# The array representation and primary children’s understanding and reasoning in multiplication

## Abstract

We examine whether the array representation can support children’s understanding and reasoning in multiplication. To begin, we define what we mean by understanding and reasoning. We adopt a ‘representational-reasoning’ model of understanding, where understanding is seen as connections being made between mental representations of concepts, with reasoning linking together the different parts of the understanding. We examine in detail the implications of this model, drawing upon the wider literature on assessing understanding, multiple representations, self explanations and key developmental understandings. Having also established theoretically why the array representation might support children’s understanding and reasoning, we describe the results of a study which looked at children using the array for multiplication calculations. Children worked in pairs on laptop computers, using Flash Macromedia programs with the array representation to carry out multiplication calculations. In using this approach, we were able to record all the actions carried out by children on the computer, using a recording program called Camtasia. The analysis of the obtained audiovisual data identified ways in which the array representation helped children, and also problems that children had with using the array. Based on these results, implications for using the array in the classroom are considered.

## Keywords

Array Multiplication Representations Reasoning Understanding## Notes

### Acknowledgements

The authors would like to thank the referees who looked at this paper, for their constructive comments and their suggestions of related references.

## References

- Ainsworth, S. (2006). DeFT: a conceptual framework for considering learning with multiple representations.
*Learning and Instruction*,*16*(3), 183–198. doi: 10.1016/j.learninstruc.2006.03.001.CrossRefGoogle Scholar - Anghileri, J. (2000).
*Teaching number sense*. London: Continuum.Google Scholar - Brinkmann, A. (2003). Graphical knowledge display—mind mapping and concept mapping as efficient tools in mathematics education.
*Mathematics Education Review*,*16*, 35–48.Google Scholar - Chi, M. T. H., De Leeuw, N., Chiu, M.-H., & LaVancher, C. (1994). Eliciting self-explanations improves understanding.
*Cognitive Science*,*18*, 439–477.CrossRefGoogle Scholar - Christou, C., & Papageorgiou, E. (2007). A framework of mathematics inductive reasoning.
*Learning and Instruction*,*17*(1), 55–66. doi: 10.1016/j.learninstruc.2006.11.009.CrossRefGoogle Scholar - Davis, R. B. (1984).
*Learning mathematics: the cognitive approach to mathematics education*. London: Croom Helm.Google Scholar - Davison, I. (2003). Using an interactive whiteboard to facilitate pupil understanding of quadrilateral definitions.
*Proceedings of the British Society for Research into Learning Mathematics*,*23*(1), 13–18.Google Scholar - Davydov, V. V. (1991). A psychological analysis of multiplication. In L. P. Steffe (Ed.),
*Psychological abilities of primary school children in learning mathematics. Soviet studies in mathematics education series, vol 6*(pp. 1–85). Reston: NCTM.Google Scholar - DfEE (1999).
*The National Numeracy Strategy: framework for teaching mathematics from reception to year 6*. Sudbury: DfEE Publications.Google Scholar - DfES (2006).
*Primary framework for literacy and mathematics: core position papers underpinning the renewal of guidance for teaching literacy and mathematics*. Norwich: DfES.Google Scholar - Dickson, L., Brown, M., & Gibson, O. (1984).
*Children learning mathematics: a teacher*'*s guide to recent research*. London: Cassell Educational Ltd.Google Scholar - Ernest, P. (1994). Social constructivism and the psychology of mathematics education. In P. Ernest (Ed.),
*Constructing mathematical knowledge: epistemology and mathematics education*(pp. 62–72). London: Falmer.Google Scholar - Flexer, R. J. (1986). The power of five: the step before the power of ten.
*The Arithmetic Teacher*,*34*(3), 5–9.Google Scholar - Goldin, G. A. (1998). Representational systems, learning and problem solving in mathematics.
*The Journal of Mathematical Behavior*,*17*(2), 137–165. doi: 10.1016/S0364-0213(99)80056-1.CrossRefGoogle Scholar - Greer, B. (1992). Multiplication and division as models of situations. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 276–295). New York: Macmillan.Google Scholar - Groβe, C. S., & Renkl, A. (2006). Effects of multiple solution methods in mathematics learning.
