More importantly, the idea was the s tudent’s own . The clas s had a nice problem to work on,

conjectures were made, proofs were attempted, a nd this is what one student came up wit h. Of

course it took se veral days, and was the end result of a long s equence of failures.

To be f air, I did paraphrase the proof consider ably. T he original was quite a bit more

convoluted, and contained a lot of unnece ss ar y v erbiage (as well a s spelling and grammatical

errors). But I think I got the feeling of it across . And these defects were all to t he good; they

gave me something to do as a teacher. I was able t o point out several stylistic and logical

problems, and the student was then able to impr ove the a rgument. For instance, I wasn’t

completely happy with the bit about both diagonals being diameters— I di dn’t think that was

entirely obvious— but that only meant there wa s m ore to think about and more understa nding to

be gained fr om the situation . And in fact the student was able to fill i n this gap quite nicely:

“Since the triangle got rotated halfway around t he circle, the tip

must end up exactly oppo site from where it started. That’s why

the diagonal of the box is a diameter.”

So a great proj ec t and a b eautiful piece of mathe matics. I ’m not sure who was mor e proud,

the student or myse lf. This is exactly the kind of experience I want my students to have.

The probl em with the standard geometry curr iculum i s that the private, per s onal exper ienc e

of being a struggling artist has vir tually been eliminated. The art of proof has been replaced by a

rigid step-by step pattern of uninspired formal deduc tions. The textbook presents a set of

definitions, theorems, and proofs, the teacher copies them onto the blackbo ard, and the students

copy them into their notebook s. They are then asked to mimic them in the exercises. Those that

catch on to the patter n quickly are the “good” stud ents.

The result is that the student becomes a passive p articipant in the cr ea tive act. Students are

making statements to fit a preexisting proof -patt er n, not becaus e they mean them. T hey are

being trained to ape arguments, not to intend them . So not only do they have no idea what their

teacher is saying, they have no i dea what they the mselve s are saying .

Even the traditional way in which definitions ar e presented is a lie. I n an effor t to create an

illusion of “clarity” befor e embark ing on the typical casca de of propositions and theor ems , a set

of definitions are provided so that state ments and their proof s can be made as succ inct as

possible. On the surface this seems fairly innocuous; why not make some abbreviations so tha t

things can be said more economically? The problem is that definitions matter . They come f rom

aesthetic de cis ions about what di stinctions you as an artist consider important. And they ar e

problem-gener ated . To make a definit ion is to highlight a nd call attention to a feature or

structural proper ty . Historically this comes out of working on a problem, not as a prelude to it.

The point is you don’ t start with definitions, you start with problems. Nobody ever had an

idea of a number being “irrational” until P ythagoras attempted to measure the diagon al of a

square and discovered that i t could not be repre sented as a fraction. Definitions make sense

when a poi nt is reached in your argument whic h makes the distinction neces s ary. T o make

definitions without motivation is more likely to caus e confusion.

This is yet another example of the way that students are shielded and excluded fr om the

mathematical process. S tudents ne ed to be able to make their own def initions as the need

arises— to frame the debate them s elves. I do n’t want students saying, “the defini tion, the

theorem, the pr oof ,” I want them saying, “my defi nition, my theorem , my proof.”