When Tom Bennett announced researchED Maths and Science I was more than a little bit excited: a subject-specific conference informed by strong academic research? Yes please! The programme for the day was packed full of sessions I wanted to see, so selecting which ones was very difficult. Here’s what happened on my day, a mixture of what I heard with my own thoughts.
Session 1: Memorable maths teaching, Peps McRae
Peps was drawing on evidence from cognitive science about how memory works in order to inform teacher instruction. He began by highlighting how fickle our memories are – fragments of what we thought about with some intelligent gap filling – and proceeded to talk about long-term and working memory. I have done a lot of reading, from various sources, on this topic recently, so it was particularly interesting to hear it all brought together in a fifty-minute summary.
Our long-term memory is made up of everything we’ve learned and can have either a low or high degree of structure. When it has a high degree of structure (it is multi-layered and hierarchical) we can treat this as understanding. Understanding is, essentially, highly-structured long-term memory. This is important for me as it defines something that is often a woolly concept.
Our working memory is what authors our long-term memory, it’s the thinking we do in the moment. We know it is very limited (for more on this, read Daisy Christodoulou‘s Seven Myths About Education and Daniel Willingham‘s Why Don’t Students Like School) so as teachers we need to consider precisely what we want our students to be thinking about. Do I want them to be distracted by too much display? Do I want them to be tackling complex instructions instead of thinking about the maths at hand? Whatever they’re thinking about is what they’re going to remember, and the gatekeeper to their thinking is their attention.
Peps explained how what students already know is the the most significant factor affecting what they can learn and that we must pitch our instruction at just the right level – too high and they can’t access it, too low and they don’t learn enough. This is why, particularly in mathematics, the ladder of abstraction (what the Chinese call concrete-pictorial-abstract) is such a good way to help students learn. He mentioned briefly the Matthew effect, the testing effect, the importance of distributed practice, the need for assessment systems built for long-term memory and more, but the thing to stress this time for me is the cognitive empathy gap: we, the teachers, are significantly more expert than our students, the novices, and it’s very easy to forget that topics are easy for us since we process them in the context of our highly-structured long-term memory, where the schemata are much more complex than those of our students. This means it is easier for us to think about maths than it is for our students. We need to eliminate distraction and reduce cognitive load. For instance, when questioning the class write everyone’s answers on the board so they don’t have to hold them in their heads, prime the students by “warming up” their prior knowledge, provide the bigger picture so they can place new knowledge easily into an existing schema, give them opportunities to look inwards and think carefully about what they already know (this could be as simple as “Tell me about this shape”) and minimise their need to filter out superfluous information.
There is a lot more elaboration to be made on many of these points, but I’ll save that for another day.
Session 2: The genius of Siegfried Engelmann, Kris Boulton
Siegfried Engelmann is little known in this country. He published his seminal work, Theory of Instruction, in 1982 but has been researching and publishing since the 60s. Engelmann’s Direct Instruction provides very clear methods of teaching which feature what he called logically faultless communication – a way of explaining something that cannot possibly be misunderstood or misinterpreted. Kris was keen to emphasise that what he introduced yesterday was just the tip of the iceberg of Engelmann’s work, this tip was the idea of differing examples.
For instance, if you are trying to explain what a surd is, instead of struggling with a definition that students may or may not understand, show them with examples, minimally different examples and non-examples:
Each of these numbers would appear one at a time (left-hand side first, then right) accompanied by the teacher narrating, “This is a surd, this is not a surd, this is a surd, this is a surd, this is not a surd, etc”. We were shown a similar sequence to demonstrate what a triangle is. The sequence is followed by more examples and non-examples with the teacher asking the whole class, “Is this a surd?”. Through carefully designed sequences, including boundary examples, the students quickly see what does and does not fall into the category.
Lots more to investigate and read about on this one, I had just started to look into Engelmann anyway so now my appetite is truly whetted.
Lots of opinions were shared here. It was made clear that there is little maths-specific research into setting and the observation was made that setting in maths, which is a hierarchical subject, is a very different animal to setting in non-hierarchical subjects (such as the humanities). Predictably, no consensus was reached, the main warning points were:
- setting is fine provided you don’t allow expectations to be lowered in lower sets and provided you don’t confound “ability” with “prior attainment”
- mixed ability teaching works best if the students take individual responsibility for their learning
Session 4: Teaching for mastery, Mark McCourt
Mark began by making the point that mastery is nothing new, indeed it was being described in different terms by Carleton Washburne at the turn of the 20th century. In the 1950s it was called the behaviourist movement and in the 80s it was diagnostic teaching.
Mark was his usual, brutally direct self, decrying the fact that 95% of our students leave school without a top grade in mathematics, without knowing what they ought to know. He rightly likens mathematics to a Jenga tower, where secure lower blocks are essential to any kind of building (the foundations being numerosity, place value, the base 10 system, proportional reasoning and arithmetic) and points out that it doesn’t matter how long a student takes to get to the end goal, as long as they get there. If you are in any way unconvinced of the need for a mastery approach in mathematics, I recommend listening to Mark speak.
Barry became a math(s) teacher upon retirement having been shocked at the state of his own children’s mathematics teaching. He set out a brief history of mathematics education in the US, where the progressive/traditional debate seems particularly fierce. He demonstrated how the fundamental features of a traditional maths education: logical sequencing, memorisation, mastery, explicit instruction, “I do, we do, you do”, get caricatured and misrepresented by progressive reformers (“drill and kill” “all rote, no understanding”) and made the case for mastery of the “most efficient” standard algorithms. In a particularly amusing analogy he said that novices trying to do problem-solving in a discovery lesson is like throwing a non-swimmer into a swimming pool, standing on the side while they flounder, and shouting “Now’s a good time to learn breaststroke!”
Some of Barry’s arguments can be read here (thanks to Jack Marwood for the link).
UPDATE: Barry has shared his presentation from the day here.
Session 6: Teaching with variation, Debbie Morgan
Debbie was talking about the way Chinese textbooks employ variation theory in order to force students to practise the idea rather than the mechanics. Take a read from slide 28 here to see the examples she used. When writing or using an activity, ask yourself “what is the activity really checking?” Is it mechanical recall of a process or understanding of the concept? A well-designed task employing variation theory can achieve both.
The whole day was hugely enjoyable, and possibly the first conference I’ve attended in a long time where my notes will be revisited regularly rather than put away and never read again. I am in the process of creating a new scheme of work with our maths department based on a mastery approach, details of which I will be sharing at some point, so the day confirmed a number of areas for me, gave me plenty more to think about and helped me to cement my understanding of cognitive load and some of the science of memory.
On top of that I met many of the people I’ve only engaged with on Twitter up to now – maths teachers really are a wonderful bunch of people! Thanks to Tom Bennett and everyone who presented, I look forward to another conference.