Around two years ago, I began to play golf. I bought a set of clubs and went to my local driving range and began to hit some balls. I used my intuition – surely it can’t be that difficult? Sure enough, eventually, I could hit the ball in a particular vague direction but had no control over the distance and power I was hitting the ball with, nor did I have any knowledge of the different types of club and when to use each one. I certainly didn’t have a high success rate or any consistency.
The importance of knowledge
We can’t think critically in the absence of background knowledge. How can I possibly know which club to use if I do not know how the distance of one club ranges from one to another, or what type of ground I can hit this from, or what happens when the wind is blowing in a certain direction? More importantly, how far can I hit the ball with each club when I use a full swing, half swing, quarter swing… when I hit it perfectly? What is perfect?
This relates to maths in the way that the clubs are our maths tools/methods… we need knowledge of what they are, how they are done perfectly, when to use them, how to adapt them when the problem in front of us changes. What seems like the same shot on the surface from one hole to another in golf requires a very different approach in most cases. No shot in golf is ever the same. In fact, I’d argue that you never hit the same shot twice in your life. You’d have to hit the same ball in the same place, with the same temperature, with the same length of grass, with the same wind, with the same clubs, with the same mind-set, the same energy, with the same flag position (the hole position changes every day on a golf course) on the same green with the same length of grass… and probably many more variables. What we must try to learn is what is the BEST way to hit this ball given the conditions to get the ball as close to the hole as possible. This is like a maths problem. On the surface they may look very similar, but they require a different approach. We need to train our students how and when the approaches are appropriate for the different problem sets they face.
The importance of deliberate practice
So I went to an expert… a golf coach. He told me there are underlying principles behind a golf swing that, if you can perfect, you will be able to hit the ball in a straight line and control your distance. These principles vary from club to club and depending on the firmness of the ground you are on. I told him straight away that I could hit the ball really far and wanted to learn how to gently pitch a shot to a short distance. I thought I was the best judge of my own ability and the best person to make the choice for my learning. I am a novice, I was wrong.
He took a video of me taking a swing with no advice and then split the screen along side a pre-recorded model example. He then played them side by side so that I could compare the similarities and differences between my swing and the perfect swing. He gave me immediate feedback on the very first part of my swing… how I hold the club. I’d been doing it all wrong and as a result had practiced this incorrectly for a significant period of time. This formed misconception was very difficult to break. The new grip felt, unnatural, uncomfortable even. As a result, I couldn’t buy into it at first and often found myself reverting back to my old grip when to going got tough. He got me to practise picking up a variety of clubs, several times and then locking the grip as he had asked me to. This became much more natural with practice and it improved the control of the direction of my club face by an incredible amount. The success I gained from this meant that I wanted to learn more… what else could this man show me that was going to improve my golf?
This relates to maths teaching in that our novice students are not always the best judges of what they need to learn. They know too little about the topic hierarchy compared to the expert teacher. They may hold misconceptions that they have practiced many times that are deeply embedded. The important thing is that these are diagnosed as early as possibly through high quality formative assessment in lesson, corrected in isolation and practiced perfectly before this misconception affects other areas of maths that it is linked to. Modelling excellence is really important to set the bar as high as possible. Showing a video/work from a student who was/is in a similar position to yourself and how they progressed is also a powerful idea. “Achievement often leads to motivation but motivation doesn’t always lead to achievement.” Once students have been given the opportunity to practice and feel the success that quality direct instruction brings, they will develop confidence in the teacher and develop an intrinsic motivation to learn more.
The importance of self quizzing and self explanation
I wanted to develop a good sense of which club to use depending on how far away from the hole I was. My irons (a type of golf club) go from a 4 iron to a 9 iron. I know I can hit a 9 iron 120 yards on flat ground and that each club below that gives me an extra 10 yards (an 8 iron is 130 yards and so on…). I know that if the wind is against me I need to “club down” and if the hole is down hill (below my feet) then I need to club up. Without this knowledge I cannot make important decisions. I would often look at clubs in a certain position and just pick one out but now I have been taught to first try to recall how far I can hit each club and then begin to self explain why I have chosen that club in this scenario and how it relates to the problem that faces me in this moment.
This process of “self-explanation” is a brilliant meta-cognition strategy because it allows us to reflect and link our knowledge together. In maths, asking students to reflect on why a particular method has been chosen, what has stayed the same and what has changed in an intelligently varied set of examples, why we took a particular step or why a certain outcome occurred when we carried out a certain operation is really valuable. We can also ask, “have I seen a similar problem before? What did I do? What happened? How is this different? So what should I do now?”
The importance of manipulatives/scaffolds
So way down the line after many lessons I know how to stand, how to hold the club, what an excellent full/half swing looks like, which clubs to use and when … and to not bother playing in January. I have a pretty good understanding of the principles and rules of golf so how do you become an expert? Well I need lots of experience of lots of different situations and lots of deliberate practice at the different types of shots.
To practice these types of shots, my coach set up lots of fictional situations. He used cones and sticks when I was swinging my club back to show me how to keep it in line and when to stop my back swing. He let me analyse my previous swings and other people’s swings that he had coached and asked me what I would say to them now that I know what I know. When I began to revert to my “system 1 thinking,” he stopped me and made me practice that thing before I moved on. I had to practice just swinging my club back 100 times at one point until I just touched where his hand was without even hitting a ball because I was “over-swinging.” He told me that with young children he gave them clubs with larger faces and lighter, larger balls to help with the main idea of swinging the club and making contact. With more experienced golfers, he made them change their back swing to go further towards their body or further away from their body to cause a “draw” or “fade” so that you can curve the ball around trees or obstacles. When “putting” on the green he asked me to putt the ball towards a cone so that the hills on the green made the ball take a certain path and eventually took these away when I became better.
