Multiplication facts or multiplicative thinking

What Maths Teachers wish Year 9 students knew
26 February 2019

We just want them to know their tables!

It is a truth universally acknowledged by high school maths teachers that students need to be fluent in multiplication facts. (Apologies to Jane Austen)

You can read more about this claim in my previous post: What Maths Teachers wish Year 9 students knew

I have been thinking about why this is the case, what is so special about multiplication facts, and whether it is more an indicator of something else.

Maths teachers like to teach algebra. Simplifying algebraic expression, and factorising quadratics are made much easier if one is at home with multiplication facts or relationships. For example if the constant value in a quadratic is 27, it is helpful to be able to instantly recognise that 27 is 3 times 9. If you have to explore all sorts of possibilities, then the working memory is taken up with the exploration, rather than the process of factorising.

Other mathematical processes that are made simpler by facility with multiplication are fraction operations, finding areas, estimation and the long multiplication and division algorithms. When we teach probability, it helps to be able to multiply. But really that is about it! I would love to know what else is helped by facility with multiplication tables.

Do students need tables or multiplicative thinking?

When teachers say they want their students to have their multiplication facts at their fingertips, is that really what is needed, or is it an indicator of multiplicative thinking? The numeracy project and its equivalent in other countries has introduced this idea to teachers, to categorise students as to whether they still think additively or have progressed to multiplicative thinking.

Multiplicative thinking includes the following: (Taken from a document from Dianne Siemon, Associate Professor of Mathematics Education.)

  • a capacity to work flexibly and efficiently with an extended range of numbers (for example, larger whole numbers, decimals, common fractions, ratio, and per cent),
  • an ability to recognise and solve a range of problems involving multiplication or division including direct and indirect proportion, and
  • the means to communicate this effectively in a variety of ways (for example, words, diagrams, symbolic expressions, and written algorithms).

I attended a fascinating keynote by Shelley Dole some years ago, where she suggested that children have a greater capacity for multiplicative thinking when they start school, but the sole emphasis on counting and adding may be detrimental. Should we use more multiplicative thinking quite early on, rather than wait until addition in all its forms is mastered?

It is quite possible to be automatic and fast at multiplication tables, without being a multiplicative thinker. Some drills and timed tests encourage speed at recall of facts, which may not be accompanied by conceptual understanding.

The aim is fluency

The NCTM definition of procedural fluency is as follows:

“Procedural fluency is the ability to apply procedures accurately, efficiently, and flexibly; to transfer procedures to different problems and contexts; to build or modify procedures from other procedures; and to recognize when one strategy or procedure is more appropriate to apply than another.”

This is what maths teachers would like from their students, to be procedurally fluent in the basic operations.

Which brings me back to my favourite teaching quote from Kevin Maxwell: “Our job is to teach the students we have. Not the ones we would like to have. Not the ones we used to have. Those we have right now. All of them.”

So though maths teachers might wish for our students to arrive at high school, procedurally fluent and multiplicative thinkers, the reality is that many will not be. Our job is not to complain about them or their teachers, but to increase our capacity to help them progress from where they are.

High school teachers are creating the future primary school teachers, just as they were influential in the formation of the current primary school teachers’ feelings towards maths. It is estimated that at least one third of primary school teachers have feelings about mathematics that exhibit the hallmarks of trauma. I discuss this in a previous post: Maths trauma can be healed

The answer is systemic.

What do you think?

Is what teachers want really multiplication tables, or is that an indicator?

What can teachers at all levels to do to help each other and the learners?

 

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