I do not list my achievements to make myself feel good (well, not entirely, anyway). I do so to illustrate just how far it is possible to come without really having a clue what you are doing.
There are two key implications from Anderson’s model that we will revisit many times throughout this book: Existing knowledge makes thinking and learning easier. Once something has been learned – whether it is correct or not – it is very difficult to unlearn. Hence, practice does not make perfect, practice makes permanent.
There are many different definitions about what learning is, but the one I am going to opt for throughout this book is provided by Kirschner, Sweller and Clark (1998), who define learning as ‘a change in long-term memory’, going on to say that if nothing has changed in long-term memory, then nothing has been learned.
For Coe (2013), learning happens when people have to think hard, and indeed making changes to long-term memory is likely to be effortful. There are two main pathways through which such
For Coe (2013), learning happens when people have to think hard, and indeed making changes to long-term memory is likely to be effortful.
For Willingham (2009), ‘understanding is remembering in disguise’ – it is taking correct old ideas from long-term memory, getting them into working memory, and rearranging them in a new order to make new connections.
As we learn, our brain architecture changes and thoughts are processed differently. This means that as we move to mastery of a given skill or concept, our brains form different links in long-term memories, and it is actually possible to observe different activation patterns during problem-solving. They conclude that ‘in addition to processing efficiency, enriched representations, and structural expansions, experts can flexibly use strategies, by recruiting the associated brain regions, to solve a range of problems, whereas novice performers can not’.
The answer of £75 probably came to you almost instantly, without you really being conscious of what you had done. You immediately and unconsciously recognised the percentage sign and what it means, and you know that 25% is the same as a quarter. You have automated this knowledge, and hence it takes up no space whatsoever in your working memory. You have also automated the procedure for finding a quarter of something, so you can quickly work out a quarter of £300 by halving and halving again without imposing too much strain on your working memory. You can do all of this because you are an expert. You could also probably do this calculation if music was playing in the background, your partner was asking why you haven’t done the washing up, and you were attempting to juggle. The fact you have automated much of the knowledge necessary to answer the maths question frees up space in your working memory to attend to other things.
Does the answer arrive instantly this time? Now, this problem is likely to be cognitively demanding for both you and a novice learner. But the difference is that you have automated all the basic knowledge involved. You do not need to expend any effort contemplating what each term in the formula means, how to square 5, how to find a third, or what to do with the pi symbol. This frees up cognitive capacity to attend to more global features of the problem, such as correctly interpreting the question and devising a strategy to get to the solution. Not only are you more likely to solve the problem, you are also more likely to learn from the experience. A novice learner, who does not have the relevant knowledge automated, is likely to get bogged down in the minutiae of the problem, experience cognitive overload, and potentially not learn anything transferable.
Understanding what problems are really about can be one of the toughest skills for students to master. But here’s the thing – it is not a skill, at least not in the same way that adding together two fractions is a skill. It is a feature of being an expert.
Your experience of answering lots of related questions, the significant volume of knowledge you have stored in long-term memory and – no less significant – the way that knowledge is organised allows you to avoid the surface features and identify the problem’s deep structure.
We suffer from a ‘curse of knowledge’ (eg Wieman, 2007), whereby when you know something it can be difficult to think about it from the perspective of someone who does not know it.
This can make it – for me, at least – very difficult to see things from a student’s perspective, empathise with the difficulties they have, and help them overcome them. I simply cannot remember what it was like to not be able to answer the questions in this section, and that is a problem.
‘memory is the residue of thought’.
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’.
Imagine that you are the Year 8 teacher who has inherited my class of students brought up on a diet of ‘I am thinking of a number’. What do you do? Well, your only real option is to teach students another way of solving linear equations that works for these new types of questions – probably some variant of the balance method. Then you have two choices: either tell students to abandon their old way, which they will probably be reluctant to do as it is so easy, and miles better than the new, tricky way you are trying to teach them; or give them a complex set of rules which enables them to spot which questions require which methods, which effectively means they need a different schema for each variation, and maths quickly becomes the disparate set of meaningless rules that countless scores of children and adults view it as. Either way, you are likely to be fighting a losing battle, and it is undoubtedly all my fault.
I think very carefully about the methods I teach my students, asking myself: How long will this method last them? How do I move them on from this method? Is it built upon solid mathematical foundations? If necessary, I teach them in a way that is more difficult in the short-term, but which leads to greater long-term gains.
In his book Drive, Pink (2011) suggests a key determinant of motivation is autonomy – the desire to direct our own lives.
Pink (2011) identifies purpose – the yearning to do what we do in the service of something larger than ourselves – as a second key determinant of motivation.
Pink’s (2011) third element of motivation is mastery – the urge to get better and better at something that matters.
Finally, if we think questions like this are in any way motivating to our students in so much as they relate to their wider lives, then I think we might be a little deluded. After all, as Wiliam points out, beer is not measured in eighths of pints, and even were it to be so measured, it is unlikely that Alan would be overly concerned about the precise measurement while enjoying his drink.
Lemov (2015) sums all this up in his strategy Without Apology: ‘embrace – rather than apologise for – rigorous content, academic challenge, and the hard work necessary to scholarship’.
Praise should be immediate and unexpected. Praise loses much of its informational and motivational impact if the teacher praises a child for having shown good effort two weeks ago. Making praise unpredictable is hard to do, but can be of huge benefit. The goal is not simply to get the child to stop asking for praise; it is to help the child to think of their work differently – as something that is done for the student’s own satisfaction, not to garner praise from the teacher.
the effort a person is willing to expend on a task is determined by the expectation that participation in the task will result in a successful outcome,
For Coe (2013), learning happens when students think hard. But I now realise that students may only be willing to think hard if they believe that effort will pay off. Too much experience of past struggle and failure will only dampen that belief.
Begin a lesson with a short review of previous learning. Present new material in small steps with student practice after each step Ask a large number of questions and check the responses of all students. Provide models. Guide students’ practice. Check for student understanding. Obtain a high success rate. Provide scaffolds for difficult tasks. Require and monitor independent practice. Engage students in weekly and monthly review.
For Chi (2000), one of the main drivers of the effect is when the learner repairs their own mental model.