Maths misconceptions – How to clear up common mistakes
Think you know what maths is all about? Think again… your assumptions might be muddying the waters of your teaching
Have you ever spoken to a maths-loving colleague about why they have an affinity for the subject, only to hear maths misconceptions like “it is always right or wrong – there are no confusing grey areas”? You wouldn’t be the first.
Such views go along with the idea that mathematical truths are timeless and absolutely certain, which can be appealing.
Conversely, when pupils’ only experience of maths is getting ticks or crosses, the subject can lose its appeal.
It contributes to making many children anxious, which affects their capacity to learn. It also contributes to long-term negative attitudes towards situations that involve numbers and quantitative reasoning; calculating bills, reading graphs, measuring, etc.
The funny thing is, maths isn’t about a singular truth at all.
Take geometry as an example; what we learn in school about angles and lines is only true for geometry on a flat surface. (Such as that there are 180 degrees in a triangle.)
There are, however, many other kinds of geometry, including geometry done on a sphere. And if we drew a triangle on a sphere, we might get different results.
A triangle drawn from the North Pole, down to the equator, around a quarter of the equator, and back up to the North Pole, for instance, will have three right angles and so an angle sum of 270 degrees.
So, our seemingly unshakeable ‘truth’ that a triangle has 180 degrees, turns out to be more contingent on context.
The idea of singular truths is one of several misleading but common maths misconceptions, or dogmas.
There are several ways we can address these dogmas. These will both reduce anxiety about doing maths and make the subject seem more relevant along the way…
One of the reasons, perhaps, that many people come to believe that maths is always right and wrong, is that a lot of teachers’ questions have single answers (and these are often questions that machines can answer more quickly!).
A simple shift in the kind of questions we ask can make a significant difference; if I shift my attention away from whether the answer I get from a pupil is correct or not, and onto the method they have used to get that answer, I am immediately out of the right/wrong paradigm.
For example, instead of asking, “What is 7 x 8?”, we might start to shift our questions in ways that invite multiple correct answers. For example: “What methods do you have for working out 7 x 8?”.
As teachers, we are more interested in methods than answers, so the latter is, for us, a more useful question.
It gives us more information about how pupils are learning. Crucially, it’s also more engaging for the children than being focused on the result.
Maths building blocks
Another of the common maths misconceptions is that we need to teach it in a building block manner.
Teachers frequently make this assumption. For example, they might say that a pupil cannot understand the material in Y4, if they haven’t yet completed ‘simpler’ learning in Y3.
However, just because we can arrange maths in a hierarchy from simple to complex, doesn’t mean that’s the most effective approach.
If we look at other areas of learning, starting simple is not what we do. Consider how children learn their home language(s) for example; it is apparent that they are immersed in complexity from the very start.
Children notice patterns and try out ways of talking and expressing themselves, getting feedback from the environment.
They don’t get restricted exposure to simple words, like monosyllabic ones, before being exposed to more complex words and grammar.
But that’s what we sometimes do in maths; we expose children to a restricted set of numbers (1-20) and shapes (plane, mostly regular ones).
We need to stop assuming that children need simplicity.
The big picture
Research shows that children can thrive when we offer them ‘the big picture’, or a sense of a coherent ‘whole’.
They can learn when the complex situation provides immediate feedback on their actions; just like they do when they learned to speak.
The National Centre for Excellence in the Teaching of Mathematics (NCETM) professional development materials offer some excellent examples of working with bigger structures from the start of a topic.
For instance, the NCETM suggests working with children on naming the numbers 20 to 99, before working on 11-19.
This works well, because the numbers above 20 have relatively regular naming conventions. This means children can use their powers of pattern-spotting and structuring to figure them out.
After becoming successful with this structure, they can then work on the exceptions. We’d also advocate a dual naming of 11-19 (also proposed by the NCETM). These work alongside the usual names, to make the structure apparent. E.g., reading 13 and ‘one-ty-three’, just like 63 is read ‘six-ty-three’.
The dogma of maths as a building block subject leads directly to grouping pupils according to their prior attainment. This offers some children a repetitive cycle of number-dominated skills to learn.
We see no reason, just because a pupil struggles with aspects of early number work, to deny them the opportunity to work with bigger numbers. Or, indeed, to learn algebra.
In fact, it may be precisely working on big numbers that helps make number structure become apparent.
Breaking down maths misconceptions
Teaching maths is a complex and fascinating task. There aren’t necessarily any easy answers for how to help children who are struggling.
However, when we no longer see maths as purely right or wrong, it becomes more human, more approachable.
And, when students come to appreciate a mathematical structure (such as how number naming works, or how addition and subtraction are related) this can open up worlds of interest and engagement.
Give it a go…
Alf Coles is professor of mathematics education at the University of Bristol. Nathalie Sinclair is distinguished university professor at Simon Fraser University.