A few mornings ago, Rufus William got in touch with an interesting request:

I’ll admit to being a little anxious, but in the spirit of enquiry, I agreed.


This was the activity: A domino is made up of 2 squares. A pentomino is made up of 5. How many different pentominoes are there?

I got out my paper and pen and duly set to work:


I quickly realised the answer was ‘loads’. At that point I gave up and told Rufus the answer was “more than 50”. He then asked me what I thought of these:


My first answer was that if you cut out the shapes you could make a box without a lid out of each of them. Although Rufus liked this answer it clearly wasn’t the right one. I dredged up some memories of transformation from GCSE maths and hazarded that maybe they could all be rotated. Rufus nodded approvingly – as much as you can on Twitter – and agreed they could be either be rotated or “reflected on to each other. Mathematically, we don’t consider them different. They’re congruent shapes.”

Ok. Makes sense. Rufus asked whether I thought this was a good way of teaching congruence. I said I thought it was interesting, but possibly inefficient.

Here’s why. At the start of the activity I had no idea what I was doing. It took me a few minutes to work out that there were loads of different ways to arrange 5 squares, but that wasn’t really the point of the activity.

When I was asked what I thought of the patterns which turned out to be congruent my first thought was imaginative perhaps, but off topic. I was remembering making something similar with my daughters. When it was clear another answer was expected I though back to what I already knew. Luckily there were enough vestiges of the maths I’d learned in school to be able to supply a broadly correct answer. I still didn’t know what the point was until the concept of congruence was explained.

What would have happened if I couldn’t remember anything – or hadn’t been taught anything – useful? I would have floundered about with guesses until I lucked on, or was told the answer.

Now, I’m no mathematician – as should be pretty clear by now, but an alternative way of teaching congruence would have been to turn the activity around:

  1. Explain the concept of congruence.
  2. Show examples and non-examples to demonstrate.
  3. Ask me to draw some congruent shapes to make sure I’d got it.

The question is, which method is best? I think it would be quicker, and therefore more efficient, to use my alternative. Less time would be wasted guessing and working out that there were an awful lot of ways to arrange squares. But, would students have a better understanding if they had to puzzle out the concept for themselves? Maybe, maybe not. They might have ended up discovering something which turned out to be wrong or inappropriate. Because of the way our minds work, this might embed misconceptions, which are hard to get rid of.

It would be fascinating to run a mini test where a class was split with one half having the discovery method and the other having worked examples and seeing which group did better in a test next lesson, then again in another test at the end of the term. There have been a number of studies which indicate which approach would get the better results, but who knows – I’ve been wrong before and will certainly be wrong again.

The other consideration is that one method might be more ‘engaging’ than the other. Maybe students would prefer playing the pentoninoes game first? Maybe this would result in a more positive view of maths? That’s certainly what we as teachers often intuitively believe. And it might be true. The alternative view is that students become more motivated the more proficient they become. Being asked to play games might feel fun at the time, but is, perhaps, unlikely to intrinsically motivate students. But getting good enough at something that the basics become easier and you can have a go a more challenging stuff might actually be the best way to motivate students.

This is certainly borne out by my experience of nagging my eldest daughter to practice her piano scales. She’s now got to the point where she can play hands together and bang some of her favourite songs and we can’t get her off the bloody thing!

Rufus has written the following in response:

I encountered the idea on my PGCE, I’d like to discuss this experience as well as my observations of its effectiveness with my students; it’s quite a bit different doing it over Twitter than doing it in the classroom.

The activity is simply for the students to find out how many different arrangements of 5 squares they can find. There is an element of setting it up as a challenge to see if there are more than ten different ways, but I wouldn’t consider my students to be ‘playing the pentominoes game’. Also, whether the students consider this to be a fun activity or not is not relevant to me beyond my desire for them to consider my lessons to be both serious and inviting, and most importantly, lessons in which they learn maths.

Inevitably, when the class work on the activity, at least a few students will start to ask whether they can consider some of the shapes the same or different, and this will lead to a class discussion. For example, these two shapes generate a lot of discussion:

Screen Shot 2016-06-16 at 08.21.56

It is now that I go through David’s stages:

  1. Explain the concept of congruence.
  2. Show examples and non-examples to demonstrate.
  3. Ask the students to draw some congruent shapes to make sure they’d got it.

Therefore I am not saying that I am not giving clear definitions and examples, it is just that I think giving the students the activity first helps them to understand and embed the concept in a stronger way than if I had started the lesson with the explanation.

The reason I feel like this is that I was given the activity to do on my PGCE course. I have an excellent degree in maths, as well as excellent school grades, however, at that time I was not familiar with the concept of congruence. When I started to have a go, I too was apprehensive, and in trying it I was confused. However, in the course of a class discussion, I experienced the desire to want to clarify whether certain shapes could be considered the same or different, and I found that the concept of congruence made perfect sense to me when it was explained.

I am aware that this does not determine that my students will feel the same way, and I wonder how I can ever know if there’s a ‘best’ way to teach something. At the moment, my judgement is that it’s a good way to teach the concept because it always instigates a class discussion which effectively asks a question that I want to answer: “do we consider these shapes to be the same or different?”