Educational Studies in Mathematics

, Volume 70, Issue 3, pp 217–241 | Cite as

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

  • Patrick Barmby
  • Tony Harries
  • Steve Higgins
  • Jennifer Suggate


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.


Array Multiplication Representations Reasoning Understanding 



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


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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  • Patrick Barmby
    • 1
  • Tony Harries
    • 1
  • Steve Higgins
    • 1
  • Jennifer Suggate
    • 1
  1. 1.Durham UniversityDurhamUK

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