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Teaching Proofs – My Plan

I’m teaching geometry this year. It’s my first time since student teaching to do this, so I don’t have a whole lot of experience with it yet.

Proofs are introduced in geometry. There are two whole chapters on it in our book. It seems that, because of the way that standards are written, that they’re taught as a separate entity.

Yet I remember, from my childhood many years ago, that the most powerful part of geometry was that we spent the whole year building up proofs – often several every day, going from the very basic to the very complex.

My kids are scared of proofs. They’ve had some exposure, enough to confuse them, but not enough to enthrall them.

So I want to revisit the process of my own geometry learning – proofs intertwined throughout the whole year, but with my upgraded teaching sensibilities.

The plan: a geometry proof battledeck1. Really, just flash cards. But they’ll get out the cards, and pick out the ones that are possibly applicable to the givens of the theorem, and then select others that may get them closer to the proof. Not only will there be tactile interaction, but the focus will be on the arrangement of the steps of the proof, fitting together the logical progression, rather than trying to remember the myriad possible theorems they have to draw on. Best of all, once they derive the proof, they get the new theorem (with the proof written out on the back) to add as another weapon in their battledeck. At the end of the year, They’ll hopefully have their entire history of proofs in one fat deck.

I’ve got 3 dozen binder clips, 1000 blank white cards, and 28 young minds to shape.

1 My first thought was to pitch this as sort of a Magic, the Gathering thing. My kids have no idea what that is. They do get Pokemon & Yu-Gi-Oh!, though, and are down with the idea of having to pick out a series of cards to achieve a particular objective. However, I fear that if I push this metaphor, that they’ll have an expectation of head to head battles, and toss it out one that fails to materialize.

{ 4 } Comments

  1. N.D. | September 27, 2008 at 11:32 am | Permalink

    This sounds interesting. I feel the same about proofs – they become this separate entity.
    I’ll be curious to see how it goes. I am going to try to intertwine them more throughout the year as well.

  2. toni | January 26, 2009 at 7:24 am | Permalink

    Hi i need to learn proofs really badly and figure out how to do them. I’m in the 10th grade so next year will be my hardest year. right now i’m in geometry n i don’t understand not a word exspecilly when i just got the handle on proofs about numbers and stuff now were in to theorems that theres lik a million different ones to pick from n my teacher does not help at all. but thanx for sharing if theres an easy way to understand it better lik how did u get to understand proofs the way u do.

  3. ian | June 5, 2009 at 12:44 pm | Permalink

    i don’t know about this.

    i’m a math and philosophy undergrad who is about to graduate with a BA in both subjects. throughout my schooling in math, though we could definitely try to memorize theorems, or even write them on flash cards and forgo memorizing them, i think the biggest thing that helped was putting the theorems into context and seeing how coming up with proofs is a huge exercise in creativity, and not in formal deduction (i just completed an introductory formal logic course, and while i can apply modus ponens and reductio ad absurdum like a pro, all of that fades in the background while doing math.)

    for example, i was always wondering why the hell green’s theorem and the divergence theorem had to be memorized and why they were so important in advanced calculus. well, i learned several years later while having a conversation with a physics professor that they were simply generalizations of the fundamental theorem of calculus! and then i realized that the fundamental theorem of calculus was a statement of so much more than just relating integrals to “antiderivatives”! (what’s the upshot of all of these theorems? if we have a smooth curve and wanted to find the area underneath it, then we simply find the “antiderivative” and subtract the end points from each other. that is, the area is wholly determined by the boundary!) and now, i’ve committed those theorems to memory, not out of rote, but out of contextualization.

    math is not simply formal deduction! when you use cards as a guideline for proofs, we are saying that these cards have no meaning other than chicken scratches.

  4. John Gabriel | August 31, 2009 at 7:43 pm | Permalink

    It is true that most mathematics professors are ignorant beyond belief. I am a self-taught mathematician.

    My proverb is:
    A true educator can only explain what he understands and can only understand what he can explain.

    I learned calculus when I was 14. Today, many decades later, I am still learning. For example, I thought most mathematicians understood that all integrals are line integrals, but realized I was the first to fully understand this concept. The following link explains:

    http://knol.google.com/k/john-gabriel/all-integrals-are-line-integrals/nz742dpkhqbi/44#

    Well, aside from my average sum theorem, I believe that certain theorems such as Green’s, have never been fully understood. In my opinion, Green’s theorem relates the boundary area of a closed region to a difference of volumes calculated over the same region. This volume difference is interpreted as area under special circumstances.

    Most proofs of Green’s theorem are pitiful. I am working on a proof based on the ideas I just mentioned. Keep watching my blog and knols.