Square Peg

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John Dabell

I trained as a primary school teacher 25 years ago, starting my career in London and then I taught in a range of schools in the Midlands. In between teaching jobs, I worked as an Ofsted inspector (no hate mail please!), national in-service provider, project...
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What is a ‘square’?

When someone says they ‘will square up to you’, it might be worth pointing out, whilst adopting a fighting pose, that strictly speaking this just isn’t good maths. What they really mean is that they ‘will square rectangle up to you’. What’s all the fuss about?


I once surveyed 100 pupils and 100 teachers and asked them to name the following shape:

Out of the 100 pupils:

1 wrote cube

1 wrote quadrilateral

98 said square

Out of the 100 teachers:

7 said “Is this a trick question?”

4 wrote rectangle

1 wrote ‘equilateral rectangle’

1 wrote ‘congruent quadrilateral’

87 said ‘square’.

In classrooms around the world you will see pictures and posters of 2D and 3D shapes that informally ‘teach’ children geometry. They are simple enough: there is a picture of a shape with its name underneath. The problem is some of these name labels are faulty and unfortunately so is the teaching that goes with it.

An equilateral rectangle is always labelled ‘square’ and a shape with two long sides and two short sides is labelled ‘rectangle’. They are of course both rectangles, one is a square rectangle and the other an oblong rectangle.

Interestingly, ‘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 because it describes a non-square rectangle.

Posters, pictures and teaching can lack precision which is why it is essential we differentiate between rectangles.  If we happily and unconsciously tell children ‘this is a square’ and ‘this is an oblong’ we are just passing on wobbly maths.

You wouldn’t say, “Look at that yellow!” so why say “It’s a square!” when what you really mean is “Look at that yellow flower!” and “It’s a square rectangle”.

Car Crash Maths

Using precise language when naming and describing shapes is just as important as getting a measurement accurate to the nearest millimetre.

Drawing and labelling ‘square’ and ‘rectangle’ isn’t good maths and it isn’t good practice. If we continually contrast squares and rectangles we convince pupils of their separateness.

Viewing each shape exclusively sets up problems later in life when someone somewhere points out that squares fit the description of rectangles.

Take a look at the educational posters in your classroom and see how a square rectangle has been drawn – it will be labelled as ‘square’. Now look at the oblong – that will be labelled ‘rectangle’. But a square is a rectangle too but children won’t recognise it as one because their poster has planted a firm image in their minds about what a ‘rectangle’ looks like.

Look at this poster. If you have one like this in your class, take it down and rip it up. It might look fun but it’s out of order. In fact, don’t rip it up, send it back to where you purchased it and demand your money back – faulty goods.



The classroom environment is a powerful ‘second teacher’ and if you cover your classroom in defective posters then expect children’s learning to be damaged too.

Instead of tearing the poster to pieces and getting angry, ask the children to write a letter to the publisher and point out the shapes that haven’t been labelled correctly. When you start to look at definitions in maths textbooks, maths dictionaries and online materials you will see that there is a major problem when it comes to their definitions because they are ambiguous. ‘Rectangle’ and ‘square’ are seen as separate entities. My experience as a teacher across Key Stages has taught me that pupils and many teachers fail to understand the inclusive nature of shape and that squares can be included within rectangles.

The ‘square’ misconception is a very common one and forms part of a large heap of misconceptions I have collected. In their book Making every lesson count, Shaun Allison and Andy Tharby talk about keeping a record of common misconceptions to use and integrate into our teaching,

“The maths and science departments in our school are now mapping out and recording common misconceptions – and the best ways to help students to avoid them – across their respective curriculums. An advantage of creating an inventory like this is that it can be shared with new, inexperienced teachers to prevent them from making the same mistakes as their colleagues.”

From Shanghai to Salford, aspects of basic geometry have a tendency to be taught with little or no conceptual understanding and poor subject knowledge. Children might think they have learnt their shapes but what they will have learnt is faulty.

3 thoughts on “Square Peg

  1. Great post. I knew that a square was a special kind of rectangle, but hadn’t thought of square and rectangle as adjectives. Noun seems to be the first definition provided in most resources. I would be interested in that list of misconceptions. One of my pet peeves is the misconception that butterflies “hatch” from “cocoons”. We can thank Eric Carle and his Very Hungry Caterpillar for spreading that one.

  2. Good post. I love the discussion of mathematical language and how we need to be more precise (especially when it comes to negative/subtract). To build further on your point shouldn’t we more accurately describe rectangle as an adjective as well since they are parallelograms (with all angles being equal) and therefore a ‘square’ should be more accurately called a squared, rectangle parallelogram or even a regular parallelogram?

    On that note what should we be calling a Rhombus?

  3. I originally agreed emphatically with all of this, being a secondary maths teacher whose students are so often unreceptive to the idea of a square being a type of rectangle. But the more I mull it over, the more I think we might be taking them through a pretty meaningful learning process in teaching them ‘wrongly’ at first. This way they have the opportunity to challenge learned convention and see the relative merits of folk language versus mathematical language. After all, your survey does prove the mathematicians wrong by at least one legitimate definition of correct language (that what is used and understood is ‘correct’).

    Whether or not ‘square’ is a legitimate noun is a brilliant gateway debate to empathising with different perspectives on language. I think you’ve just provided me with a starter I can share with my English teaching colleagues, too!

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