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.