Monday, March 28, 2011

Top-Down Approach to Proofs

I am trying a new approach to teaching proofs this year. To give some background, last year teaching proofs was awful. We basically worked from the bottom up: we took each of the unthinkably boring triangle congruence theorems and learned them separately (doing hands-on investigatory blah blah and then some basic textbook practice), which took us well over a week. And then we finally started doing proofs, which I did by giving them pieces of incomplete proofs and asking them to fill in the missing pieces using theorems from memory, like our textbook had recommended. Kids absolutely hated it for two reasons:
1. the learning was drawn out (for both the strugglers and the high-achievers),

2. the proofs they were able to do in entirety were visually "obvious", therefore they lacked basic motivation to do them.
I hated it as well, because I was struggling with teaching kids basic theorems, proof construction, and even basic geometry naming convention and notations all at once! It was uber-frustrating and ineffective.

This year, I sort of said, "Screw all that." The proofs can wait until the end of the year, until we've more or less mastered most basic geometric calculations and formulas and notations. Because if we wait until the end of the year, then
1. I can spend some extra time on proofs if I need to, without worrying about running out of time for other topics,

2. they will already have a more solid foundation for visualization and reasoning, on which we can build their proof reasoning,

3. their geometric vocabulary and ability to name angles / segments precisely will be ready as well.

4. we can work on "juicier" proofs, which eliminates the motivation issues I saw before.

...So, here we are in Q4 and we're just getting to proofs now. This year, I also made a huge decision that maybe memorization of all the base theorems isn't so important. For my honors kids -- yes, they'll still need to memorize most of them before the test. But for my regular kids? Is it really important that they can pull "Reflexive property of congruence" off of the top of their head, instead of pulling it off of a list of theorems? The answer, I decided, is no. (I have the blessing of my students not needing to take a standardized Geometry test down the road.) I decided that the true worth of doing the proofs is that they get to practice the formality of their reasoning techniques, so I made some huge strategic changes:
1. We don't do the two-column format. Every proof still needs to start off with a "Given" and a "Goal" and a diagram, and they still need to write a numbered list of statements, each accompanied by a reason. But they don't need to do the two-column thing, which never made much sense to the kids anyway. Instead, I make them go through each completed proof to color-code their reason and their statement separately, in order to show that each statement in their sequential logic is fully substantiated with a justification, right there within the same numbered step! The kids find this to be very natural. So far I have not had a single kid complain about the requirement of a stepwise justification being confusing, in my regular or honors classes! (In fact, some of the artistically inclined kids can hardly wait to color code their proof at the end.)

2. They get a reference sheet of basic theorems to use, for now and on the test. (Well, regular kids do, anyway. Honors kids will have to learn them before the test.) So far, I've been extremely pleased with the tremendous difference I've seen in their ability to construct proofs! Instead of feeling frustrated each time they need to look up a theorem in the textbook like looking for a needle in the haystack, having a reference sheet means that they can quickly look it up and feel very confident when they know that it's the next necessary building block in their logical sequence. We've just started this late last week, and it's going very well. I think that toward the end of the week, I'm going to switch out their diagrammed reference sheets with a flat list of theorems with definitions, and then sometime next week I'll further remove the definitions so that by the time of their next formal assessments, they're used to only referring to a list of theorem names -- no definitions or descriptions will accompany those theorem names. Slowly removing the scaffold will, I believe, help keep the focus on the logical process, while building their independence and their ability to recall basic theorems...

3. I'm not going over why SSS and SAS and ASA and HL theorems work. I'm sorry, but if I had to teach each of them separately again, either I'd kill myself or my kids would kill me, out of sheer boredom. I think that a smart 9th-grader who stops to think about those theorems can figure out for the most part why they work, and a struggling 9th-grader won't remember anyway; we don't need to have everyone suffer through the tedium of those lessons.

4. No more "fill-in-the-missing-step" proofs! They're kind of pointless, in my opinion. Kids can always fill in a blank. That doesn't mean they know the first thing about writing proofs. This year, we write every proof from scratch, right from the beginning.

Anyway. Just thought I'd share my approach. This whole top-down approach to proofs is making it a whole lot less of a nightmare for me. We've worked on some triangle similarity and congruence proofs so far, and we're going into circle proofs and coordinate plane proofs next. Kids are doing fabulously with looking through those reference sheets and putting logical steps in order and justifying each step... and I think they even think they're kind of fun (like a puzzle). YAY!

Addendum 3/29/11: Since a couple of you requested, here's a take-home proof that I assigned a "fake grade" to, to informally assess the kids (and so that they can see what grade they would have gotten, had it been a real quiz or test). You can see the color-coding for statements vs. reasons. If you are looking closely at the proofs, you should know that I crossed out angle and segment notations where they weren't naming the points in the correct order (for example, saying angles ADE and EBC are congruent. Yes they're referring to the correct angles, but no they should have said angles ADE and CBE instead). For those details, I took off only 10% total in their "fake points", but I corrected them everywhere on their paper so that they'd start to pay attention. In other words, if you look at this first proof below, that girl's proof is splendid besides that minor issue, even though it looks like I marked it up a bunch. --And let's keep in mind that these are my regular Geometry kids, only 2 or so days after we started doing proofs! :)

1. WOW, sounds GREAT!! I am teaching it at the end of the year too, and I had no idea where to start (I'm teaching Geo for the first time).
Are you giving them the reference sheet that you made, or are you making them create one? If you are giving them one, would you mind sharing your version w/ me, just so I have some focus of where to go?
I was struggling with the idea of fill in the blank, I think I am going to try your idea and "hope" it goes well.

2. OK. I'll share what I've got - can you wait until the end of the week so that I can have more stuff to give you? (I prefer having tried them out first myself before posting.)

3. For proofs, I used an ACT unit that worked really well. I'm interested in seeing what your color coded version looks like. Could you take a picture of a sample maybe?

4. YES! So well said. "If two triangles share two pairs of congruent sides, and the included angles are respectively congruent, then" aarrggghhh!!! The much better question is, if I know the SAS of a triangle, can I figure everything else out? How?

I love color coding too. Please share examples if possible!

5. Great Post Mimi. I have a post on basic proofs here:

http://mathandmultimedia.com/2009/12/30/intro-to-math-proofs/

It's algebraic but your students may find it useful.

Keep it up.

6. Ooh, I like those. I've given those statements to my kids before as conjecture exercises. Definitely will give it to them next week to illustrate the difference between making conjectures and proving!

Thanks! Any other good ones that you've come across? I'll link to all the good easy proofs I used this weekend.