Fields. Springer. pp. 27–58. https://doi.org/10.1007/978-3-030-65632-4_2
[56]Nathan, M. J., Walkington, C., Boncoddo, R., et al. (2014). Actions speak louder with words: The roles of action and pedagogical language for grounding mathematical proof. Learning and Instruction, 33, 182–193. https://doi.org/10.1016/j.learninstruc.2014.07.001
[57]O'Donnell, C. L. (2008). Defining, conceptualizing, and measuring fidelity of implementation and its relationship to outcomes in K–12 curriculum intervention research. Review of Educational Research, 78(1), 33–84. https://doi.org/10.3102/0034654307313793
[58]Osana, H. P., Adrien, E., & Duponsel, N. (2017). Effects of instructional guidance and sequencing of manipulatives and written symbols on second graders’ numeration knowledge. Education Sciences, 7(2), 52–74. https://doi.org/10.3390/educsci7020052
[59]Osana, H. P., & Lafay, A. (2019). The relationship between analogical reasoning and second-graders’ structural knowledge of number. In: Proceedings of the Forty-First Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education; 14–17 November 2019; St Louis, MO, USA. pp. 201–205.
[60]Petersen, L. A., & McNeil, N. M. (2013). Effects of perceptually rich manipulatives on preschoolers’ counting performance: Established knowledge counts. Child Development, 84(3), 1020–1033. https://doi.org/10.1111/cdev.12028
[61]Pouw, W. T. J. L., Van Gog, T., & Paas, F. (2014). An embedded and embodied cognition review of instructional manipulatives. Educational Psychology Review, 26(1), 51–72. https://doi.org/10.1007/s10648-014-9255-5
[62]Raven, J. C. (1977). Standard Progressive Matrices. Paris: Editions Scientifiques et Psychologiques.
[63]Rittle-Johnson, B., Loehr, A. M., & Durkin, K. (2017). Promoting self-explanation to improve mathematics learning: A meta-analysis and instructional design principles. ZDM Mathematics Education, 49(4), 599–611. https://doi.org/10.1007/s11858-017-0834-z
[64]Rojo, M. M., Knight, B., & Bryant, D. P. (2021). Teaching place value to students with learning disabilities in mathematics. Intervention in School and Clinic, 57(1), 32–40. https://doi.org/10.1177/1053451221994827
[65]Ross, S. H. (1989). Parts, wholes, and place value: A developmental view. Arithmetic Teacher, 36(6), 47–51. https://doi.org/10.5951/AT.36.6.0047
[66]Rousselle, L., & Noël, M. P. (2007). Basic numerical skills in children with mathematics learning disabilities: A comparison of symbolic vs non-symbolic number magnitude processing. Cognition, 102(3), 361–395. https://doi.org/10.1016/j.cognition.2006.01.005
[67]Siegler, R. S., & Ramani, G. B. (2009). Playing linear number board games—but not circular ones—improves low-income preschoolers’ numerical understanding. Journal of Educational Psychology, 101(3), 545–560. https://doi.org/10.1037/a0014239
[68]Sweller, J., van Merrienboer, J. J., & Paas, F. (2019). Cognitive architecture and instructional design: 20 years later. Educational Psychology Review, 31(2), 261–292. https://doi.org/10.1007/s10648-019-09465-5
[69]Träff, U., Olsson, L., Östergren, R., et al. (2017). Heterogeneity of developmental dyscalculia: Cases with different deficit profiles. Frontiers in Psychology, 7(2000), 1–16. https://doi.org/10.3389/fpsyg.2016.02000
[70]Uttal, D. H., Amaya, M., del Rosario Maita, M., et al. (2013). It works both ways: Transfer difficulties between manipulatives and written subtraction solutions. Child Development Research, 2013(1), 216367. https://doi.org/10.1155/2013/216367
[71]Witzel, B. S., Riccomini, P. J., & Schneider, E. (2008). Implementing CRA with secondary students with learning disabilities in mathematics. Intervention in School and Clinic, 43(5), 270–276. https://doi.org/10.1177/1053451208314734
[72]Yuan, L., Prather, R., Mix, K., et al. (2021). The first step to learning place value: A role for physical models? Frontiers in Education, 6, 683424. https://doi.org/10.3389/feduc.2021.683424
