# SPECIALISED CONTENT KNOWLEDGE: THE CONVENTION FOR NAMING ARRAYS AND DESCRIBING EQUAL GROUPS’ PROBLEMS

## Abstract

Specialised Content Knowledge (SCK) is defined by Ball, Hoover-Thames, and Phelps (2008) as mathematical knowledge essential for effective teaching. It is knowledge of mathematics that is beyond knowledge which would be required outside of teaching; for instance, the capacity to determine what misconception(s) may lie behind an error in calculation. Such knowledge should be core business of teachers of mathematics, and any perceived shortfall in SCK viewed as problematic. The research reported on here is part of a large study about multiplicative thinking involving approximately two thousand children between nine and twelve years of age and their teachers. Data were generated from semi-structured interviews and a written diagnostic assessment quiz. As part of that large project, forty-four Australian and New Zealand primary and middle school teachers were asked to respond to student work related to multiplicative thinking, particularly to concepts of numbers of equal groups and commutativity. Participants’ responses reflected confusion about a pivotal piece of SCK, the convention for naming arrays. As well, questionable assumptions about the children’s work samples were made. Given that there is not unanimous agreement amongst mathematics educators about naming conventions, these observations may not be surprising. Due to the sample size, broad generalisations cannot be made, but the results suggest that many teachers may have limited SCK with regards to the important mathematical area of Multiplicative Thinking (MT).

**Keywords:** Conventions, arrays, multiplicative, teacher content knowledge.

**REFERENCES **

Anthony, G., & Walshaw, M. (2002). Swaps and switches: Students’ understandings of commutativity. In B. Barton, K. C. Irwin, M. Pfannkuch, & M. O. J. Thomas (Eds.). *Mathematics Education in the South Pacific (Proceedings of the 25 ^{th} annual conference of the Mathematics Education Research Group of Australasia, Auckland*, (pp. 91-99). Sydney: MERGA.

Askew, M. (2018). Multiplicative reasoning: teaching primary pupils in ways that focus on functional relations. *The Curriculum Journal, 29*(3), 406-423.

Ball, D. L., Hoover-Thames, M., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? *Journal of Teacher Education, 59*(5), 389-407.

Barmby, P., Harries, T., Higgins, S., & Suggate, J. (2009). The array representation and primary children’s understanding and reasoning in multiplication. *Educational Studies in Mathematics, 70*, 217-241.

Barmby, P., Harries, T., & Higgins, S. (2010). Teaching for understanding/understanding for teaching. In I. Thompson (Ed.), *Issues in teaching numeracy in primary schools, (2nd. ed.).* Maidenhead, U.K.: Open University Press.

Benson, C. C., Wall, J. T., & Malm, C. (2013). The distributive property in Grade 3? *Teaching Children Mathematics, 19*(8), 498-506.

Behr, M., Harel, G., Post, T., & Lesh, R. (1994). Units of quantity: A conceptual basis common to additive and multiplicative structures. In G. Harel & J. Confrey (Eds.). *The development of multiplicative reasoning in the learning of mathematics* (pp. 121-176). Albany, NY: State University of New York Press.

Boaler, J., Chen, L., Williams, C., & Cordero, M. V. (2016). Seeing as understanding: The Importance of visual mathematics for our brain and learning. *Journal of Applied and Computational Mathematics, 5*, 1-6.

Brown, G., & Quinn, R. J. (2006). Algebra students’ difficulty with fractions: An error analysis. *Australian Mathematics Teacher, 62*, 28-40.

Bruner, J. S. (1966). *Toward a theory of instruction.* Cambridge, MA: Belkapp Press.

Clements, D., & Sarama, J. (2019) From Children’s Thinking to Curriculum to Professional Development to Scale: Research Impacting Early Maths Practice. In G. Hine, S. Blackley, & A. Cooke (Eds.). *Mathematics Education Research: Impacting Practice (Proceedings of the 42 ^{nd} annual conference of the Mathematics Education Research Group of Australasia) pp. 36-48.* Perth: MERGA. Retrieved from: https://merga.net.au/Public/Publications/Annual_Conference_Proceedings/2019-MERGA-conference-proceedings.aspx

Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. *Journal for Research in Mathematics Education, 26*(1), 66-86.

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

Creswell, J. W. (2007). *Qualitative inquiry and research design: Choosing among five approaches (2nd ed.).* Thousand Oaks: Sage publications.

Delaney, S., Ball, D. L., Hill, H. C., Schilling, S. G., & Zopf, D. (2008). Mathematical knowledge for teaching: Adapting U.S. measures for use in Ireland. *Journal of Mathematics Teacher Education, 11*(3), 171-197. http://dx.doi.org/10.1007/s10857- 008-9072-1

Depaepe, F., Verschaffel, L., & Kelchtermans, G. (2013). Pedagogical content knowledge: A systematic review of the way in which the concept has pervaded mathematics educational research. *Teaching and Teacher Education 34*, 12-25

Downton, A. (2008). Links between children’s understanding of multiplication and solution strategies for division. In M. Goos, R. Brown & K. Makar (Eds), *Navigating currents and charting directions (Proceedings of the 31st annual conference of the Mathematics Education Research Group of Australasia*, pp 171-178). Brisbane: MERGA Inc.

Goonen, B., & Pittman-Shetler, S. (2012). The struggling math student: From mindless manipulation of numbers to mastery of mathematical concepts and principles. *Focus on Basics, 4*(5), 24-27.

Haylock, D. (2010). *Mathematics explained for primary teachers. (4th. Ed.).* London: Sage.

Hill, H. C., Ball, D. L., & Schilling, S. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers' topic-specific knowledge of students. *Journal for Research in Mathematics Education, 39*(4), 372-400.

