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.

Huh.

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.

Reblogged this on The Echo Chamber.

Nice post. I agree that the obsession with making maths applicable can be really damaging for maths education. I have a couple of comments:

1. I don’t see why the issue of making maths ‘real’ needs to be tied up with the whole constructivism debate. For one thing, the obsession with everyday applicability is by no means the preserve of constructivists. From experience, it seems that very many ordinary maths teachers — whose teaching style is broadly traditional by default — think that applicability is super-important.

Conversely, there’s no reason why constructivists wouldn’t want students to do discovery-type tasks in a pure maths setting. As a case in point, Paul Lockhart’s famous ‘A mathematician’s lament’ seems to suggest a broadly constructivist approach to maths education, but seems pretty keen on keeping it pure (following the Hardy essay from which he takes his title).

Perhaps constructivists tend to like real world applicability, but it strikes me that this is a mere sociological fact, rather than anything intrinsic to constructivism per se.

2. I suspect that the sheer pervasiveness of the ‘making maths real’ point of view is what makes ‘but when will I ever use this?’ so common in maths: Students are told from a young age that the point of learning maths is for its everyday usefulness, so when they come across something (e.g. quadratic equations) which just isn’t useful in day-to-day life, they immediately reach for the question. Nobody asks ‘when will I every use this?’ when learning about symbolism in Shakespeare in English lessons, or the formation of stars in science lessons; for most subjects, students are perfectly capable of understanding that *that’s not the point*.

This then leads to people constructing ridiculous examples of what Dan Meyers calls pseudo-contexts (whatever your other differences, I hope you can agree with him on the ridiculousness of these) — attempts to shoehorn maths into real life in thoroughly nonsensical ways. These just reinforce the impression on students that maths is pointless.

I agree with your second point. I’ve written something similar before. On your first point, I don’t think that real-world application is something that is restricted to constructivism but I was quoting from a definition of constructivism when I made that comment. I think you will find other definitions that make similar points about real and/or authentic contexts.

Fair enough. In that case my beef is with those constructivists if they mean that to be an essential part of the definition. I always took it that the core claim of constructivism is the (intuitively appealing but empirically falsified) claim that knowledge is best learn when constructed rather than when transmitted. And these definitions conflate that claim about how knowledge is best learnt (constructed vs. transmitted) with a wholly different claim about what context it should be constructed/transmitted in (real world vs. pure).

In psychology, constructivism regards the way the brain makes sense of available information. ‘Transmitted information’ is not meaningful in itself: it needs sense making, an autonomous brain process (actually: lots of processes) that is not open to our own introspection. In this psychological context, it is a ridiculous idea to ask pupils to do the construction themselves (by little homunculi in their heads?) in inquiry learning (itself a ridiculous idea for obvious reasons that many gurus and their followers refuse to see). Reading a few pages from Fibonacci main work might be a cure 😉 [now available for the first time in translation. This sunday and monday on offer by Springer: 9.99€ for the eBook]

Couple of thoughts:

1. PISA’s report seems like poor research. It shows that to do better on their tests you should study questions like on their tests. Of course exposure to test questions will help students do better but better research has shown this rarely is then transferrable to non-test situations. So we get a bunch of students that can do tests well and can’t figure out the change required when buying something.

2. Teaching to context especially if the students are encountering that context in multiple ways will help with recognition and improve results. So if the maths is real world market sure it links with other subjects and help each other engage the students.

3. Most people learn things as required, very few have the character strength of ‘love of learning’ and the extrinsic motivation of a test doesn’t work for many students. So creating a reason to learn or develop a skill is great. This is were maths books get it wrong placing real world problems at the end of a topic or set of questions and contorting the world to fit theory. Know you aren’t going to like it but this were Dan Meyer’s ideas can help by first creating a hook to interest students and then provide them with the tools to solve the puzzle and make students want to learn. With the sell in follow up with developing the skills further or in more abstract forms. If Dan is to on the nose have a look at mathspickle.com for some great tasks that get students to engage with more abstract maths ideas.

I think you have missed something important with your first point. PISA is a contextualised test – students have to work out the dimensions of a revolving door and so on. It is based upon the realistic mathematics education approach from the Netherlands. Yet the students who did best were the ones who studied the most pure mathematics. So the study shows the reverse of your claim that, “to do better on their tests you should study questions like on their tests.” That’s why it is interesting.

Thumbs up to Jonathan’s attempt above to split curriculum from instruction. As you note, they’re often conflated, but let’s not continue making the same mistake just because everybody else does.

In any case, while I agree with a lot of your analysis, I think you’re reaching past the data to conclude that making mathematics real is a bad thing. (Tentative titles aren’t your usual style, Greg.) I wrote on it here:

You will see in the post that I acknowledge that this is correlational. But I find it very interesting, given the overwhelming narrative in maths education that making maths ‘real’ is a good thing. We might ask where the evidence is for

thisproposition. I am also reminded of research (eg Bryan et al) that finds that U.S. teachers prioritise applications whereas those in the Far East tend to be more comfortable in the abstract. It doesn’t seem to be damaging Chinese students all that much.Greg, I like your point about the need to torture reality to fit it into real world math classes. Imagine a homework assignment where students had to get an example of a 21st century math problem using trigonometry that their parents had to do at work. This would quickly show how out of touch the real world problem crowd is with the real world use of math.

What the proponents of real world math problems miss is that people who actually use a lot of math in their work typically spend 3 or more years at university before they start work as a novice scientist, economist or engineer. So any attempt to help students into these careers would be far better designed if they worked backwards from preparing students for first year university.

On the other hand if they want interesting problems that involve a bit of arithmetic and geometry, some thinking and a bit of research they could get a copy of What If and work through that. It would be more motivating and a little more humble about claiming to prepare students for real world math.

A sample here:

https://what-if.xkcd.com/

Enjoyed reading your article.

This harks back to a comment I read on Quora very similar to the dilemma posed within this post.

The doctoral graduate responded to a question asking where Mathematics is being used in the modern world. He elaborated that whilst he cannot get into the details of how he uses Graph Theory to solve modern day problems in Physics and Chemistry, Maths’ purpose should never be solely based on application.

In fact, what made him enjoy it is the fact that it was a puzzle, it was fun and it was just something to play around with. I cannot feel enough sentiment with this thought, and your post only strengthens this position for me.