“Numbers are not objects of study just because they are numbers already constituting a branch of learning called mathematics, but because they represent qualities and relations of the world in which our action goes on, because they are factors upon which the accomplishment of our purposes depends. Stated thus broadly, the formula may appear abstract. Translated into details, it means that the act of learning or studying is artificial and ineffective in the degree in which pupils are merely presented with a lesson to be learned. Study is effectual in the degree in which the pupil realizes the place of the numerical truth he is dealing with in carrying to fruition activities in which he is concerned. This connection of an object and a topic with the promotion of an activity having a purpose is the first and the last word of a genuine theory of interest in education.” John Dewey, Democracy and Education, 1916

Maths teachers are often exhorted to teach mathematics through real-life contexts. This is a key part of the progressive tradition and a feature of reform mathematics teaching. There appear to be two main reasons, the first is affective and the second cognitive.

We presume that real-life contexts will be motivating for students. Most maths teachers will have had a student ask if the maths that they are learning will be useful in real life. There is a tendency to accept the premise of the question and conjure a lame response, something that I am not prepared to do. We don’t tend to judge other subjects in this way. Education is not a purely utilitarian pursuit; a means for producing future employees. Even if it were, we are terrible at predicting the future. The key principles that inhabit the mathematical canon have endured and this is why we teach them. We cannot know how or when or to whom they might prove valuable.

The idea of motivation is also fraught. How far must we stray from optimal learning in order to accommodate it? I can engage a group of 12-year-olds in a poster making activity but they won’t be learning any maths and when they realise this then they are likely to view the subject less favourably. In my new book, I show that the linear idea that you must first motivate students in order to get them to learn is simplistic and flawed.

The other reason for the use of real-world contexts is cognitive. There is a body of research that suggests that students do not transfer what they learn in maths class to situations where they really could apply this knowledge in real-life. Jean Lave is probably one of the foremost researchers in this area. Perhaps if we used real-life contexts more in mathematics classrooms then students would be able to transfer their learning better to problems they meet elsewhere? It seems reasonable.

It was therefore with interest that I read a comment on my previous blog post by a regular contributor known as “Chester Draws” – I assume it’s not his real name. He linked to this paper in the journal, “Frontiers in Psychology: Cognition”. It details a set of experiments where content was held the same but the degree of contextualisation was varied. The more contextual the instruction, the less able students were to transfer the learning to new situations. This may be because the contexts added more information for the students to process. It’s only one paper but it should at least raise some doubt about the reform narrative.

I was reminded of Dan Willingham’s comments that learning tends to adhere quite strongly to the context in which we first learn it. Furthermore, I recalled research that showed that maths teachers in China are much more comfortable working in the abstract than maths teachers in the U.S. When you contemplate the idea of transfer then it becomes quite clear that this is the purpose of creating abstract rules and generalisations in the first place. An abstraction can potentially be picked-up and dropped down somewhere else. Concrete examples do not lend themselves to transfer in the same way.

Yet I must avoid presenting a false choice here. I think that many traditional maths teachers will relate concepts to contexts. It’s more a question of the degree to which this is a priority. Reform advocates will argue against ‘pseudo-contexts’ that are meant to represent real-life but are not credible. I think that traditionalists would be more relaxed about these examples provided that they furthered an understanding of the maths. So it’s more a case of a kind of fundamentalism versus a more pragmatic approach. Indeed, just last week I was teaching some abstract theory regarding binomial random variables and I gave an example of what one might be, “You have a biased coin with a probability of 0.7 of tossing a head. You toss it 50 times. The number of times you get a head would be a binomial random variable.”

Is this useful? Can it be applied to a real-life contexts that my students will experience? Who cares.

Reblogged this on The Echo Chamber.

In Physics of course, context is king.

I’ve recently been teaching momentum & unusually have been doing it at the end of the GCSE course because the board has added it in. What has surprised me is the boys saying”Sir, this is the same maths as Boyles Law” or “Sir, this is the same as Moments”. I’m not sure whether to be delighted that they are sophisticated enough to recognise the shape of the Maths, or horrified that they risk mixing up the contexts & making mistakes in the process!

Regarding the probability thing, this is the time to introduce them to the ideas around sampling and significance. Without the maths!

Greg,

Have you read Pradeep Mutalik’s essay in The Best Mathematical Writing of 2015. Despite best in its title the book is a mixed bag. But this one is worth a read or watch at https://www.youtube.com/watch?v=2qwROzmqAuc

Mutalik is a math puzzle guy and the article discusses the psychological positive buzz you get when solving a puzzle or getting some nice part of a mathematical problem. It is worth some thought on many levels. The idea supports the mastery leads to motivation idea on a very chemical level: you solve a problem you get a hit of pleasure.

It certainly answers why mathematical problems don’t have to be any more real world than a Sudoku puzzle to motivate joy in mathematics. There is probably a dark side too though. I think you would have to be incredibly fortunate to get long term learning in mathematics through a series of these little moments of joy with nothing feeling like work in between.

Put another way, sometimes the math itself is the context. Math on its own should be treated as interesting enough, without always being dressed up in fake problem contexts. Kids like to think about primes, and infinity, and Pythagorean triangles, in my experience.

The trick is knowing when, and when not, to use real world contexts. When will it help, and when will it hinder the learning?

Yeah, my parents were not cruel enough to name me Chester Draws. I use that moniker because I like to make comments on blogs that can be a touch acerbic, and I don’t want students to find them if they search my name (which they do, because I have a website with their homework and practice exercises).

As you say, we practice Maths in contexts, just like we always have. Because in the end most people will use it to work out how much paint to buy to cover a room, or how much concrete will pave the drive.

But we need to teach it initially context free, and only then apply it, once the basic skill is learned.