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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.