title: Does the cue help? Children's understanding of multiplicative concepts in different problem contexts.
abstract: BACKGROUND
  Understanding arithmetical principles is a key part of a conceptual understanding of mathematics. However, very little attention has been paid to children's understanding of multiplicative, as compared to additive, principles.
  
  
  OBJECTIVE
  This study investigated (a) children's ability to use commutative and distributive cues to solve multiplication problems, (b) whether their ability to use these cues depends on the problem context, and (c) whether separate mechanisms might underlie children's understanding of commutativity and distributivity.
  
  
  METHODS
  Twenty-seven 9-year-olds (Year 5) and thirty-two 10-year-olds (Year 6).
  
  
  METHODS
  Forty-eight multiplication problems (with a multiple-choice response format) were presented to children. There were four types of problem: Commutative, Distributive, Combined commutative-distributive (all preceded by a cue) and No cue problems. Each type of problem was presented in three different contexts: Isomorphism of measures, Area, and Cartesian product.
  
  
  RESULTS
  Children demonstrated a good understanding of commutativity but a very poor understanding of distributivity. A common mistake in the distributive problems was to select the number that was one more, or one less, than the answer in the cue. Children's understanding of distributivity (but not commutativity) seemed to depend on the problem context. Factor analysis suggested that separate factors underlie the ability to solve commutative and distributive problems.
  
  
  CONCLUSIONS
  Nine- and 10-year-olds understand commutativity, but are unable to use the distributive principle in multiplication. Their errors suggest that they may confuse some of the principles of multiplication with those of addition. When children do begin to understand the principle of distributivity, they most easily apply it in the context of Isomorphism of measures multiplication problems. The implications for mathematical education are discussed.
authors.affiliation: Department of Experimental Psychology, University of Oxford, Oxford, UK. sarah.squire@psych.ox.ac.uk
authors.forename: Sarah
authors.initials: S
authors.lastname: Squire
authors.forename: Charlotte
authors.initials: C
authors.lastname: Davies
authors.forename: Peter
authors.initials: P
authors.lastname: Bryant
mesh_terms: D002648:Child; D003071:Cognition; D003210:Concept Formation; D003463:Cues; D005260:Female; D006801:Humans; D008297:Male; D008433:Mathematics; D011340:Problem Solving