Primary maths: Are you using confusing shortcuts?

We try to give our pupils a quick route to understanding with mathematical shortcuts, but we could be doing the exact opposite, writes Jon Parsons

Primary maths: two boys taking shortcut across field

A quick look through my Twitter feed lately suggests that two of the hottest education topics at the moment are cognitive load theory and the role of vocabulary and long-term memory in learning.  

As my brain often focuses on such things from a mathematics teaching standpoint, I have found myself wondering why we still teach children incorrect mathematical vocabulary and confusing shortcuts, such as:

When is a diamond not a diamond?

When it’s a rhombus. Or a rotated square. Or a kite. The problem with teaching the children the use of the word "diamond" in a mathematical context is that there is no mathematical diamond shape.

In everyday life we use the word to describe the gem, or perhaps the suit in a deck of cards, or the playing area in baseball.

These are all everyday uses, but are not interchangeable or even the same shape. Therefore using the word "diamond" to describe a specific quadrilateral is at best confusing and at worst incorrect.

Quick read: How we almost eliminated marking in maths

Quick listen: 3 myths about project-based learning

Want to know more? The truth about homework

We know that prior knowledge is recalled and used when building new understanding; if that prior knowledge is incorrect it can actively hinder the learning of new concepts.

Furthermore, using diamond instead of either square or rhombus will also make the statement of all squares being rhombuses but not all rhombuses being squares particularly difficult, as we are then saying all diamonds are diamonds but not all diamonds are diamonds.

One of the often cited reasons for using the word "diamond" is that it is easier for a child to learn instead of "rhombus"; actually, if you teach a child it is a rhombus, it is no more difficult to remember than square or triangle, and it prevents cognitive overload in later learning, caused by saying to a child “for the last two years we taught you that this was a diamond; actually, it’s called a rhombus”. 

Vocabulary and meaning matter in mathematics, so let’s get this one right from the beginning.

No more alligators

It seems intuitive to teach the inequality symbols in a child-friendly way, using alligators or crocodiles. However, what actually happens is that the cognitive load of trying to remember both the principles of the concept, as well as which alligator eats which numbers, can make fully embedding this understanding difficult.

The longer that pupils spend in the mode that relies on the shortcut, the more this becomes a poor proxy for actually understanding when, why and how the symbol should be used.

For children experiencing inequality symbols for the first time, there will be little or no prior knowledge to build on. It is therefore both sensible, and builds a better, long-term understanding, to use the correct terminology from the start.

Explicit modelling of the use of the symbols in context, not only in number sentences but also as spoken and written sentences, is going to be far more useful than adding confusing shortcuts.

To round up or down?

Although we use the term “rounding down”, this can be extremely misleading for pupils, especially when no change is made to the digit.  

As I have mentioned previously, rhyme and chant (“five to nine, climb the vine, zero to four, down to the floor”), although potentially helpful to remember the procedure, can often become poor proxies for embedding concepts and actual learning. 

Ask a child why the rhyme/chant works, and they may find it difficult to articulate that the number in question is closer to x than y.  

When recently discussing rounding with a group of perfectly capable Year 6 mathematicians, they were able to perfectly clearly articulate what to do when rounding, but generally had no clear idea as to why.

The understanding of the concept, which I know they had been taught, had been replaced with the recall of the shortcut.  

As teachers, we need to be aware that repetition of a rhyme, song or shortcut is not evidence that a concept has been learned and stored in the long term memory.

Some practical ideas to try:

Modelling how to use symbols, in a variety of contexts, will help to embed the understanding of them. This includes the modelling of correct vocabulary and usage in problems, as well as thought processes.

If pupils have little or no prior understanding (when you’re introducing new shapes or symbols, for example), then using the correct vocabulary and imagery from the start of their learning process will help to cut through the noise and lighten the learning load. 

Start your teaching by establishing what prior understanding pupils bring to new concepts. Do they have deeply embedded understanding of concepts, or are they using shortcuts as methods to mask misconceptions?

Carefully plan the questions that you ask. “What do you understand about rounding numbers?” will give you a better sense on pupil learning than “How do we round numbers?”

Jon Parsons is deputy headteacher at White Meadows Primary Academy

Log in or register for FREE to continue reading.

It only takes a moment and you'll get access to more news, plus courses, jobs and teaching resources tailored to you