The numeracy process: how numeracy ideas are learnt (by John Munro)

8 types of math knowledge:

Conceptual Knowledge that is, what ideas mean, knowing ‘when to’ and ‘why’, ‘what’ two or more ideas share, recognising examples of an idea,

Algorithmic or Procedural Knowledge algorithms, procedural knowledge, knowing ‘how to do’ things in mathematics, for example, subtract by decomposing, use a calculator to add count down

Mathematical Relationships relating ideas, eg, ordinal, equality and inequality

Language Knowledge talk, read, write, use maths vocabulary and the symbolism of maths

The Task and Outcome

Automatized Factual Knowledge for example, recall the four times table, subtraction ‘facts’. Information recalled automatically demands less mental attention

Problem-Solving Strategies in maths; ways of thinking about mathematical ideas

How to think about and learn maths ideas the actions they use to learn maths such as visualisation, talking about ideas they are learning, and asking ‘What does this remind me of?’ This helps them link with what they know.

Beliefs and Attitudes about learning maths

what maths is like (e.g. dull and boring), how maths is learnt (e.g. you mustn’t guess or take risks vs risk-taking and guessing important in learning), and themselves as maths learners (e.g. I could never learn maths vs I find it hard but I can learn it)


The development of arithmetical abilities (Butterworth, 2005):

Piaget (as cited in Butterworth, 2005) argues that children need three basic capacities:

  • Capacity to Reason Transitively: the child should be able to reason from the facts that if A is bigger than B, and B is bigger than C, then A is bigger than C.
  • Number is only affected by adding or taking away objects. Moving objects around, using different objects (cars, cats, dots, pens, etc) does not change the number.
  • Ability to ignore features of the objects being counted (that shape, colour, size doesn’t matter). A set of three cats has the same numerosity as a set of three chairs.  Numerosity is the number of things in a set.

Butterworth (2005) identifies three main stages in the development of counting as an addition strategy:

  1. Counting All – 3 + 5 is counted separately as 1,2,3 and then 1,2,3,4,5 then counting them all together.
  2. Counting on from first – is starting with the first number (3 in 3 + 5) and count 4, 5, 6, 7, 8 (usually using an aid like fingers).
  3. Counting on from larger – selecting the larger number (5 in 3 + 5) and counting 6, 7, 8.

Full text available here:

More resources available at:

Copyright Dr. John Munro 2013