Why Symmetry Can Be A Game-changer For Your Students

Some ideas get a WHOLE LOT more air time in the maths classroom than others. These are the biggerbeefier, seemingly more important concepts (think: multiplication, area of shapes, and linear equations).

Then there are the smaller ideas, which might take up a lesson, here or there – but nothing more.

I want to share one example that, as you’ll see, offers far more than we usually think.

Nestled away, in a room labelled Geometry, symmetry is often ticked off and forgotten.

Yet, viewed differently, it becomes a game-changing concept that can powerfully change the way students see maths, find meaning and solve problems.

Symmetry: it's not just geometric!

My views on symmetry changed dramatically, when I read this throwaway line in an article on problem solving-

One of the more important strategies that can be used to approach a problem...Remember that symmetry is not just geometric.  Emily McCullough & Tom Davis

That one moment forced me to check all of my prior assumptions – and opened up a completely new way of viewing symmetry's mathematical possibilities.

What followed was this-

If symmetry isn’t just geometric, where else can we see it?

To answer this question, check out the following example (it's one we discussed with our Circles Members)-

Consider the symmetry in these 3 items. What information does that symmetry tell you?

Before you read what other teachers observed, take a moment to think about how you see symmetry in those 3 items.

The information that symmetry unlocks

Here’s what teachers noticed about the symmetry in the 3 items-

1. Symmetry helps to deepen conceptual understanding

  • Renee: These images encourage more reasoning to explain that symmetry must be balanced
  • Jeny (on the mobile balance): Developing the concept that the = sign means equality, and not just ‘the answer is'.
  • Nathan (on the number line): positive integers have their negative equivalents that are symmetrical around zero. Leads to the zero pairs as a useful tool when exploring integer operations.

2. Symmetry helps to identify the features of an object

  • Susan: In a clock, 7 is opposite 1… etc
  • Colin: The clock is a grab bag of symmetries (including rotational & reflection symmetries).

3. Symmetry is a ‘gateway’ to other, more complex concepts

  • Mathew: Great for introducing new concepts with a concrete visual. Clock -> Modular arithmetic, Number line -> Parity, Scales -> Equivalence, etc.
  • Jacqui: It’s important for juniors to have strong visual symmetry to understand something like normal distribution.

Where else can we find symmetry?

Using the definition of symmetry, we can start to see where else it shows up…

What is symmetry?  1. a sense of harmonious and beautiful proportion and balance.  2. a property where a mathematical object is not changed under a set of operations or transformations.

The first meaning refers to the symmetry that we commonly notice outside of mathematics, e.g. in music, nature, science and art.

The second meaning is all about:

  • doing ‘things’ (e.g. rotating, adding, reflecting, splitting)
  • to mathematical objects (e.g. functions on a Cartesian plane, building a probability tree, simplifying a sum or some other expression).

Through this lens, we can see symmetry as a powerful problem solving tool – one that allows us to make the mathematics we’re working with far simpler and much more meaningful.

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