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Wednesday, February 10, 2010

Error-Finding in Geometry Proofs

This got kids asking some really good questions and they seemed to like the activity.  I'm required to teach two-column proofs, but wanted to elevate the drudgery with some higher order thinking.

I asked them to choose two proofs from recent assignments, write them out, except make one major error.

Their ideas of what constitutes a "major error" in a 2-column proof were enlightening. They wrote a list on the board including missing a whole step, using an invalid reason, using parallel lines to conclude the wrong angles are congruent, and assuming something from the way the diagram looks that wasn't given. They decided not to include things like omitting the horizontal bar from a line segment or the congruency squiggle.

Many of them wanted to discuss their idea for messing up their proof. They kept their voices low and hid behind a folder to talk to me. Some of them are very devious. I will be contacting them when I plan my next bank heist.

Before they handed them in, I asked them to describe the error on an attached post-it. They turned in something like this:

I told them I would copy and redistribute the proofs the next day for them to find the errors.

Notes for next time:

Many of the proofs actually had more than one error. I left this alone. When I passed out the error-ridden handouts, I just told them that lots of the proofs had more than one error, and their job was to find them all.

I had them write on blank copier paper. Next time I should probably hand out a template instead. I think a standard format would be easier for everyone to correct, and also to ensure they leave some room between each line.

The post-its were unnecessary. I don't know why I thought that was a good idea.

I have 60 students in Geometry. Checking 120 proofs would be unreasonable, and there were plenty that would have turned out repetitive. I ended up printing four per page, and giving the kids one page to check for errors.

The error checking takes a while, but they didn't need much help with that part. Finding the errors would work fine as a take-home assignment. Then in class we could project a select few and have the kids describe what corrections they would make.

The arguments over when and how to decide a proof was okay were amazing, and worth ten days of plain-old Geometry class. Consider this format for other things, too.

A way to do this same exercise with less paper and more tech? I don't know. They need to be able to create the proofs and then write on the proofs to correct them. Proof-writing kind of defies computer use because of all the symbols and formatting and drawing diagrams. Unless you have like tablets with stylii for all the students, which we don't. Now I'm thinking of maybe like a smart notebook template with all the weird symbols they would need, infinitely cloned, sitting around the edge of the page so they could just drag them into place. But, still. That would definitely take more time and probably be a total disaster.

The students liked that none of the tasks felt super-difficult, and they loved creating something they knew was going to be shared with their peers. A few seized the opportunity to write ASS as a triangle congruence theorem. Whatever. It could have been worse. Fifth period thinks it's hilarious to say "schlongbongler" and I don't know what that is, but it doesn't sound good.

Phew. I'm glad I got all that out. I think I enjoyed writing f(t) more when I mostly used it to reflect on my way-imperfect practice. I need to get back to that.