I quote from a recent article by Brown and Porter in the De Morgan Journal, published online by the LMS. http://education.lms.ac.uk/wp-content/uploads/2012/02/Brown_and_Porter.pdf and commented on subsequently by David Wells. I feel the following idea needs advertising.
"A technique widely used by psychologists and trainers is error-less learning. This falls into two types. One is where large hints, props, and supports to a specific course of action are given, and the action is rewarded as a symbol of success. Then the various props are gradually withdrawn. The other type uses reverse chaining: the easiest way to see to this is to think of encouraging a child to put on a vest. You do not throw him or her a vest and say put it on; instead, you put it almost on, and then ask the child to do the final action. Subsequently, you gradually put the vest less and less fully on, till the whole action can be done.
One way of using the last technique in university mathematics is to write out a formal proof and then erase bits of it. The student has to fill in the bits, using clues from the rest of the proof. This has some analogies with the practice of a professional mathematician, who may have an idea and outline for a proof, but needs to work on details. The student also gets an idea of the structure of a proof. Such an exercise is also very easy to mark!
The general feeling about error-less learning is that it works like a dream!
In either method, the fact long verified by psychologists is used, that we learn from success. We can also learn to accommodate failure if that is gradually introduced, and strategies are available for dealing with failure."