This relates to maths in that there are many different manipulatives/representations that we can use to help students re-align their thinking/scaffold their way to excellence. My golf coach showed me what a perfect putt on the green looks like and expected me to be able to achieve that level eventually but he knew he may need to provide some structure initially until I got it correct. Yes, I could discover this for myself eventually by trying different shots and hoping the ball found the hill and rolled into the hole but this would have been completely inefficient compared to his expert knowledge and scaffolding. The manipulatives can be visual, verbal, analogies or physical objects. They can be used in reception or year 13 but they must be faded eventually once this concept has been grasped and practiced perfectly.
The importance of retrieval and spaced practice
According to Ebbinghaus’ forgetting curve and Bjork’s new theory of disuse, our memories decay over time if they are not used. We can forget as much as 40% of content the very next day – even when it is taught and practiced well. Everything is competing for children’s attention – teachers, pupils, the class clown, displays, adverts, social interactions, tone, body language, hand writing, emotion, hunger, noise, adverts, parents, computer games, scents, time… MOBILE PHONES. The list is endless. To remember content we need to make sure we attend to it. Because I know what I know about cognitive science, I ensure that my £40 golf lesson is not wasted. I practice one or two days later at the driving range and put everything into practice that he explained. There are gaps in my knowledge that always crop up so I have to ask him again the next session to clarify. This is what happens to me, a semi-intelligent, intrinsically motivated individual with a keen interest in getting better at golf. What happens to our novice, unmotivated maths students the second the bell goes….? They forget. We must ensure we provide explicit opportunities to re-activate what they have learned previously that is going to help them progress. Just because content has been taught, does not mean it has been learnt. If teaching is hard and learning is difficult then we must treat it this way!
The importance of interleaving
This is the bit that blew my mind. He told me that studies in sport show that by interchanging between your clubs and swings has a positive effect on your game. In the initial stages of learning the swing it is important to hit the same type of shot multiple times so that your muscle memory develops for that swing and you perfect it.
Once all the different types of swing have been practiced and embedded it is then more beneficial to inter-change between swings and clubs. So you hit a driver first, the a pitching wedge, then a 5 iron and so on. This makes PERFECT sense to me now. After all, in a game of golf you rarely use the same club/make the same type of shot twice in a row. As you move from one club to another, you must ensure that you become familiar with the similarities and differences that both of those clubs and types of shot bring. He told me to practice playing the same shot but imagining that the wind was different – same surface structure but different approach.
Robert Bjork on Craig Barton’s podcast said that the most surprising piece of research that he had been involved in was around interleaving. Observers were asked to pick out paintings by a certain artist based on the characteristics of the painting. The first group was shown paintings artists in blocks and the second group were shown different paintings by different artists in an interleaved fashion. The second group was able to better identify he artists by the characteristics of the paintings. This is entirely counter-intuitive but the reason why this occurs is that observers were not only able to make comparisons between similarities but they were also able to spot the differences much more clearly. He then went onto say that in many studies that are observed blocking content can be as good as interleaving but is never better.
This has huge implications on teaching maths from the way we often block topics together in our schemes of work and curriculum design to the way we design activities/examples/questions sets in lessons. It would, perhaps, be better to do 2 lessons on fractions, then move onto ratio, then 2 more lessons on fractions, then 2 lessons on something else. It would, perhaps, be more beneficial to design a set of practice questions where the first two questions were dividing fractions, then the same fractions were involved but you have to add them or subtract them for question 3 and 4.
- 3/4 ÷ 2/3
- 4/5 ÷ 2/9
- 3/4 + 2/3
- 4/5 – 2/9
Is your school adopting a mastery approach? Is the content blocked because the word “mastery” is making staff spend 5 weeks on a topic until students have mastered it? Is mastery not also the idea of comparing and contrasting approaches, inducing regular rapid recall of underpinning knowledge, applying different concepts to different surface structures and understanding the deep structure to a problem? All of this may well be achieved with interleaving.
What does a full round of golf tell me?
It tells me that on that day, in those conditions, I got that particular score. There are many reasons why I got that score. Maybe I didn’t choose the right club. Maybe I made a daft mistake. Maybe I didn’t concentrate when I was putting the ball into the hole from 1 foot. It gives me a summative idea of where I am at. It does not diagnose properly particular skills that a drill with my coach can identify. It serves the purpose of providing me with the experience of what a full round feels like and maybe helps me with that situation into the future.
This is related to the idea of summative vs formative assessment. We can’t make an accurate diagnosis of pupil understanding based on high stakes summative assessments. Just because I made a certain mistake in my round of golf, does that mean that I do not understand? We can make accurate assessments in our day to day interactions through daily reviews/multiple choice questions/low stakes quizzes and really isolate which skill we need to improve through careful assessment opportunities in lessons. Let’s not use summative assessments for formative assessment purposes.
What I am trying to say is that my experience of being a novice learner in one domain has made me remember how it feels. As teachers, we have only really recently, through our higher education, experienced being an expert relative to a school student.
We must cure our own expertise blindness and realise that novice learners learn better from high quality explicit instruction in the initial skill/knowledge acqusition phase.