### Distributivity, partitioning, and the multiplication algorithm

Chris Hurst^{(1*)}, Ray Huntley

^{(2)}

(1) School of Education, Curtin University

(2) Independent researcher, Playmouth

(*) Corresponding Author

#### Abstract

Multiplicative thinking underpins much of the mathematics learned beyond the middle primary years. As such, it needs to be understood conceptually to highlight the connections between its many aspects. This paper focuses on one such connection; that is how the array, place value partitioning and the distributive property of multiplication are related. It is important that students understand how the property informs the written multiplication algorithm. Another component of successful use of the standard multiplication algorithm is extended number facts and the paper also explores students’ ability to understand and generate them. One purpose of the study was to investigate the extent to which students used the standard multiplication algorithm and if their use of it is supported by an understanding of the underpinning components of the array, partitioning, the distributive property, and extended number facts. That is, we seek to learn if students have a conceptual understanding of the multiplication algorithm and its underpinning mathematics that would enable them to transfer their knowledge to a range of contexts, or if they have procedurally learned mathematics. In this qualitative study, data were generated from the administration of a Multiplicative Thinking Quiz with a sample of 36 primary aged students. Data were analyzed manually and reported using descriptive statistics. The main implications of the study are that the connections among the multiplicative array, place value partitioning, base ten property of place value, distributive property of multiplication, and extended number facts need to be made explicit for children in terms of how they inform the use of the written algorithm for multiplication.

#### Keywords

#### Full Text:

PDF#### References

Anthony, G., &Walshaw, M. (2009). Effective pedagogy in mathematics. Belley, France: United Nations Educational, Scientific and Cultural Organisation. Retrieved from: http://www.ibe.unesco.org/publications.htm

Askew, M. (2012). Transforming primary mathematics. Abingdon, U.K.: Routledge.

Australian Curriculum, Assessment, Reporting Authority (ACARA). (2020). The Australian Curriculum: Mathematics, v 8.3. Retrieved from http://www.australiancurriculum.edu.au/mathematics/key-ideas

Benson, C.C., Wall, J.T., &Malm, C. (2013). The distributive property in Grade 3? Teaching Children Mathematics, 19(8), 498-506. https://doi.org/10.5951/teacchilmath.19.8.0498

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.

Cotton, T. (2016). Understanding and teaching primary mathematics. (3rd. ed.) Abingdon, U.K.: Routledge.

Downton, A., Russo, J., & Hopkins, S. (2019). The case of the disappearing and reappearing zeros: A disconnection between procedural knowledge and conceptual understanding. 2019. In G. Hine, S. Blackley, & A. Cooke (Eds.). Mathematics Education Research: Impacting Practice (Proceedings of the 42nd annual conference of the Mathematics Education Research Group of Australasia) pp. 236-243. Perth: MERGA.

Haylock, D. (2014). Mathematics explained for primary teachers. (4th. ed.). London: Sage.

Hurst, C. (2017). Children have the capacity to think multiplicatively, as long as . . . European Journal of STEM Education, 2(3), 1-14. https://doi.org/10.20897/ejsteme/78169

Hurst, C., & Huntley, R. (2018). Algorithms and multiplicative thinking: Are children ‘prisoners of process’? International Journal for Mathematics Teaching and Learning, 19(1), 47-68.

Hurst, C., & Hurrell, D. (2018a). Algorithms are useful: Understanding them is even better. Australian Primary Mathematics Classroom, 23(3), 17-21.

Hurst, C., & Hurrell, D. (2018b). Algorithms are great: What about the mathematics that underpins them? Australian Primary Mathematics Classroom, 23(3), 22-26.

Hurst, C., & Hurrell, D. (2016). Multiplicative thinking: much more than knowing multiplication facts and procedures. Australian Primary Mathematics Classroom, 21(1), 34-38.

Hurst, C., & Hurrell, D. (2014). Developing the big ideas of number. International Journal of Educational Studies in Mathematics, 1(2) 1-18. https://doi.org/10.17278/ijesim.2014.02.001

Jazby, D., &Pearn, C. (2015). Using alternative multiplication algorithms to offload cognition. In M. Marshman, V. Geiger, & A. Bennison (Eds.). Mathematics education in the margins (Proceedings of the 38th annual conference of the Mathematics Education Research Group of Australasia), pp. 309-316. Sunshine Coast: MERGA.

