A toy problem is a simplified problem meant to be a warm-up to a more complicated problem. I worked on a project earlier this year that was so complex that the write-up of the toy version grew to over 100 pages. We had to make a toy version of the toy version in order to have something easy to wrap your head around.

In Handbook of Markov Chain Monte Carlo Charles Geyer gives a warning about using toy problems in teaching.

It’s hard to know what lessons to learn from a toy problem. Unless great care is taken to point out which features of the toy problem are like real applications and which unlike, readers may draw conclusions that do not apply to real-world problems.

A large part of math education consists of toy problems, and that may be why so few people with a math degree are prepared to do applied math. Even people with an applied math degree.

College education should involve a lot of toy problems. But as Geyer says, it helps a great deal to point out why they’re toy problems, which aspects are realistic and which are not.

Very interesting insight there.

I suppose I’m the opposite to those mathematicians who won’t do applied maths, I’m a mathematical modeller who shied away from pure maths.

I have had to learn, though, that there are usually ways to simplify the target problem until tractable in order to at least find test cases for a more complex solution. My tendency was to try to solve the whole thing immediately, but then had no way of testing it.

In reality often the toy model permits a better grasp of the dynamics of the system, making it easier to see which effects can or should be discarded, and which ones are critical to a useful solution.

Problems which require good work are generally complex and in need of some simplification; a lot of the skill is using whatever domain knowledge is available to tease apart the problem as posed.

John, thanks for the information on MCMC. I work in the cost and schedule estimating domain aa well as probabilistic reliability and safety. The “toy problem” concept was very useful.

I was a “pure” mathematics major, and (…a long time ago) tutored people from elementary school kids to senior undergraduates in mathematics, and I had not seen the term “toy problem” used before now – it’s a fantastic term! I have often decried the overuse of “fake” problems, “artifical” problems, “insanely neat answer” problems in math education, but “toy problem” captures the essence.

The problem is we use toy problems so much in early mathematics education that a majority of kids grow up thinking there are nothing *but* toy problems in math – and come to the somewhat reasonable conclusion that math is nothing but a toy. And that’s awful. I do wish we taught math at the primary school level as a far more applied thing, and kept the abstractions to a minimum, introducing them as necessary when there was a logical, problem-solving need for them as opposed to “abstraction first, applications later (or never).” – the techniques suitable for teaching motivated undergrad math majors don’t work well for most people!

John:

I don’t know how useful is the phrase, “which features of the toy problem are like real applications and which unlike.” The trouble is that real applications vary! Some are like toy problems, some are not. Often a challenge in applied work is to decide how much of the complexity of the real world can be left out of the modeling.