Maths Misconceptions: Squares and Rectangles

Editor note: This blog post was originally published on 23 July 2018

There is always plenty of buzzy stuff to discuss in maths especially when it comes to misconceptions, muddles and mix-ups. It’s important to tackle maths misconceptions head-on.

In fact, this is one of the recommendations in the ‘Improving Mathematics at Key Stages Two and Three‘ report from 2018. Simply pleading ignorance and continuing to teach them is no longer an option. Instead, we can positively use these misunderstandings and misconstructions during formative assessment, enabling us to help pupils “develop richer and more robust conceptions“.

It’s time to go back to square one

The real problem with misconceptions is that when they’re unwittingly taught, children are being told faulty facts. Take the square, one of the first 2D shapes we’re taught, and something you’d imagine there wouldn’t be much confusion around. But how well do we really know it? Here’s the thing, a ‘square’ isn’t really a square. It’s actually a lazy label we have attached to it and, like super glue, it’s hard to shift.

When we say ‘square’ we use it as a noun when in reality ‘square’ is an adjective that describes a type of rectangle. What we should be telling children is that this shape is ‘a square rectangle’. It’s illuminating to see how prevalent the ‘square’ misconception is and that maths learners young and old are often taught to categorise rectangles and squares separately.

Pupils should be learning that a square is a more specific classification of a rectangle just as a rectangle is a more specific classification of a parallelogram, and a parallelogram is a specific classification of a quadrilateral.

Why misconceptions become entrenched

Children have a firm and fixed view of shapes and the stereotypes they have are deeply entrenched and remarkably persistent. The reason why is simple. It’s because we’ve taught them.

But, as teachers, we can’t take all the blame.

Respected maths dictionaries, trusted resource books, gold standard websites and maths posters all play a part and are long overdue for an upgrade. If you have a shapes poster in your class, see how it conforms to stereotypical images and narrow definitions of shapes. A square rectangle is labelled a ‘square’, an oblong rectangle is just a ‘rectangle’, a parallelogram is always pictured as a pushed over rectangle, and so on. It’s time to challenge these traditional images and refine our own shape definitions so that children are getting correct information from the beginning. It’s time for a maths makeover!

In most maths classrooms, children learn the names for shapes without considering their essential properties. For instance, show colleagues or your class a ‘square rectangle’ and ask them to name it. Most will automatically say ‘square’. You might get some who respond with ‘quadrilateral’, ‘polygon’, ‘rectangle’ and ‘parallelogram’. A few might even venture a ‘rhombus’, ‘tetragon’ or ‘quadrangle’ but this will be rare. If you place the shape so it rests on one of its vertices then ‘diamond’ is sure to come up, as changing the orientation of a shape can dramatically alter our perception of it.

Discussing misconceptions

Maths talk is vital to the process of judging the quality and depth of your pupils thinking. Why not try looking at some maths statements in small groups and getting your pupils to place each one into a column: either true, false, or it depends.

• A rectangle has four lines of symmetry
• A square is half the size of a rectangle
• An oblong is another name for a rectangle
• A rectangle has four congruent sides
• Every square is a rectangle
• Every rectangle is a square
• The diagonals of a rectangle cross at right angles
• A rectangle has rotational symmetry order of 4
• A parallelogram is a pushed over rectangle

Well-chosen true false statements are excellent training devices to develop mathematical thinking at a higher level. Their purpose is to challenge stereotypes and achieve mathematical insights based on conjecture and proof. They give pupils a challenge and promote thinking, disagreement and dollops of discussion when facilitated by expert teacher scaffolding.

Once children have articulated their ideas, you can listen and share them as a whole class and debate any similarities and differences. This ‘unpacking’ process is so important to our maths work and is recommended by Jeremy Hogden and Dylan Wiliam in their book ‘Mathematics Inside The Black Box’. It shows what children know, don’t know and partly know. If we ignore this way of working we could miss what is being said.

Maths facts

What children talk about usually dictates which direction you move in, but providing definitions to support them and help shape their thinking and progress their ideas is also crucial. When teaching squares we need to set the record straight. By letting pupils know that there are two types of rectangle — square and non-square (oblong) — we can start to mathematise with more insight and accuracy.

The best maths teaching helps children to think about ‘rectangle‘ as the family name and ‘square‘ or ‘oblong‘ as the first name. Rectangle refers to any quadrilateral shape 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 with a couple of extra bits: 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.

Square‘ is an adjective to describe the type of rectangle so separating squares and rectangles and making them seem different is wobbly thinking. How is a rectangle different to a square? It isn’t. A square is a rectangle.

Getting maths definitions right

The real issue is that children in primary classrooms do little more than learn the names of shapes rather than identifying properties. Their maths diet has been lacking.

In order to develop richer understanding in our pupils we need to do more. Why not get your class to design a wanted poster for a square rectangle, or write the definition of a square rectangle for a class maths dictionary. You could even ask the children to write to a dictionary company to get their definition. Children have recently got the definition of “bullying” changed in the Oxford English Dictionary so why can’t your pupils do the same for ‘square‘?

If our children are going to think independently and deeply about their maths we need to get the basics right first and challenge the way we have previously taught. Only then can we be sure that our pupils are learning correctly.

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John Dabell