Fostering conceptual knowledge of procedures by explicating structures in multiple representations: Classwise randomized controlled trial for the case of unit conversion

Sofia Peters; Susanne Prediger; Juliane Weiss

doi:10.29333/iejme/19642

Full Text (PDF)

Sofia Peters ¹ ^* , Susanne Prediger ¹ ² , Juliane Weiss ²

More Detail

¹ TU Dortmund University, Vogelpothsweg 87, 44227 Dortmund, GERMANY² IPN Leibniz Institute for Science and Mathematics Education, Berlin, GERMANY^* Corresponding Author

Abstract

Instructional approaches with multiple representations and rich discourse practices can effectively foster conceptual knowledge of mathematical procedures. Yet, qualitative studies indicate that supporting students in explicitly articulating the underlying mathematical structures might even increase learning gains. This hypothesis is tested in a classwise randomized controlled trial (with N = 434 students) for the example topic unit conversion in digitally assisted teaching-learning environments. Both intervention groups (using multiple representations and rich discourse practices) had significantly higher learning gains than the control group in conceptual knowledge and content-related language proficiency. The structure-explicating intervention showed significant additional conceptual and language-related learning gains. These findings contribute to theorizing on principles of conceptually focused instruction and the high relevance of engaging students in explicating structures.

Keywords

unit conversion
procedural and conceptual knowledge
controlled trial
digitally-assisted teaching
learning environment
structure explication

