“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.