Hurst, C. (2017). Children have the capacity to think multiplicatively, as long as …. *European Journal of STEM Education, 2*(3), 1-14.

Hurst, C. (2018). A tale of two kiddies: A Dickensian slant on multiplicative thinking. *Australian Primary Mathematics Classroom, 23*(1), 31-36.

Hurst, C., Hurrell, D., & Huntley, R. (in press). Factors and multiples: Important and misunderstood. *International Journal on Teaching and Learning Mathematics. *

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

Jacob, L., & Mulligan, J. (2014). Using arrays to build towards multiplicative thinking in the early years. *Australian Primary Mathematics Classroom, 19*(1), 35-40.

Jacob, L., & Willis, S. (2001). Recognising the difference between additive and multiplicative thinking in young children. In J. Bobis, B. Perry & M. Mitchelmore. (Eds.), *Numeracy and beyond (Proceedings of the 24th Annual Conference of the Mathematics Education Research Group of Australasia, Sydney* pp. 306-313). Sydney: MERGA.

Jacob, L., & Willis, S. (2003). The development of multiplicative thinking in young children. In: L. Bragg, C. Campbell, G. Herbert, & J. Mousley (Eds.), *Mathematics education research: Innovation, networking, opportunity.* *Proceedings of the* *26th Annual Conference of the Mathematics Education Research Group of Australasia, 6 - 10 July 2003*, Deakin University, Geelong.

Lesh, R., Cramer, K., Doerr, H. M., Post, T., & Zawojewski, J. S. (2003). Model development sequences. In R. Lesh, & H. M. Doerr (Eds.), *Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching* (pp. 3-33). Mahwah, NJ: Lawrence Erlbaum

McKenna, S. (2019). Discussion, conjectures, noticing. *Mathematics Teaching, 267*, 38-39.

Mulligan, J., & Watson, J. (1998). A developmental multimodal model for multiplication and division. *Mathematics Education Research Journal, 10*(2), 61-86.

Mutodi, P., & Ngirande, H. (2014). The nature of misconceptions and cognitive obstacles faced by secondary school mathematics students in understanding probability: A case study of selected Polokwane Secondary Schools. *Mediterranean Journal of Social Sciences, 5*(8), 446-455.

Nunes, T. & Bryant, P. (1996). *Children doing mathematics*. Oxford, UK: Blackwell.

McKenna, S. (2019). Discussion, conjectures, noticing. *Mathematics Teaching, 267,* 38-39.

Reys, R. E., Rogers, A., Bennett, S., Cooke, A., Robson, K., Ewing, B., & West, J. (2020). *Helping children learn mathematics, (3rd. Aust. Ed.)*. Milton, Qld: Wiley.

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.

Schneider, M., Grabner, R. H., & Paetsch, J. (2009). Mental number line, number line estimation, and mathematical achievement: Their interrelations in grades 5 and 6. *Journal of Educational Psychology, 101*(2), 359.

Shulman, L. S. (1986). Those who understand, knowledge growth in teaching. *Educational Researcher 15*(2), 4-14.

Siegler R. S., Duncan G. J., Davis-Kean P. E., Duckworth K., Claessens A., Engel M., Susperreguy M. I., & Chen M. (2012). Early predictors of high school mathematics achievement. *Psychological Science, 23*(7), 691-697.

Siemon, D., Warren, E., Beswick, K., Faragher, R., Miller, J., Horne, M., Jazby, D., Breed, M., Clark, J., & Brady, K. M. (2021). *Teaching mathematics: Foundations to middle years – 3rd ed. *Melbourne: Oxford University Press.

Siemon, D., Beswick, K., Brady, K. M., Clark, J., Faragher, R., & Warren, E. (2015). *Teaching mathematics: Foundations to middle years – 2nd ed. *Melbourne: Oxford University Press.

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

Sophian, C., & Madrid, S. (2003). Young childrens' reasoning about many-to-one correspondences. *Child Development, 74*(5), 1418-1432.

Sriraman, B. (2006). Conceptualizing the model-eliciting perspective of mathematical problem solving. In M. Bosch (Ed.), *Proceedings of the Fourth Congress of the European Society for Research in Mathematics Education (CERME 4)* (pp. 1686-1695).

Stott, D. (2016). Using arrays for multiplication in the Intermediate Phase. *Learning and teaching mathematics, 21*(6-10).

Swan, P., & Marshall, L. (2010). Revisiting mathematics manipulative materials. *Australian Primary Mathematics Classroom, 15(*2), 13-19.

Tipps, S., Johnson, A., & Kennedy, L. M. (2011). *Guiding children’s learning of mathematics. (2nd ed.).* Belmont, California: Wadsworth. (p. 203; p. 237)

Vale, C., & Davies, A. (2007). Dean’s great discovery: Multiplication, division and fractions. *Australian Primary Mathematics Classroom. 12*(3), 18–22.

Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2013). *Elementary and middle school mathematics: Teaching developmentally (8th. Ed.).* Boston: Pearson.

Way, J. (2011). Multiplication series: *Number arrays*. Retrieved from https://nrich.maths.org/2466

Waller, P. P., & Marzocchi, A. S. (2020). From rules that expire to language that inspires. *The Mathematics Teacher, 113*(7), 544-550.

Wright, V. J. (2011). *The development of multiplicative thinking and proportional reasoning: Models of conceptual learning and transfer*. (Doctoral dissertation). University of Waikato, Waikato. Retrieved from http://researchcommons.waikato.ac.nz/.

Young-Loveridge, J. (2005). Fostering multiplicative thinking using array-based strategies. *The 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 pp. 635-643).* Palmerston North, NZ: MERGA

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