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Shaking up traditional geometry

March 29, 2013 am31 11:29 am

My proof-based geometry course is a proof-based geometry course. But I can still shake things up: Logic, Non-standard Theorems, Construction, Construction, Student-generated reference materials (for use on tests)

Were I to stick closely to my text (Jugensen/Brown/Jurgensen), most of my readers would recognize the course instantly as the more or less standard geometry course that’s been taught in the United States for a century. Of course the amount of proof has been substantially reduced from fifty years ago, but the idea, the sequence, they are the same. This is a course in proof, but also in reasoning. It is the only axiomatic system that most high school students explore.

And I hate teaching it. Ugh. So I “innovate” – though I suspect that all of my innovations are quite old, and have been done before.

1. Open with a unit on logic, and logic proofs. For those of you from NY State who recall the proofs in Course II, no, more, harder. Include extraneous statements. Teach more rules of replacement and rules of inference, and prove the rules before using them. Venn Diagrams and Euler Diagrams and truth tables. Consistency. And indirect proofs. This was a big unit.

2. Have students create their own glossaries/reference sheets. Allow/insist on constant revisions and updates. Allow/insist that the students bring their reference sheets to each quiz and test.

3. Construction. Fully one quarter of the class periods devoted to construction. Some standard construction. A lot of more creative stuff. We have a set of Michael Serra’s geometry books, and his opening chapter has been a nice resource.

4. Construction. Students must have the tools with them at all time. Quick constructions often become parts of ordinary non-construction lessons.

5. Oddball theorems. There are two types of deductive proof that students encounter.

The kind of deductive proof we more often associate with high school geometry presents a diagram with some given information and asks the student to prove another piece of information. What is being proven is usually already clearly true to the eye.

The other kind is to prove a theorem. The book does this for the students. Or I do it in class. And then we use the theorem. Sometimes the proof of a second version of the same theorem is offered as an exercise.  And then, if this were the 1970s or earlier, we would ask the students to memorize these proofs, and recite them on a test.

But this is wrong!  Proving theorems is at the core of what mathematicians do. The students need to be asked to prove theorems. But all the good ones are taken. So I will ask students to prove less-known, less-useful theorems. Practice doing the real thing.

(Getting to oddball theorems came out of discussions with math bloggers 2 – 4 years ago. Don’t remember exactly who, and exactly when, but the list of helpful suspects includes Ben-Blum Smith, Pat Bellew, f(Kate), PO’ed Teacher, and the Math Curmudgeon. I think Ben is the likeliest to have hosted this sort of discussion.)

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4 Comments leave one →
  1. Geoff Slack permalink
    March 29, 2013 pm31 6:09 pm 6:09 pm

    Halabi-you’ll never believe this, but I have a unique opportunity to teach a small geometry class next fall, that I just found out about today. So, I have taught the standard geometry course (and stil do) but I was thinking, how could I do this so that is not so “ugh”. Then I thought, let me contact halabi- and ,, ready made to order, look at this post!

    Thank you for the great writing, as always.

    I like the idea of starting with logic instead of the definition of a point, line, etc.

    I also agree with your point about proving theorems, not applying them to some obvious situation.

    Any specific thoughts on scope and sequence for a unique/motivated group?

    Any open source text on subject you/people like for geometry?

    Thanks for your ideas

    Peace!

    • March 29, 2013 pm31 7:22 pm 7:22 pm

      One of the powerful possibilities with logic is that laws can be proven (use a truth table to show that all four or all eight cases work), and then used to build subsequent laws. (Can also use Euler diagrams and Venn diagrams, at least sometimes). Proving something you will use later – that’s similar to proving theorems.

      Most text books get to triangle congruence and stall. They spend a lot of time proving non-theorems… practicing application of SSS, SAS, ASA, rt angle SS… I think we spend too much time there.

      I find that the extra time on constructions comes back to me, as they “see” and “feel” much more instinctively how the stuff works.

      But no, no ideas for anything open source.

      I’m curious to hear what you come up with.

      By the way, did you know that no ear of corn has a prime number of rows?

      • Geoff permalink
        March 30, 2013 am31 6:50 am 6:50 am

        Re:corn

        Don’t believe it

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