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

INT ELECT J MATH ED, Volume 21, Issue 4, November 2026, Article No: em0897

https://doi.org/10.29333/iejme/19642

Publication date: 20 Oct 2026

Article Views: 12

Article Downloads: 6

Open Access References How to cite this article

References

Battista, M. T., & Clements, D. H. (1996). Students’ understanding of three-dimensional rectangular arrays of cubes. Journal for Research in Mathematics Education, 27(3), 258–292. https://doi.org/10.2307/749365
Battista, M. T. (2007). The development of geometric and spatial thinking. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 843–908). Information Age.
Blazar, D., & Archer, C. (2020). Teaching to support students with diverse academic needs. Educational Researcher, 49(5), 297–311. https://doi.org/10.3102/0013189X20931226
Cheeseman, J., McDonough, A., & Golemac, D. (2017). Investigating children’s thinking about suspended balances. New Zealand Journal of Educational Studies, 52(1), 143–158. https://doi.org/10.1007/s40841-016-0073-9
Cheung, A. C. K., & Slavin, R. E. (2016). How methodological features affect effect sizes in education. Educational Researcher, 45(5), 283–292. https://doi.org/10.3102/0013189X16656615
Clark, F. B., & Kamii, C. (1996). Identification of multiplicative thinking in children in grades 1-5. Journal for Research in Mathematics Education, 27(1), 41–51. https://doi.org/10.2307/749196
Cramer, K. A., Post, T. R., & delMas, R. C. (2002). Initial fraction learning by fourth- and fifth-grade students: A comparison of the effects of using commercial curricula with the effects of using the rational number project curriculum. Journal for Research in Mathematics Education, 33(2), 111–144. https://www.jstor.org/stable/749646
Dietrichson, J., Bøg, M., Filges, T., & Klint Jørgensen, A.-M. (2017). Academic interventions for elementary and middle school students with low socioeconomic status: A systematic review and meta-analysis. Review of Educational Research, 87(2), 243–282. https://doi.org/10.3102/0034654316687036
Dincer, E. O., & Osmanoglu, A. (2018). Dealing with metric unit conversion: An examination of prospective science teachers’ knowledge of and difficulties with conversion. Science Education International, 29(3), 174–182. https://doi.org/10.33828/sei.v29.i3.5
Erath, K., Ingram, J., Moschkovich, J., & Prediger, S. (2021). Designing and enacting instruction that enhances language for mathematics learning – A review of the state of development and research. ZDM – Mathematics Education, 53(2), 245–262. https://doi.org/10.1007/s11858-020-01213-2
Freudenthal, H. (1981). Major problems of mathematics education. Educational Studies in Mathematics, 12(2), 133–150. https://doi.org/10.1007/BF00305618
Fuson, K. C. (2019). Overcoming errors in fraction computation by emphasizing unit fractions, length drawings, and student explanations. Universal Journal of Educational Research, 7(8), 1663–1678. https://doi.org/10.13189/ujer.2019.070805
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/10.2307/749373
Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, A. I., Carpenter, T. P., & Fennema, E. (1997). Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education, 28(2), 130–162. https://doi.org/10.2307/749759
Gibbons, P. (2002). Scaffolding language, scaffolding learning. Teaching second language learners in the mainstream classroom. Heinemann.
Glade, M., & Prediger, S. (2017). Students’ individual schematization pathways. Educational Studies in Mathematics, 94(2), 185–203. https://doi.org/10.1007/s10649-016-9716-5
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
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Lawrence Erlbaum.
Hiebert, J., & Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and subtraction. Cognition and Instruction, 14(3), 251–283. https://doi.org/10.1207/s1532690xci1403_1
Kilpatrick, J., Swafford, J., & Findel, B. (2001). Adding it up. Helping children learn mathematics. National Academy Press. https://doi.org/10.17226/9822
Lamon, S. J. (1996). The development of unitizing: Its role in children’s partitioning strategies. Journal for Research in Mathematics Education, 27(2), 170–193. https://www.jstor.org/stable/749599
Lenhard, W., & Schneider, W. (2006). ELFE 1-6. Ein Leseverständnistest für Erst- bis Sechstklässler [ELFE 1-6: A reading comprehension test for first to sixth graders]. Hogrefe.
Lorange, A., & Rinvold, R. A. (2014). Students’ strategies of expanding fractions to a common denominator – a semiotic perspective. Nordic Studies in Mathematics Education, 19(2), 57–75.
Meteyard, L., & Davies, R. A. I. (2020). Best practice guidance for linear mixed-effects models in psychological science. Journal of Memory and Language, 112, 104092. https://doi.org/10.1016/j.jml.2020.104092
Moschkovich, J. (2015). Academic literacy in mathematics for English Learners. The Journal of Mathematical Behavior, 40(A), 43–62. https://doi.org/10.1016/j.jmathb.2015.01.005
National Council of Teachers of Mathematics [NCTM]. (2014). Principles to actions. Ensuring mathematical success to all. NCTM.
Nguyen, H., An, T., & Maher, E. M. (2023). Impact of designed task-based interventions on k-8 preservice teachers’ knowledge and ability to solve measurement unit conversion problems. International Journal on Studies in Education, 5(3), 232–244. https://doi.org/10.46328/ijonse.126
Organisation for Economic Co-operation and Development [OECD]. (2023). PISA 2022 results. The state of learning and equity in education (Volume I). OECD. https://doi.org/10.1787/53f23881-en
Paetsch, J., Radmann, S., Felbrich, A., Lehmann, R., & Stanat, P. (2016). Sprachkompetenz als Prädiktor mathematischer Kompetenzentwicklung von Kindern deutscher und nicht-deutscher Familiensprache [Students’ growth trajectories in mathematics: The role of language proficiency]. Zeitschrift für Entwicklungspsychologie und Pädagogische Psychologie, 48(1), 27–41. https://doi.org/10.1026/0049-8637/a000142
Paulus, C. (2009). Die “Bücheraufgabe” zur Bestimmung des kulturellen Kapitals bei Grundschülern [The “book-at-home index” for capturing primary students’ cultural capital]. http://hdl.handle.net/20.500.11780/3344
Peters, N., Prediger, S., & Weiss, J. (2025). Fostering students’ measurement estimation skills in a digital teaching-learning environment: A class-wise randomized controlled trial in Grade 5. Education Sciences, 15(2), 238: 1–25. https://doi.org/10.3390/educsci15020238
Peters, S., & Prediger, S. (2026). Enhancing students' understanding of unit conversion by explicating underlying structures: Design research on schematization in Grade 5. Educational Studies in Mathematics, 121(2), 283–307. https://doi.org/10.1007/s10649-025-10437-6
Pöhler, B., Prediger, S., & Neugebauer, P. (2017). Content- and language integrated learning: A field experiment for percentages. In B. Kaur, W. K. Ho, T. L. Toh, & B. H. Choy (Eds.), Proceedings of the 41st Annual Meeting of the International Group for the Psychology of Mathematics Education (PME 41) (Vol. 4, pp. 73–80). PME.
Prediger, S. (2022). Enhancing language for developing conceptual understanding: A research journey connecting different research approaches. In J. Hodgen, E. Geraniou, G. Bolondi, & F. Ferretti (Eds.), Proceedings of Twelfth Congress of the European Society for Research in Mathematics Education (CERME12) (pp. 8–33). University of Bolzano / ERME. https://hal.archives-ouvertes.fr/hal-03756062/
Prediger, S., Erath, K., Weinert, H., & Quabeck, K. (2022). Only for multilingual students at risk? Cluster-randomized trial on language-responsive instruction. Journal for Research in Mathematics Education, 53(4), 255–276. https://doi.org/10.5951/jresematheduc-2020-0193
Renkl, A., Berthold, K., Große, C. S., & Schwonke, R. (2013). Making better use of multiple representations: How fostering metacognition can help. In R. Azevedo & V. Aleven (Eds.), International handbook of metacognition and learning technologies (pp. 397–408). Springer. https://doi.org/10.1007/978-1-4419-5546-3_26
Rittle-Johnson, B., & Schneider, M. (2015). Developing conceptual and procedural knowledge of mathematics. In R. C. Kadosh & A. Dowker (Eds.), Oxford Handbook of Numerical Cognition (pp. 1118–1135). Oxford University Press.
Ross, S. H. (1989). Parts, wholes and place value: A developmental view. The Arithmetic Teacher, 36(6), 47–51.
Selling, S. K. (2016). Making mathematical practices explicit in urban middle and high school mathematics classrooms. Journal for Research in Mathematics Education, 47(5), 505–551. https://doi.org/10.5951/jresematheduc.47.5.0505
Smith, J. P., & Barrett, J. E. (2017). Learning and teaching measurement: Coordinating quantity and number. In J. Cai (Ed.), Compendium for Research in Mathematics Education (pp. 355–385). National Council of Teachers of Mathematics.
Steinbring, H. (2005). The construction of new mathematical knowledge in classroom interaction. An epistemological perspective. Springer. https://doi.org/10.1007/b104944
Tan Sisman, G., & Aksu, M. (2016). A study on sixth grade students’ misconceptions and errors in spatial measurement: length, area, and volume. International Journal of Science and Mathematics Education, 14(7), 1293–1319. https://doi.org/10.1007/s10763-015-9642-5
Tondorf, A., & Prediger, S. (2022). Connecting characterizations of equivalence. Educational Studies in Mathematics, 111(3), 399–422. https://doi.org/10.1007/s10649-022-10158-0
Walshaw, M., & Anthony, G. (2008). The teacher’s role in classroom discourse: A review of recent research into mathematics classrooms. Review of Educational Research, 78(3), 516–551. https://doi.org/10.3102/0034654308320292
Wessel, L. (2020). Vocabulary in learning processes towards conceptual understanding of equivalent fractions—specifying students’ language demands on the basis of lexical trace analyses. Mathematics Education Research Journal, 32, 653–681. https://doi.org/10.1007/s13394-019-00284-z
Wortman, P. M. (1983). Evaluation research: A methodological perspective. Annual Review of Psychology, 34, 223–260. https://doi.org/10.1146/annurev.ps.34.020183.001255
Zahner, W., Calleros, E. D., & Pelaez, K. (2021). Designing learning environments to promote academic literacy in mathematics in multilingual secondary mathematics classrooms. ZDM – Mathematics Education, 53(2), 359–373. https://doi.org/10.1007/s11858-021-01239-0

