1.3. What are they thinking about?
What I used to think
I used to think that lessons needed to be memorable to make them…well, memorable. This led to ‘The Swiss Roll Incident’: Bolton, Lancashire, 2014.
That year I taught a delightful Year 7 class, and we had just come to the end of our unit on fractions. We had covered all of the basics, and now it was time for them to apply their newly acquired skills to some different contexts. Hence, I set them the following problem to think about overnight:
Imagine you have 7 Swiss rolls, and you can stack them on top of each other. What is the fewest number of cuts you must make to ensure 12 people get the exact same amount of Swiss roll, and there is nothing left over?
That evening, whilst my students were pondering this tasty problem, I took a trip to Tesco. £12.96 later (these were deluxe Swiss rolls, I will have you know), I was tooled-up with all I needed for what was sure to be an amazing lesson.
And indeed it was.
Armed with a plastic knife and a bundle of paper towels, I set about presenting a real-life, practical solution to the Swiss Roll problem. Cream and jam went everywhere, and a particularly tasty looking quarter of a Swiss Roll was lost forever to the dusty classroom floor. But in the end we had done it. Maths in action, and the kids were loving it.
At the time of writing, these Year 7s are now in Year 11, and unfortunately I no longer teach them. However, I happened to see one of them in the corridor the other day, and after exchanging a few pleasantries, I asked her how she was getting on in maths:
‘Not bad, sir, but it’s a bit boring.’
‘What do you mean? Maths can never be boring!’
‘Well, it’s alright, I guess, but we don’t do anything fun like that Swiss roll lesson we did in Year 7.’
‘Veronica, just out of interest, do you remember what that lesson was about?’
‘Swiss rolls!’
And there we have the problem.
Sources of inspiration
My takeaway
As soon as the ink has dried on my ‘experts and novices think differently’ tattoo, the next on my list is ‘memory is the residue of thought’. I do not think there are two more important phrases for teachers to be aware of. Both will feature prominently throughout this book, but it is the second of these delights, coined by Willingham (2003), that is the cornerstone of this section.
Memory is the residue of thought, so students remember what they think about.
Consider the model of thinking introduced in Section 1.1. Before we start worrying about things like working memory capacity, relevant schemas, transfer and problem-solving, we need to ensure that the items being processed in working memory are the items that need to be processed. Learning is a change in long-term memory, but if the relevant ideas, skills and concepts have not found their way into working memory in the first place, then learning is not going to happen – or at least not the type of learning we intend. It is like having a top of the range juice-maker, but continually feeding it parsnips.
In the lesson I described above, I had hoped my students would have been considering fractions. Instead, they were undoubtedly thinking about jam. Whereas I wanted their working memories to be filled with common denominators, factors of 12 and families of equivalent fractions, instead it was filled up with thoughts of cream, how hungry they were, and the fact Mr Barton managed to get some jam on his glasses. Students remember what they think about. So, was it any wonder that, four years on, Veronica remembered Swiss rolls but not the content of the lesson, let alone the solution to the problem I posed (which is 3 cuts, by the way)?
Once I have the mantra ‘memory is the residue of thought, so students remember what they think about’ tumbling around in my head when I am weighing up lesson activities, I start to view things in a whole new light. Hence, when I came across across a five-star-rated resource on TES entitled ‘Experimental Probability – Bottle Flipping’, I shuddered.
This is the resource description provided by the author:
I created this PowerPoint as an engaging end of term activity for my low attaining year ten class to investigate the relationship between amount of water in a bottle and the chances of it being flipped successfully. I used the resource to revise experimental probability but the wider investigation covers collecting data, a bit of FDP etc. I know bottle flipping is a bit of a sore point … for some but my class were fully engaged and it helped give them an appreciation of experimental probability.
I am sure the students were engaged, but engaged in what? The finer points of experimental probability, or the best technique to flip a bottle of water?
Which brings us nicely onto the subject of engagement. It is an oft-cited claim that students need to be engaged to be learning, and hence attempting to engage students can become an overriding aim of teaching. But does engagement actually lead to more learning? According to Coe (2013), not necessarily. In his much-discussed list of ‘Poor Proxies for Learning’, Coe includes occasions when ‘students are engaged, interested, motivated’.
Now, Coe is not saying that engagement prohibits learning taking place, nor is he dismissing the possibility that engagement can support learning. He is saying that observing engagement alone does not imply learning. And the Swiss roll and Bottle Flipping lessons are classic examples of this. I am sure if someone walked into the room whilst those lessons were taking place they would have been met with a sea of engaged, interested and motivated students. The problem is, of course, what exactly were they engaged in? Unless we have further evidence (a test of retention, for example), we must be extremely careful in concluding that learning is taking place.
All of this still leaves us with a question: would Veronica have remembered that fractions lesson four years on had I left my Swiss rolls at home? Probably not. But would she have remembered the maths involved? I think so. I cannot remember the specific moment I learned to add fractions, or solve equations, or draw tree diagrams. But I can do them. And that is because of lessons where as much of my attention as possible was directed to thinking – and thinking hard – about the maths the teacher intended me to think about.
So, am I saying that maths lessons should be boring? No. I firmly believe – and as I hope to demonstrate in the chapters that follow – that all students can draw motivation and pleasure from learning and achieving in mathematics. But I am saying that there is a real danger that students can latch onto the surface structures of the experiences we provide, so much so that they fill up their working memories and leave little space for anything else. And if this is the case, then we had better be extremely careful when deciding what those surface structures are, for they are the things students are likely to remember.
What I do now
Willingham (2009) offers the following advice: ‘review each lesson plan in terms of what the student is likely to think about. This sentence may represent the most general and useful idea that cognitive psychology can offer teachers’.
When I interviewed Peps Mccrea for my podcast, he suggested what I feel is a useful refinement to Willingham’s original statement: students remember what they attend to. Whilst the sentiment is the same, Peps made the point that as teachers we may have more success directing what students attend to rather than what they think about.
So, each time I plan a lesson I ask myself one question: at this stage of the lesson, what will my students be attending to? And if the answer is not what I need them to be attending to, I change my plan. Of course, I cannot control the focus of my students’ attention, and even the most expertly planned lessons can fall victim to students’ wandering minds. However, I can certainly boost my chances via the activities I choose, and also via the general principles of good instructional design discussed in Chapter 4.
I am also far less concerned about engagement – it comes a far distant second when planning lessons to considerations of learning, because I have realised that the path from engagement to learning is not as clear-cut as I previously thought. Of course I want my students to be interested in the work we are doing, but I feel there is a more reliable and sustainable way of achieving this, which will be covered in Chapter 2.
Hence, no more Swiss rolls or bottle flipping. No more ‘make a poster or a PowerPoint’ for revision, with students’ working memories more consumed with thoughts of colours, fonts and animations than the mathematical content. No more Tarsia jigsaws that the students need to cut out themselves, lest concerns about the neatness of the triangular pieces divert their attention away from the maths. And no more loop cards, with students possibly more concerned about when their turn is coming, suffering anxiety about speaking in public, or switching off when their turn has passed.