Common Misunderstandings - Level 3 Multiplicative Thinking

Although most students at this Level have some knowledge of the multiplication facts to 100 and can perform simple multiplication and division procedures correctly, many rely on rote learning and/or a naïve, groups of understanding for multiplication based on repeated addition (often counting equal groups by ones). With little or no access to a broader range of ideas for multiplication they find it difficult to develop efficient mental strategies, and as a consequence, tend to rely on memorised procedures for multiplying and dividing larger whole numbers and decimals.

This could be due to/associated with:

  • an inability to trust the count and see numbers as countable units in their own right, that is, view 6 items as 1 six (“a six”) rather than 6 ones (see Level 2.2 Efficient Counting Tool);
  • poorly developed or non-existent mental strategies for addition and subtraction;
  • an over-reliance on physical models to solve simple multiplication problems; and/or
  • a limited exposure to alternative models of multiplication.

By the end of Level 3 students need to be able to think about multiplication in a number of different ways to recognise when multiplication is required and how it relates to division, support efficient mental and written computation, and solve a wider range of problems involving equal groups, simple proportion, combinations, and rate. To do this they need to recognise the numbers 2 to 10 as countable units, count large collections more efficiently, and appreciate the advantages of representing multiplicative situations in terms of arrays and regions. That is, that arrays and regions

  • more neutrally represent all aspects of the multiplicative situation, that is, the number of groups, the equal number in each group, and the product (last two not as evident in groups of models);
  • can be used to relate the two ideas for division, partition (or sharing) and quotition (or how many groups in), to multiplication;
  • support commutativity (eg, 3 fours can be rotated to show that it is the same as 4 threes) so halving the amount of learning required for the multiplication facts;
  • support more efficient, generalisable mental strategies for multiplication; and
  • provide a basis for moving from a count of equal groups (eg, 1 six, 2 sixes, 3 sixes, 4 sixes, …) to a constant number of groups (eg, 6 ones, 6 twos, 6 threes, 6 fours, 6 fives …) which supports more efficient mental strategies (eg, 6 groups of anything is double 3 groups or 5 groups and 1 more group).

More importantly, arrays and regions support the shift from an additive groups of model to a factor-factor-product model which is needed to support fraction representation, the multiplication and division of larger whole numbers, fractions and decimals, and algebra. An awareness of the for each idea or Cartesian product is also needed at this Level to support work in Chance and Data (eg, problems involving combinations), measurement (including problems involving rates), and fraction representation. For example, if a diagram showing thirds is halved and halved again, there are 4 smaller parts for each third, this is not a groups of idea that corresponds to students experience.

A key indicator of the extent to which students have developed a broader range of ideas to support multiplicative thinking is the extent to which they manipulate both the size of the group and the number of groups to meet specific needs (eg, instead of committing 6 eights to memory in a meaningless or rote way, recognise that this can be thought of as 5 eights and 1 more eight, or 3 eights doubled).