To help teach the skill of abstracting in Math students are given “word puzzles” which ultimately lead to a single “right” answer. The students are required to work out how to put the data from question together correctly the find the solution. These types of problems in Math can lay a poor groundwork for teaching Operations Research and (to some extent) Statistics.
One major issue with these types of questions is that the word puzzle often appears to be put in the context of the “real world” but the problem is really what a colleague calls “applicable math” rather than “applied math”. The distinction he draws is that “applied math” should start from a real problem some industry of business faces. “Applicable math”, on the other hand is the archetypal nail constructed for the particular hammer being taught. The math problem is devised first then some context is imagined to set the problem in the “real world”.
Some examples of “applicable math” I have seen include a BMX rider travelling along a linear path then following a parabola (why!). A farmer wanting to maximise the grass area contained by a fixed amount of fencing (what does he do with the rest of the grass?). A woman (of course!) maximising ‘taste’ when choosing how much ice cream and snack bars to eat for dessert (huh?). No wonder students have trouble abstracting. How can students develop intuition when the problem situations presented are counter-intuitive!
Of course finding a real-world context that exactly fits the math we want to teach is hard. And often these contexts spawn models that are much bigger and more complex than we could use. If math application seems purposeless, the students will see the underlying math as pointless.
In statistics students might be expected to apply a line fitting or regression to six data points collected from friends comparing their height with their shoe size. I have no problem with illustrating techniques on small data sets or with small examples – but don’t pretend anyone would do this in practice, or that the results will have meaning. Better to collect enough data to enable the results to have some validity, select a small subset of data to illustrate the underlying math, then show results for the full sample. (And whatever you do, don’t claim that height causes shoe size – we all know its the other way around!)
Dr Shane Dye – Guest Blogger