It is natural to focus on the differences between us. This is productive when it comes to analysing approaches to teaching because such discussions allow us to clarify our own thinking or the thinking of others. These arguments often throw light on unexamined assumptions and allow the possibility – and it usually is just a possibility – of changing minds.
Publicly, I often found myself disagreeing with the late Grant Wiggins. We differed on the validity of E D Hirsch’s arguments and specifically those around reading comprehension strategies. I also criticised Wiggins’ views on the nature of understanding.
Yet Wiggins will most be remembered for his work on ‘backwards design’. Forget designing lessons on the basis of activities that you would like to do. Instead, think of what you want to achieve and select your activities according to that. As is the case with many powerful ideas, it is deceptively simple and yet you can quickly find many examples of practices that ignore this principle.
I found myself reaching for the concept of backwards design when contemplating the approach that Jo Boaler sets-out in her latest book, “Mathematical Mindsets”. There is a lot to analyse in this book, not least the application of Carol Dweck’s mindset theories to maths teaching. This is perhaps for another post (although I can’t help remarking that Boaler’s argument that Einstein was slow to learn to read is not supported by his official biographers). Instead, I will focus on Boaler’s views on maths teaching.
In a chapter called “Rich Mathematical Tasks”, Boaler introduces us to a lot of cool activities. She talks of visiting a Silicon Valley startup and knocking the socks off the bright young entrepreneurs by asking them how to solve 18 x 5. Apparently, they did this in many different ways and were so amazed by this fact that they wanted to make “18 x 5” t-shirts. The focus here is on the activity. The activity continues to hold centre-stage throughout this section. We read, for instance, of a task that requires students to draw different rectangles with the same fixed area.
The mathematical objectives behind these activities are never clear. Instead, the implication is that they will show students that maths is a creative subject that isn’t all about right answers. It will help them with the aforementioned mindsets and things like that. It is also suggested that these tasks are highly motivating. I am deeply sceptical that this kind of activity-based ‘situational’ interest will necessarily lead to long-term motivation in mathematics and I wrote about it here.
I think it is relatively simple to come up with cool maths activities and that this is something that is valorised a lot more that perhaps it should be. It is far more valuable to be able to structure instruction so that students can learn and understand complex, abstract concepts. Such instruction will not necessarily be cool and funky but it will allow students to grow their expertise and, perhaps, their sense of their own ability or ‘self-concept’.
Boaler also promotes productive failure: students should struggle with trying to solve a problem before being given explicit instruction in how to solve it. There is some evidence for this idea but I am not convinced that it is sufficient to support this as a general maths teaching strategy. We have lots of evidence to support explicit instruction and even the most casual of observers will see that the concept of productive failure could backfire. The productive failure studies tend to have problems with the controls that they employ, specifically in their interpretation of explicit instruction. There is nothing disingenuous about this – it really is hard to design well-controlled trials that do equal justice to the different conditions that they try to compare. Read the paper cited by Boaler for yourself and see what you think.
The other evidence that Boaler cites are her own studies in the UK and US. These have generated much comment and so I will make just a few observations. In the UK research, Boaler studied two schools. One used a traditional approach and the other used more of a problem-based learning approach. We don’t know the identity of these schools and it is quite possible that one or both are outliers. We could have simply happened upon an unusually effective problem-based learning school and an unusually ineffective traditional one. We don’t know whether the maths instruction was more important than other environmental factors, classroom behaviour and so on. The sample size of N=2 is not really sufficient for drawing any firm conclusions. The US study was of three schools which hardly improves the situation.