One of the assumptions that is held by many educators is that maths should be taught, where possible, through real-world examples and applications. Some trace this idea back to John Dewey and it certainly follows the kind of naturalistic logic that asks why learning to do arithmetic can’t be like learning to walk or speak (there is actually a good reason why it can’t): If only we could motivate students to learn mathematics by showing them its utility in reaching some other goal such as baking a cake or making a go-cart then all the pain of learning would go away.
This is an instrumental view of mathematics: maths is a tool for achieving another purpose rather than something of value in its own right. As with many ideas that originate in the progressive movement of the early twentieth century, this view has been subsumed in a verbose way into the later theory of constructivism where, for instance, “Challenging, open-ended investigations in realistic, meaningful contexts need to be offered which allow learners to explore and generate many possibilities, both affirming and contradictory.”
The problem with the instrumental view of maths is that it gradually evaporates everything in the maths curriculum, including the kinds of investigatory activities favoured by constructivists. You can function just fine in society living a rich a fulfilling life without much maths at all. Need to do some sums? Grab a calculator. And there’s certainly no need to learn anything as ethereal as algebra. When was the last time you had to solve for x outside of a maths classroom?
If you try to bring the real-world into the classroom then you probably won’t succeed very well – the best you can hope for is a simulation of real life and, at worst, you will be torturing reality to try to fit the maths.
Whilst the motivational posters of Twitter still valorise ‘real-world’ and ‘authentic’ without explaining what these mean, the more thoughtful constructivists have moved on, tying themselves in knots as they try to hold on to the principle of authenticity whilst avoiding its absurdity.
In David Perkins’ book “Future Wise,” he seeks to define a criterion that he names “lifeworthy”. This isn’t about learning only those concepts that have a direct application in real life but it also sort-of is. What emerges is essentially an idiosyncratic list of what Perkins thinks is important. He is a fan, for instance, of the French revolution. When it comes to mathematics, it’s out with quadratic equations – not lifeworthy enough – and in with statistics.
The late Grant Wiggins made a similar turn when trying to define ‘authenticity’, specifically in the form of assessment tasks. For Wiggins, authentic tasks must be, “representative challenges within a given discipline. They are designed to emphasize realistic (but fair) complexity; they stress depth more than breadth. In doing so, they must necessarily involve somewhat ambiguous, ill structured tasks or problems.” Yet this doesn’t mean they have to be ‘real-world’ or ‘hands-on’. The CAN be. But the don’t have to be.
Wiggins goes on to outline an example of the kind of open-ended mathematics task favoured by constructivists. This is authentic, he asserts, because it involves doing real maths.
When you look at Perkins’ proposals for maths tasks, despite the definitional hi-jinks, they involve students doing things like planning, “for their town’s future water needs or model its traffic flow.” Which sounds real-world, mundane and dull.
The concept of real-world maths is so ingrained – particularly through it’s adoption by constructivists – that the massive multi-national testing program run by the OECD and known as the Programme for International Student Assessment (PISA) adopted the principle for its maths test. PISA maths is based upon Realistic Mathematics Education (RME) – a maths philosophy from The Netherlands that takes an instrumental view of mathematics. In RME, students first work in real-world contexts, using their intuition to solve problems before developing more formal approaches.
It is therefore ironic that the OECD has found evidence in its own data that pure, abstract mathematics teaching is linked to higher performance on its own tests of supposedly real-world problems than teaching that focuses on real-world contexts and applications.
The OECD nails the instrumental view:
“It is hard to find two scholars holding the same view about how mathematics should be taught, but there is general agreement among practitioners about the final goal: mathematics should be taught ‘as to be useful’.”
I disagree with that. The study goes on to look at how maths is taught in different countries. It is essentially a study of correlations and so you could wave it away for that reason but the authors have tried to control for a number of factors. Crucially, they find the following:
As the report suggests, this finding is consistent with cognitive science and the fact that learning is often tied to the contexts through which it is learnt. Indeed, one of the most powerful aspects of mathematics is that it is abstract and therefore can be generalised across diverse contexts.
It is odd but not entirely surprising to see how these results have been spun. Andreas Schleicher, education boss at the OECD decided to somewhat miss the point. To him, it was not the contexts that were the problem but the way that they must have been used. He assumed that teachers of applied maths must be teaching students tips and tricks and asking them to mechanically learn simple mathematical procedures because Schleicher knows, a priori, that this would be bad.
Diane Briars of the National Council of Teachers of Mathematics in the U.S. (NCTM) also took the opportunity to criticise the idea of teaching children to memorise rules as well as having a bit of a rant about ‘flip-and-multiply’ – a method for dividing fractions.
It’s almost as if they had been asked to comment on something else.