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

## 5 Comments

I recall a ridiculous example of inappropriate-applicable-math coming from a “write an equation for the tangent line” problem in an intro calculus textbook. The trouble is the word problem narrative had a spaceship was travelling along the path described by a cubic polynomial, with the tangent line was supposed to be the path followed by an object tossed overboard at one moment. Such an object would go on a tangent line, all right, but what spacecraft would fly a cubic polynomial path?

Another was from an old probability-and-statistics book which presented a question about given the probability of a transistor failing of (something not very near zero), what was the chance of at least two out of the five transistors in a radio failing … which certainly made sense in the original context, when it was unquestionably about the probability of vacuum tubes failing. (A five-tube design was common enough for radios to be credible, and their failure rate was high enough for this to make sense as a binomial distribution problem.)

What irks me most about these types of examples is that the underlying concept or analysis has appropriate interesting and realisitic examples that could be used. How many students do we turn off by perpetuating such examples?

Or am I being too harsh? Many of us clearly see past them and persist.

The other big danger here is that when we pose unrealistic story problems, we teach students not to bother sanity-checking their work – which is one of the most powerful quality checks around.

[…] colleague wrote a guest post about “applicable mathematics” which he separates from “applied mathematics”. Applicable maths appears when teachers make up […]

I remember one problem in which the protagonist’s spa bath was heating at an exponentially increasing rate, and we were asked to figure out how long it would take to reach his preferred temperature.

I can think of one or two rather contrived scenarios where that could happen, but all of them would lead to me running for the hills as fast as possible.