How to cite this article

APA

Peters, S., Prediger, S., & Weiss, J. (2026). Fostering conceptual knowledge of procedures by explicating structures in multiple representations: Classwise randomized controlled trial for the case of unit conversion. International Electronic Journal of Mathematics Education, 21(4), em0897. https://doi.org/10.29333/iejme/19642

Vancouver

Peters S, Prediger S, Weiss J. Fostering conceptual knowledge of procedures by explicating structures in multiple representations: Classwise randomized controlled trial for the case of unit conversion. INT ELECT J MATH ED. 2026;21(4):em0897. https://doi.org/10.29333/iejme/19642

AMA

Peters S, Prediger S, Weiss J. Fostering conceptual knowledge of procedures by explicating structures in multiple representations: Classwise randomized controlled trial for the case of unit conversion. INT ELECT J MATH ED. 2026;21(4), em0897. https://doi.org/10.29333/iejme/19642

Chicago

Peters, Sofia, Susanne Prediger, and Juliane Weiss. "Fostering conceptual knowledge of procedures by explicating structures in multiple representations: Classwise randomized controlled trial for the case of unit conversion". International Electronic Journal of Mathematics Education 2026 21 no. 4 (2026): em0897. https://doi.org/10.29333/iejme/19642

Harvard

Peters, S., Prediger, S., and Weiss, J. (2026). Fostering conceptual knowledge of procedures by explicating structures in multiple representations: Classwise randomized controlled trial for the case of unit conversion. International Electronic Journal of Mathematics Education, 21(4), em0897. https://doi.org/10.29333/iejme/19642

MLA

Peters, Sofia et al. "Fostering conceptual knowledge of procedures by explicating structures in multiple representations: Classwise randomized controlled trial for the case of unit conversion". International Electronic Journal of Mathematics Education, vol. 21, no. 4, 2026, em0897. https://doi.org/10.29333/iejme/19642