One fundamental tension in teaching mathematics is between the goal of understanding at an abstract, context-free level (this is the power of mathematics as a tool) and the need of students for some kind of context. A well-established phenomenon illuminating this tension is given by the following experiment, known in cognitive science as the Wason selection task. Subjects are tested on whether they can solve two logically equivalent problems. The first is the following.
"Suppose you are given cards which have a letter on one side and a number on the other, and you want to verify the rule `if there is a consonant on one side, there will be an even number on the other.' Suppose in front of you are four cards with faces showing A, B, 1, and 2. Which cards do you need to turn over to make sure the rule is followed?"The second, logically equivalent problem is the following.
"Suppose you are the bouncer at a bar and you are trying to verify that patrons of the bar are following the rule `if someone is drinking alcohol, they must be over 21 years of age.' Suppose seated at the bar in front of you are four people, one who is over 21, one who is under 21, one who is drinking a beer, and one who is drinking a lemonade. Which people do you need to approach to make sure the rule is followed?''
Cognitive scientists have observed, in repeated and varied experiments over the past forty years, that people fare much better at knowing who to approach at the bar - the person under 21 and the person drinking beer - than knowing which cards to turn over - namely B and 1. I have informally tried this experiment in some of my classes and have had striking results. In a class covering formal logic, not one student correctly answered the card question while everyone knew what to do as the bouncer. While there are competing theories as to how this discrepancy in ability to reason arises, the basic finding is clear: context matters in most people's ability to do logical and mathematical reasoning. Thus, while our ultimate goal in mathematics is a context-independent understanding, context and metaphor can be critical for some students in reaching that understanding, by linking with knowledge the students already have.
In response to these findings, I try to pepper my lectures with "everyday examples" and metaphors which illustrate different aspects of the subject matter. From roller coasters in calculus to games with mirrors in linear algebra to facts about weather maps in topology, such examples generally take little time but help many students. In some cases, these "examples" play a central role, such as in the computation of the fundamental group of the circle in topology. Here the key idea of proof is illustrated by two people, one on a spiral staircase and one on a circular track below with the person below always staying in the shadow of the person on the staircase. In my experience, connecting such a picture with the formalism leads students to internalize this proof.