Rational students, incentives and expected returns

In my last post, I contrasted George Pólya's humanistic style of teaching with the scientific management approach favored by most movement reformers. The focus of that post was more on the underlying philosophy (I'd contend that George Pólya had a very different view of people than did Frederick Taylor), but there are some extremely practical reasons for adopting Pólya's approach, particularly when it comes to building students' self-confidence.

Every investment involves a comparison of costs vs. expected value. We never know how a decision will turn out, so we have to balance money, time and trouble against expected returns. I realize this point seems to border on too-obvious-to-mention but it's surprising how often people forget there are random variables in their arguments, sometimes with disastrous results.

Take performance-based incentives. Let's say I'm going to offer to pay you a certain sum if you accomplish a task but nothing if you fail. In order for you to agree, your time and effort will have to be valued less than the product of my offer times the likelihood of success. Once again at the risk of stating the obvious, as that likelihood approaches zero, your idea of a reasonable offer will have to approach infinity. Of course, in real life, there are always bounds on the amount of money I can offer but your estimate of the likelihood of success can always get closer to zero.

In a business context, we normally deal with the small perceived likelihood problem by finding someone else or opting for a different compensation plan or simply walking away from the deal. This is yet another reason why it's dangerous to have people who don't thoroughly understand both business and education try to transplant ideas from one field to another (it also reminds us of Pólyas warning that "it is foolish to answer a question you do not understand").

In education, where we should try to reach every student, low perceived likelihoods of success can be deadly. Any reward you offer for an apparently unattainable success will seem worthless; any penalty for apparently inevitable failure will seem brutally unfair. If you want to motivate these students, you will have to convince them that, with reasonable time and effort, the odds of success are pretty good (this happens to be true for the vast majority of students but that's a topic for another post).

Pólya said "If the student is not able to do much, the teacher should leave him at least some illusion of independent work." This was, of course, meant as a last resort, but his point was that students absolutely have to think of math as something they are capable of doing.