# A design study to develop young children’s understanding of multiplication and division

- 939 Downloads
- 1 Citations

## Abstract

This design study investigated the use of multiplication and division problems to help 5-year-old children develop an early understanding of multiplication and division. One teacher and her class of 15 5-year-old children were involved in a collaborative partnership with the researchers. The design study was conducted over two 4-week periods in May–June and October–November. The focus in this article is on three key aspects of classroom teaching: instructional tasks, the use of representations, and discourse, including the mathematics register. Results from selected pre- and post-assessment tasks within a diagnostic interview showed that there were improvements in addition and subtraction as well as multiplication and division, even though the teaching had used multiplication and division problems. Students made progress on all four operational domains, with effect sizes ranging from approximately two thirds of a standard deviation to 2 standard deviations. Most of the improvement in students’ number strategies was in moving from ‘counting all’ to ‘counting on’ and ‘skip counting’. The findings challenge the idea that learning experiences in addition and subtraction should precede those in multiplication and division as suggested in some curriculum documents.

## Keywords

Multiplication Division Instructional tasks Language Representations Early primary## Notes

### Acknowledgments

The data for this paper was extracted from a larger project that was made possible by funding from the Teaching and Learning Research Initiative (TLRI) administered through the New Zealand Council for Educational Research and the interest and support of the teachers and children involved in the project.