Kaminski, E. (2002). Promoting mathematical understanding: umber sense in action. Mathematics Education Research Journal, 14(2), 133-149. https://doi.org/10.1007/BF03217358

Kinzer, C.J., & Stanford, T. (2013). The distributive property: The core of multiplication. Teaching Children Mathematics, 20(5), 302-309. https://doi.org/10.5951/teacchilmath.20.5.0302

Lemonidis, C. (2016). Mental computation and estimation: Implications for mathematics education research, teaching and learning. Abingon, U.K.: Routledge.

Matney, G.T., & Daugherty, B.N. (2013). Seeing spots and developing multiplicative sense making. Mathematics Teaching in the Middle School, 19(3), 148-155. https://doi.org/10.5951/mathteacmiddscho.19.3.0148

National Governors Association Center for Best Practices, Council of Chief State School Officers (NGA Center). (2010). Common core state standards for mathematics. Retrieved from; http://www.corestandards.org/the-standards

Norton, S., & Irvin, J. (2007). A concrete approach to teaching symbolic algebra. In J. Watson & K. Beswick (Eds). Proceedings of the 30th annual conference of the Mathematics Education Research Group of Australasia (pp. 551-560). Hobart: MERGA.

Pickreign, J, & Rogers, R. (2006). Do you understand your algorithms? Mathematics Teaching in the Middle School 12(1), 42-47.

Richland, L. E., Stigler, J. W., Holyoak, K. J. (2012). Teaching the Conceptual Structure of Mathematics, Educational Psychologist 47(3), 189-203. https://doi.org/10.1080/00461520.2012.667065

Rittle-Johnson, B., & Schneider, M. (2015). Developing conceptual and procedural knowledge in mathematics. In R. Cohen Kadosh& A. Dowker (Eds.), Oxford handbook of numerical cognition (pp. 1102-1118). Oxford, UK: Oxford University Press.

Rittle-Johnson, B., Fyfe, A.M., & Loehr, E.R. (2016). Improving Conceptual and Procedural Knowledge: The Impact of Instructional Content Within A Mathematics Lesson. British Journal of Educational Psychology. 86(4), 576-591. https://doi.org/10.1111/bjep.12124

Ross, S. (2002). Place value: Problem solving and written assessment. Teaching Children Mathematics, 8(7), 419-423.

Siemon, D., Breed, M., Dole, S., Izard, J., &Virgona, J. (2006). Scaffolding Numeracy in the Middle Years – Project Findings, Materials, and Resources, Final Report submitted to Victorian Department of Education and Training and the Tasmanian Department of Education, Retrieved from http://www.eduweb.vic.gov.au/edulibrary/public/teachlearn/student/snmy.ppt

Siemon, D., Bleckly, J. and Neal, D. (2012). Working with the Big Ideas in Number and the Australian Curriculum: Mathematics. In B. Atweh, M. Goos, R. Jorgensen & D. Siemon, (Eds.). (2012). Engaging the Australian National Curriculum: Mathematics – Perspectives from the Field. Online Publication: Mathematics Education Research Group of Australasia pp. 19‐45.

Skemp, R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching 77 (20-26).

Squire, S., Davies, C., & Bryant, P. (2004). Does the cue help? Children’s understanding of multiplicative concepts in different problem contexts. British Journal of Educational Psychology, 74, (515-532). https://doi.org/10.1111/bjep.12124

Thompson, I. (2003). Place value: the English disease? In I. Thompson, (Ed.). Enhancing Primary Mathematics Teaching. Berkshire, U.K.: Open University Press, pp. 181-190.

Turton, A. (Ed.). (2007). The Origo handbook of mathematics education. Brisbane: Origo.

Young-Loveridge, J. (2005) Fostering multiplicative thinking using array-based materials. Australian Mathematics Teacher, 61 (3), 34-40.

Young-Loveridge, J., & Mills, J. (2009). Teaching multi-digit multiplication using array based materials. In R. Hunter, B. Bicknell, & T. Burgess (Eds.), Crossing divides: Proceedings of the 32nd annual conference of the Mathematics Education Research Group of Australasia (Vol. 2). PalmerstonNorth, NZ: MERGA.

#### Article Metrics

Abstract view(s): 689 time(s)PDF: 577 time(s)

### Refbacks

- There are currently no refbacks.