Making Your Own Sense

Reflections on maths, learning and maths learning support, by David K Butler

Tag: applications

  • When will I ever use this?

    “When will I ever use this?” is possibly a maths teacher’s most feared student question. It conjures up all sorts of unpleasant feelings: anger that students don’t see the wonder of the maths itself, sadness that they’ve come to expect maths is only worthwhile if it’s usable for something, fear that if we don’t respond right the students will lose faith in us, shame that we don’t actually know any applications of the maths, but mostly just a rising anxiety that we have to come up with a response to it right now.

    There’s an interesting discussion in this pdf article   [1] of the various responses that are commonly given to this question and their various drawbacks. The author is mainly concerned that we often inadvertently confirm the uselessness of maths by our very attempts to make it seem useful. While this is a legitimate concern, I have another one: in our attempts to justify the mathematics, we forget to listen to what the student actually needs.

    In my experience, when a student asks this question, it’s a sign that they are starting to lose faith. They are having trouble motivating themselves and are seeking a reason to keep working at it. Being able to use it someday is the first thing they think of to motivate themselves, so they ask the question. But really most students will be happy with any reason that encourages them to stick at it today.

    I had been thinking about this for a couple of days, after following a Twitter conversation and the comments on a post on Dan Meyer’s blog . Then one one of the students in the MLC actually asked the question, so I was all ready with my response. I said, “Actually, I’m not going to answer that question, but instead I have my own question to ask: how are you feeling about this topic right now?”

    It is a testament to the trust I’ve built up with the students that he answered my question honestly! He said that he couldn’t see how the bits fit together or how they related to other things in the course. So I talked about how this topic fit in with the big ideas in maths, and how it connected with what they did last semester and last week. Then I helped him to organise some of the information in the topic so it was clearer how it was structured.

    And you know what? After this discussion it didn’t matter so much that he might never use it. He had what he needed to have the courage to keep going, because I took the time to find out what was really bothering him.

    [1] Otten, S. (2011) Cornered by the Real World: A Defense of Mathematics, Mathematics Teacher, 105-1, 20-25 

    Alexandre Borovik 27 April 2016:

    It is like learning to swim: how many people actually have to use swimming for *practical* purposes?

  • But I don’t like cricket

    When I was in primary school, one of my teachers once tried to teach us averages using cricket, and it is one of my strongest memories of being thoroughly confused in maths class.

    I’m pretty sure my teacher thought that using cricket to teach averages was a great idea, but (for me at least) it was a very bad idea, for three main reasons. First, I didn’t actually know the all rules of how cricket was scored. I had played cricket before, but this amounted to hitting when I was supposed to hit, running when I was supposed to run, and trying to catch when I was supposed to catch. I had never actually scored anything or been told how this was done. So all his discussion of average scores was basically meaningless to me. Second, there’s this technical detail in cricket batting averages that has to include “not out” somehow, which makes it not like normal averages. He spent most of his lesson discussing this detail and I ended up not knowing what a traditional average was, letalone a cricket average. Third, and most importantly, I didn’t like cricket. As an exercise-induced asthmatic, the running wasn’t pleasant. As someone with low coordination, I tended to be out pretty quickly as a batter, and so spend a lot of time just sitting on the bench. And as a fielder, well, the chance of actually interacting with the game as a fielder in primary-level cricket is quite low. So the mere mention of cricket turned me off. If cricket is what averages are for, then I really didn’t want to know about averages.

    And this story embodies the dangers of using “real life applications” to teach maths:

    • Students don’t know the context: If students aren’t familiar with the context of the application, the discussion will be meaningless to them, which often leaves you teaching the context itself rather than the maths.
    • The context is too complex: Most contexts are more complex than the thing you are trying to teach, and to deal with this complexity, you often cloud whatever it was you were trying to teach (or end up changing the context so much it doesn’t make sense any more).
    • Students might be turned off by the context: The application itself has a high chance of simply not being interesting to the students at hand, and they will transfer this disinterest to the maths.

    All three dangers are real and present in every classroom, especially the third one. Yet I have lost count of the number of people who have responded to the question of “how do I motivate my students to learn topic X” with “just tell them about application Y”. No-one seems to recognise the possibility of disengaging students by telling them about application Y.

    I’m not entirely sure what to do about it, unfortunately. If you have a group of students at university who are all studying the same degree (say Mechanical Engineering), then you have a good chance of picking an application they will be interested in, but even then almost always you have the second danger of complexity getting in the way. You could conceivably get the students themselves to seek out applications of the concept to things they personally are interested in, but some maths concepts simply aren’t used in varied enough places. And you could just show them a huge number of different applications so that they are sure to be interested in at least one of them (a linear algebra lecturer recently did this with eigenvalues). But of course, you yourself would have to know all these applications.

    In the end, I think we need be aware of the dangers so we can keep an eye out for students disengaging. Also, I think we need to make sure that the students are comfortable with the maths itself, and we need to be excited about the maths itself, whether we use a real-life application or not. Then the students who don’t like cricket might be able to be interested in just the maths.