John Dabell is a teacher with over 20 years teaching experience across all key stages. He has worked as a national in-service provider and is a trained OfSTED inspector.
As teachers we need to be acutely aware that the language of mathematics is not always precise and that pictorial images and descriptions have a powerful effect on the development of a shared understanding of meaning.
The last thing we want is for resources and teaching to actually support misconceptions and overgeneralisations but this happens.
If you have a classroom poster showing maths shapes then take a good look at it. Can you spot any errors? Are there any shapes that have been mislabelled?
It might not necessarily be obvious but my betting is that a couple of 2D shapes won’t be quite right. It could well be that what we are teaching isn’t quite right either.
The chances are ‘square’ and ‘rectangle’ are staring you in the face but these shape names are mocking us and making fools of us.
This is because they aren’t actually named correctly and unfortunately our teaching sometimes dovetails and reinforces these faulty representations as we fail to draw attention to the rectangle family.
Many children think a rectangle has to have two long sides and two short sides. This is because these are the only examples they see.
When Is A Square Not A Square?
There is no such thing as ‘a square’.
If we point to shapes and say ‘this is a square’ and ‘this is a rectangle’ then we are at fault because this is mislabelling and we are misleading children.
There are two types of rectangles within the rectangle family: the ‘square’ should be labelled ‘square rectangle’ and the ‘rectangle’ should be labelled ‘oblong rectangle’ or a ‘non-square rectangle’ but you won’t see any posters like that.
‘Rectangle’ refers to any quadrilateral whose corners are all right-angled, opposite sides are equal and parallel and its diagonals bisect each other.
A square rectangle is all those things but all four sides are equal and its diagonals cross at right-angles.
So a square is definitely a rectangle but it is equilateral and equiangular too. All the other rectangles are non-square rectangles because they have one pair of sides longer than the other. These are oblong rectangles. A rectangle can be tall and thin, short and fat or all the sides can have the same length. So, a square is a special kind of rectangle.
Commonly you will see that ‘square’ is used as a noun when it should in fact be used as an adjective to describe the type of rectangle it is. The same goes for ‘oblong’ too. ‘Oblong’ is the describing word.
If children hear the word ‘square’ they won’t be thinking ‘rectangle’. In their mind’s eye they see ‘square’ and ‘rectangle’ as different and distinct because that’s what the posters tell them along with images in maths textbooks, maths dictionaries and online materials. They are surrounded by imprecision and ambiguity where relationships are left unexplored and shapes are viewed exclusively.
The classroom environment and the resources we use act as second teachers and so they play a crucial role in children’s learning.
If you’ve got a defective poster then isn’t it time to bin it?
The Shape Of Things To Come
Some might say that we are ‘splitting hairs’ or ‘making a fuss over nothing’ but I disagree. We have to get the language spot on as not being laser-sharp is lazy and hazy maths.
Maths is a precise and unambiguous language so we need to tread very carefully.
If we don’t get our mathematical discourse right then we risk doing harm because somewhere along the line someone will have to pick up the pieces.
Children are disadvantaged if they do not have a rich mathematical vocabulary or if their vocabulary is faulty.
Sometimes their faulty knowledge and understanding is because they have been taught a misconception and they have swallowed it whole.
Building a facility with mathematical language involves being taught an accurate mathematical register from the outset. We shouldn’t be saying ‘square’ when actually what we mean is square rectangle.
Images of a ‘square’ and a ‘rectangle’ have become so entrenched and fixed in our minds that to challenge them almost seems odd but it isn’t. But challenge we must and that starts with teaching children to ‘see’ the rectangle family from the early years onwards.
Language is central to mathematics and the better children are at using words with precision, the better they will be able to show their maths knowledge.
Rather than simply ‘name’ shapes we should help children to identify the properties of shapes, collect all the shapes that fit these properties and then name this set.
We need to steer clear of categorising ‘rectangles’ and ‘squares’ separately as it does not allow for growth in understanding that ‘square’ is a specific classification of a rectangle.
Simply ‘naming’ shapes means children fail to appreciate the inclusive nature of shape and their inter-relationships.
We need to employ the power of precision and help children realise the naming and describing power of the mathematical register.