*Learning and Instruction*,*16*, 122–138. doi: 10.1016/j.learninstruc.2006.02.001.CrossRefGoogle Scholar - Harries, T., & Barmby, P. (2007). Representing and understanding multiplication.
*Research in Mathematics Education*,*9*, 33–45. doi: 10.1080/14794800008520169.CrossRefGoogle Scholar - Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 65–97). New York: Macmillan.Google Scholar - Izsák, A. (2004). Teaching and learning two-digit multiplication: coordinating analyses of classroom practice and individual student learning.
*Mathematical Thinking and Learning*,*6*(1), 37–79. doi: 10.1207/s15327833mtl0601_3.CrossRefGoogle Scholar - Lawson, A. E. (1994). Research on the acquisition of science knowledge: epistemological foundations of cognition. In D. Gabel (Ed.),
*Handbook of research on science teaching and learning*(pp. 131–176). New York: Macmillan.Google Scholar - Lawson, M. J., & Chinnappan, M. (2000). Knowledge connectedness in geometry problem solving.
*Journal for Research in Mathematics Education*,*31*(1), 26–43. doi: 10.2307/749818.CrossRefGoogle Scholar - Leighton, J. P. (2004). Defining and describing reason. In J. P. Leighton, & R. J. Sternberg (Eds.),
*The nature of reasoning*(pp. 3–11). Cambridge: Cambridge University Press.Google Scholar - McGowan, M., & Tall, D. (1999). Concept maps and schematic diagrams as devices for the growth of mathematical knowledge. In O. Zaslavsky (Ed.)
*Proceedings of the 23rd conference of PME, Haifa, Israel, July 1999 (vol 3)*, 281–288.Google Scholar - Meyer, J. H. F., & Land, R. (2003) Threshold concepts and troublesome knowledge: linkages to ways of thinking and practising within the disciplines.
*Enhancing teaching-learning environments in undergraduate courses project, occasional report 4.*[Accessed 27th July 2007 at http://www.ed.ac.uk/ etl/publications.html] - Moseley, B. (2005). Students’ early mathematical representation knowledge: the effects of emphasising single or multiple perspectives of the rational number domain in problem solving.
*Educational Studies in Mathematics*,*60*, 37–69. doi: 10.1007/s10649-005-5031-2.CrossRefGoogle Scholar - NCTM (2000).
*Principles and standards for school mathematics*. Reston: National Council of Teachers of Mathematics.Google Scholar - Nickerson, R. S. (1985). Understanding understanding.
*American Journal of Education*,*93*(2), 201–239. doi: 10.1086/443791.CrossRefGoogle Scholar - Niemi, D. (1996). Assessing conceptual understanding in mathematics: representations, problem solutions, justifications, and explanations.
*The Journal of Educational Research*,*89*(6), 351–364.Google Scholar - Nunes, T., & Bryant, P. (1996).
*Children doing mathematics*. Oxford: Blackwell.Google Scholar - Powell, A. B., Francisco, J. M., & Maher, C. A. (2003). An analytical model for studying the development of learners’ mathematical ideas and reasoning using videotape data.
*The Journal of Mathematical Behavior*,*22*, 405–435. doi: 10.1016/j.jmathb.2003.09.002.CrossRefGoogle Scholar - Seufert, T. (2003). Supporting coherence formation in learning from multiple representations.
*Learning and Instruction*,*13*, 227–237. doi: 10.1016/S0959-4752(02)00022-1.CrossRefGoogle Scholar - Sfard, A. (1991). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin.
*Educational Studies in Mathematics*,*22*(1), 1–36. doi: 10.1007/BF00302715.CrossRefGoogle Scholar - Sierpinska, A. (1994).
*Understanding in mathematics*. London: Falmer.Google Scholar - Simon, M. A. (2006). Key developmental understandings in mathematics: a direction for investigating and establishing learning goals.
*Mathematical Thinking and Learning*,*8*(4), 359–371. doi: 10.1207/s15327833mtl0804_1.CrossRefGoogle Scholar - Skemp, R. (1976). Relational understanding and instrumental understanding.
*Mathematics Teaching*,*77*, 20–26.Google Scholar - Steinbring, H. (1997). Epistemological investigation of classroom interaction in elementary mathematics teaching.
*Educational Studies in Mathematics*,*32*(1), 49–92. doi: 10.1023/A:1002919830949.CrossRefGoogle Scholar - Thompson, I. (1999).
*Issues in teaching numeracy in primary schools*. Oxford: Open University Press.Google Scholar - White, R., & Gunston, R. (1992).
*Probing understanding*. London: Falmer.Google Scholar - Williams, C. G. (1998). Using concept maps to assess conceptual knowledge of function.
*Journal for Research in Mathematics Education*,*29*(4), 414–421. doi: 10.2307/749858.CrossRefGoogle Scholar - Wittmann, E. C. (2005). Mathematics as the science of patterns—a guideline for developing mathematics education from early childhood to adulthood.
*Plenary Lecture at International Colloquium ‘Mathematical learning from Early Childhood to Adulthood’ (Belgium, Mons).*Google Scholar