I remember a key moment early in my career. I had been receiving target grades that students were supposed to aim for in their exams in my subject. I knew these were based on a set of cognitive tests students completed when starting secondary school, but I wasn’t clear how. Now I was staring at a set of graphs, one for each child. They had bars at different grades representing the different percentage probabilities of obtaining each of those grades. None were greater than about 30% and every child had some non-negligible probability of obtaining any of the possible grades.
This made a nonsense of target grades which were simply based on the grade with the largest percentage plus a bit of ‘aspiration’. Yet if a child had a 10% chance of an A and their largest bar was at C, did B represent an aspiration? I realised that, although there was some powerful statistics sitting behind this data that could predict relative proportions of grades for large numbers of students, at the level of the individual student, they were pretty meaningless.
I’ve tried to stay out of the divination business ever since. Are some people just not cut out for maths? I don’t know. There’s a lot of interest in genetic factors in education in recent times, but I suspect the predictive power of genes at the individual level is even less than those cognitive tests. We can fool ourselves into making up narratives about our students and even ourselves, but I suspect they are just stories.
One issue with mathematics is that people tend to misunderstand it. It is a sequence of rigidly hierarchical knowledge, much of which becomes automatised and therefore feels like a skill. Yet if any steps in that sequence are missing, there will be trouble. If a student came to me and said they wanted to study Mathematical Methods at VCE, but they could not manipulate basic linear algebra, then I would be frank that success was unlikely. That’s not to write that student off as not a maths person. It is to state, quite logically, that first they would need to learn some linear algebra and see how that goes. In time, they could be as capable as anybody else of tackling Methods at VCE.
However, we all know the casualties – the kids who opt out of maths as soon as they can, convinced that it is not for them. And we know those who, despite years of maths lessons, don’t learn basic linear algebra. Is there a inevitability to this or are we, as maths teachers, contributing to the problem?
Jo Boaler, Professor of Mathematics Education at Stanford University, has written a blog post that attempts to address this question. Boaler is convinced that everyone can learn higher level mathematics and makes a number of suggestions about what might be going wrong.
Some of these are non-sequiturs that attempt to draw lessons from neuroscience. For instance:
“The brains of the “trailblazers” show more connections between different brain areas, and more flexibility in their thinking. Working through closed questions, repeating procedures, as we commonly do in math classes, is not an approach that leads to enhanced connection making.”
Is it not? How come? There’s no reference to turn to.
However, the major thrust is an argument Boaler has made before about the value of struggle. By struggling and making mistakes, we cause our brains to grow and change. Students therefore need to be presented with situations that make them struggle with maths, but in a supportive environment where they understand that it is okay to make mistakes.
For this to be a good plan, we need to accept two propositions. Firstly, we need to accept that struggle is an effective way of learning mathematics and, secondly, we need to accept that we can intervene effectively to frame that struggle in a positive way for students.
The second of these propositions involves what might be called a ‘growth mindset intervention’. Such interventions have been proposed based upon the work of Carol Dweck, a psychology professor also at Stanford who has researched different mindsets. Currently, the evidence for the value of such interventions is weak. Students may well have different mindsets but that does not necessarily mean that as teachers, we can tinker with and somehow improve them.
Without such a backstop, inducing struggle in the maths classroom represents a significant risk. It will increase cognitive load in a way which is likely to be demotivating for many students. Indeed, observational studies of teachers suggest that more effective teachers obtain a high success rate from their students, indicating that they carefully modulate the level of challenge and ensure it does not become too high.
So perhaps struggle is effective but potentially demotivating? Not so, the same approaches that manage cognitive load appear to be both more effective and more motivating. This is not a surprise because we can intuitively grasp that it is motivating to improve at something.
I’m not sure what brain scans can really tell us about that.
However, I like Boaler’s optimism. Whether everyone can manage higher level maths or not, we should do our best to give students a strong grounding, using the evidence as our guide. They may then make their own decisions about what to pursue.