What if students were only in classes that they wanted to take? What if teachers weren't required to grade, and knew that the only way students would retain anything is if it were presented clearly and interestingly? What if parents and politicians set their sights on lifelong learning rather than short-term, numerically oriented goals?
Theoretically, one might be able to introduce several interesting mathematics topics, chosen exclusively for their ability to interest students, and then let the students decide how far they wanted to go with each. Then, perhaps after several activities done as a class, the students might be encouraged to choose one of the introduced topics to study in greater depth, culminating in some presentation to be given to the class at the end of the course.
If my hypothetical approach to this class sounds suspiciously as if it were a description of a real event rather than an theoretical one, that may be because I did, in fact, have the wonderful opportunity to perform this exact experiment in conditions very close to those stated above. What happened was every bit as incredible as I was predisposed to believe it could be.
First, it should be noted that the conditions for the experiment were not exactly those referred to above. The students in question were selected based on very high-percentile SAT scores. The class took place in a fairly relaxed atmosphere, but the students were given a final evaluation (very flexibly defined, but it was there, nevertheless). However, I was able to set the content of the course myself (except that the course title was "Discrete Math", which is a restriction, but not much of one), and thus was under no external pressure to cover any particular body of material. Students were under very little pressure from me to learn anything, except that their final evaluation was based on the presentation. However, even considering this there was evidence that their mindsets changed to one of voluntary, enthusiastic, interested learners.
The topics introduced to the class included
Groups were introduced by means of the symmetries of the equilateral triangle, and then immediately applied to the analysis of juggling three balls, then eventually six clubs passed between two jugglers. Permutation groups were introduced, applied to "musical chairs", the Rubik's cube, and another "physical rearrangement puzzle" called TopSpin.
Introduced the general concepts and applied them with a "can you draw this without retracing the lines or lifting the pencil?" puzzle. Talked about the Konigsburg bridge and introduced a few theorems. Proved a theorem in class together, with no prior preparation on my part, so they could observe an participate in the real problem-solving activity that mathematicians engage in.
Monty hall problem, conditional probability and its propensity for leading people to false conclusions about statistics, a few formulas.
Lots of counting problems, poker hands, etc.
basic ideas of set theory, discussion of set cardinality, and infinities with different cardinalities
general introduction, live experiments with human knots
paradoxes, truth tables and circuits
Line deformation fractals (a subset of L-system fractals), demonstrated and explored with interactive software.
various properties of numbers and theorems
Would that I could do justice to what happened! To describe the incredible feeling of liberation at not being forced to force the students into an externally-defined agenda--what it meant to be able to fully exploit the synergy of my fascination and expertise coupled with their curiosity and thirst for knowledge. The chaotic beauty of a classroom full of students working in small teams or individually on a problem of their own choosing. The sheer elation at being contacted after the course was over by students still working on their problems. I feel that if I could just show people what happened there they would just stand up en masse and walk away from the tired old notions of rigid curricula, required classes, grading, and everything else that contributes to changing the teaching and learning of mathematics from sheer joy into a chore with the occasional bright spot.
However, we'll have to take what we can get. True to the form of how the class went, I'll just go on from here to describe whatever I think of next.
During one of our sessions on abstract algebra, we saw how to represent permutations and "multiply them" (perform the operation of function composition). The students attacking the mini-cube went right to work translating combinations of motions on the cube to permutations. At one point, one memeber of the team came to me excitedly showing how a combination of three moves worked out to be a five-cycle and a two-cycle--meaning that if he repeated the three moves five times, he would transpose two of the cubes without changing the positions of the others! Getting the first four cubes in place is easy, and (also during the course) I had figured out a similar combination of moves that got one more in place. Their discovery was the final (and arguably the most difficult to find) piece of the puzzle. Nobody, including me, knew whether they would be able to solve it all. I had intentionally avoided working on it before the course in order to both prevent myself from being tempted to give away the answer, and (more importantly) to make the process of discovery as authentic as possible. Part of the fun of doing mathematics is that you don't know where your explorations will take you, and I wanted them to have a sense of that.