## References

- Ambrose, R. C. (2002). Are we overemphasizing manipulatives in the primary grades to the detriment of girls?
*Teaching Children Mathematics, 9*, 16–21.Google Scholar - Anghileri, J. (2001). Intuitive approaches, mental strategies and standard algorithms. In J. Anghileri (Ed.),
*Principles and practices in arithmetic teaching: innovative approaches for the primary classroom*. Philadelphia: Open University Press.Google Scholar - Arcavi, A. (2003). The role of visual representations in the learning of mathematics.
*Educational Studies in Mathematics, 52*, 215–241.CrossRefGoogle Scholar - Artzt, A. F., & Armour-Thomas, E. (1999). A cognitive model for examining teachers’ instructional practice in mathematics: a guide for facilitating teacher reflection.
*Educational Studies in Mathematics, 40*, 211–235.CrossRefGoogle Scholar - Ball, D. L., & Forzani, F. M. (2007). What makes educational research “educational”?
*Educational Researcher, 36*, 529–540.CrossRefGoogle Scholar - Ball, D. L., Lubienski, S. T., & Mewborn, D. S. (2001). Research on teaching mathematics: the unsolved problem of teachers’ mathematical knowledge. In V. Richardson (Ed.),
*Handbook of research on teaching*(4th ed., pp. 433–456). New York: Macmillan.Google Scholar - Barab, S., & Squire, K. (2004). Design-based research: putting a stake in the ground.
*The Journal of the Learning Sciences, 13*(1), 1–14.CrossRefGoogle Scholar - Bicknell, B., & Young-Loveridge, J. (2015).
*Using multiplication and division contexts to enhance young children’s part-whole thinking in mathematics. Teaching and learning research initiative: summary*. http://www.tlri.org.nz/tlri-research/research-completed/school-sector/using-multiplication-and-division-contexts-enhance.Google Scholar - Bobis, J., Clarke, B., Clarke, D., Thomas, G., Wright, R., Young-Loveridge, J., & Gould, P. (2005). Supporting teachers in the development of young children’s mathematical thinking: three large scale cases.
*Mathematics Education Research Journal, 16*(3), 27–57.CrossRefGoogle Scholar - Borasi, R. (1986). On the nature of problems.
*Educational Studies in Mathematics, 17*, 125–141.CrossRefGoogle Scholar - Carpenter, T. P., Fennema, E., & Franke, M. L. (1996). Cognitively guided instruction: a knowledge base for reform in primary mathematics instruction.
*The Elementary School Journal, 97*, 3–20.CrossRefGoogle Scholar - Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999).
*Children’s mathematics: cognitively guided instruction*. Portsmouth: Heinemann.Google Scholar - Chapman, O. (2006). Classroom practices for context of mathematics word problems.
*Educational Studies in Mathematics, 62*, 211–230.CrossRefGoogle Scholar - Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education.
*Mathematical Thinking and Learning, 6*(2), 81–89.Google Scholar - Clements, D. H., & Sarama, J. (2014). Learning trajectories: foundations for effective research-based education. In A. P. Maloney, J. Confrey, & K. H. Nguyen (Eds.),
*Learning over time: learning trajectories in mathematics education*(pp. 1–30). Charlotte: Information Age.Google Scholar - Cobb, P., Confrey, J., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research.
*Educational Researcher, 32*(1), 9–13.Google Scholar - Cobb, P., Jackson, K., & Dunlap, C. (2016). Design research: an analysis and critique. In L. D. English & D. Kirshner (Eds.),
*Handbook of international research in mathematics education*(3rd ed., pp. 481–503). New York: Routledge.Google Scholar - Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2010). Participating in classroom mathematical practices. In E. Yackel, K. Gravemeijer & A. Sfard (Eds.), A journey in mathematics education research: insights from the work of Paul Cobb (pp. 117–163). Dordrecht Springer. doi: 10.1007/978-90-481-9729-3.
- Fan, X. (2001). Statistical significance and effect size of educational research: two sides of a coin.
*The Journal of Educational Research, 94*, 275–282.CrossRefGoogle Scholar - Goldin, G., & Shteingold, N. (2001). Systems of representations and the development of mathematical concepts. In A. Cuoco & F. R. Curcio (Eds.),
*The roles of representations in school mathematics*(pp. 1–23). Reston: National Council of Teachers of Mathematics.Google Scholar - Gravemeijer, K. P. E. (1999). How emergent models may foster the constitution of formal mathematics.
*Mathematical Thinking and Learning, 1*, 155–177.CrossRefGoogle Scholar - Halliday, M., & Hassan, R. (1991).
*Language in a social semiotic perspective*. Oxford: Oxford University Press.Google Scholar - Hattie, J. A. C. (2009).
*Visible learning: a synthesis of over 800 meta-analyses relating to achievement*. London: Routledge.Google Scholar - Kelly, A. E. (2004). Design research in education: yes, but is it methodological?
*The Journal of the Learning Sciences, 13*(1), 115–128.CrossRefGoogle Scholar - Lampert, M. (1986). Knowing, doing, and teaching multiplication.
*Cognition and Instruction, 3*, 305–342.CrossRefGoogle Scholar - Marsh, H. W. (1990). Influences of internal and external frames of reference on the formation of math and English self-concepts.
*Journal of Educational Psychology, 82*, 107–116. doi: 10.1037/0022-0663.82.1.107. - Marshall, L., & Swan, P. (2008).
*Exploring the use of mathematics manipulative materials: is it what we think it is? Proceedings of the EDU-COM 2008 International Conference. Sustainability in Higher Education: Directions for Change*(pp. 19–21). Perth: Edith Cowan University. November 2008.Google Scholar - Mercer, N., & Sams, C. (2006). Teaching children how to use language to solve maths problems.
*Language and Education, 20*, 507–528.CrossRefGoogle Scholar - Meyer, M. R., Dekker, T., & Querelle, N. (2001). Context in mathematics curricula.
*Mathematics Teaching in the Middle School, 6*, 522–527.Google Scholar - Ministry of Education. (2007).
*The New Zealand curriculum*. Wellington: Author.Google Scholar - Ministry of Education. (2008).
*Book 2: The diagnostic interview*. Wellington: Author.Google Scholar - Moschkovich, J. (1999). Supporting the participation of English language learners in mathematical discussions.
*For the Learning of Mathematics, 19*(1), 11–19.Google Scholar - Moschkovich, J. (2002). A situated and sociocultural perspective on bilingual mathematics learners.
*Mathematical Thinking and Learning, 4*, 189–212.CrossRefGoogle Scholar - Moyer, P. S. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics.
*Educational Studies in Mathematics, 47*, 175–197.CrossRefGoogle Scholar - Nunes, T., & Bryant, P. (1996).
*Children doing mathematics*. Oxford: Blackwell Publishers Ltd.Google Scholar - Pape, S. J., & Tchoshanov, M. A. (2001). The role of representation(s) in developing mathematical understanding.
*Theory Into Practice, 40*, 118–127.CrossRefGoogle Scholar - Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics. In A. Gutiérrez & P. Boero (Eds.),
*Handbook of research on the psychology of mathematics education: past, present and future*(pp. 205–235). Rotterdam: Sense.Google Scholar - Puchner, L., Taylor, A., O’Donnell, B., & Fick, K. (2008). Teacher learning and mathematics manipulatives: a collective case study about teacher use of manipulatives in elementary and middle school mathematics lessons.
*School Science and Mathematics, 108*, 313–325.CrossRefGoogle Scholar - Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: an iterative process.
*Journal of Educational Psychology, 93*, 346–362.Google Scholar - Schleppegrell, M. J. (2007). The linguistic challenges of mathematics teaching and learning: a research review.
*Reading and Writing Quarterly, 23*, 139–159.Google Scholar - Setati, M., & Adler, J. (2000). Between languages and discourses: language practices in primary multilingual mathematics classrooms in South Africa.
*Educational Studies in Mathematics, 43*, 243–269.CrossRefGoogle Scholar - Sigley, R., & Wilkinson, L. C. (2015). Ariel’s cycles of problem solving: an adolescent acquires the mathematics register.
*The Journal of Mathematical Behavior, 40*, 75–87.CrossRefGoogle Scholar - Smith, S. Z., & Smith, M. E. (2006). Assessing elementary understanding of multiplication concepts.
*School Science and Mathematics, 106*, 140–149.CrossRefGoogle Scholar - van den Heuvel-Panhuizen, M. (2005). The role of contexts in assessment problems in mathematics.
*For the Learning of Mathematics, 25*(2), 2–23.Google Scholar - van der Ven, S. H. G., van der Maas, H. L. J., Straatemeier, M., & Jansen, B. R. J. (2013). Visuospatial working memory and mathematical ability at different ages throughout primary school.
*Learning and Individual Differences, 27*, 182–192.CrossRefGoogle Scholar