## 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.

Alison Blank said...

Kate, I actually do this in my class as well, only I don't tell the students to intentionally make errors. I think I will try that in the future - it sounds like it makes it more fun for the students.

Kate Nowak said...

I figured lots of you superstars already do something similar :) I just wanted to get all my thoughts down so I do a better job next year.

The intentional errors were fun. Also, nobody will feel like I'm picking on them. (Which even if you don't name names, they might, you know?) With this it's like we're all in on the game.

Elissa Miller said...

I'm interested in how you teach writing proofs in the first place. We are doing triangle congruence statements where they feel in the missing last step but that's as far as I've gotten. I've intro'd the idea of proofs but have no idea in how to go about teaching how to write them or why they are important.

josh g. said...

Thanks for posting good stuff as well as the way-imperfect. This sounds great and I will steal it next time I'm anywhere near geometry proofs.

Kate Nowak said...

Hey Elissa -

I don't have a good explanation for why it's important. I just do it because I have to. Recently Ben had some thoughts about proof that resonated with me.

I've tried different things to convince the kids that it's not a boring waste of time...but they always end up thinking that anyway. This year I assigned them a few rounds of the Eyeballing Game and in previous years I've shown parts of a documentary about Andrew Wiles.

I have a whole process for starting very small and building up into triangle congruence, and then continuing from there into longer proofs. It's kind of involved for a blog comment. Maybe this is a good excuse to finally put some materials on BetterLesson.

Lsquared said...

I would love to see your geometry proof stuff if you put it up. I teach geometry to college students (pre-teachers), and I'm always on the look out for ways to make the teaching part of it more real for those students. Would you be willing to share some of those error-laden proofs too? Finding someone's errors is always a good higher order thinking task.

I don't know if you do the sorts of proofs where students make errors like saying that triangles are congruent by SAS when really they meant to say ASA, but if you do, I may have a trick for you. When my students get sloppy I have them start writing 3 column proofs. The first two columns are the ones you're used to, but you number each line, and the third column lists the line numbers that have the facts you're using, so if you are saying triangles are congruent by SAS, then column 3 will have generally have 3 line numbers in it: one that cites congruent sides, one that cites congruent angles, and another for congruent sides: SAS. It really cuts down on some of the sloppy mistakes.

r. r. vlorbik said...

ahem.

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.

*this* is why proofs are important.

Sarah Cannon said...

I'm curious. Why are the post-its unnecessary?

I'm guessing, and this is pure speculation, that if I were doing this, I'd try the post-its with the plan to go through and check students' errors. Then I"d get home, realize that was a stupid, time-consuming idea, and lesson plan for every other class instead. I'd still have to remove all the post-its and I wouldn't be doing anything with them.

Kate Nowak said...

Yeah Sarah. That's pretty much what happened.

vlorbik, I know, but 2-column Euclidian geometry proofs don't seem to lend themselves very effectively to that goal. There has been much discussion in recent years about why we make kids do these painstaking proofs in Geometry, where it's the only time they're likely to see the concept of proof at all. I think it's a dumb way to incorporate proof into a K-12 curriculum.

Lsquared, it's on my list to upload it. The biggest obstacle will be that many of my materials were inherited and I only have paper copies. It's going to be a lot of work unless I just scan to pdf and upload those.

I've see the third column used before and I can see where it would prevent sloppiness; thanks for sharing. Our honors-level teachers do it that way. I'm not sure it's appropriate for the level of students I have, but maybe I should try it out next year and see.

r. r. vlorbik said...

There has been much discussion in recent years about why we make kids do these painstaking proofs in Geometry, where it's the only time they're likely to see the concept of proof at all. I think it's a dumb way to incorporate proof into a K-12 curriculum.

far *too* much discussion by my guess.
elaborating on "keep it simple"
seldom accomplishes its
stated ends, duh. and yet.
here goes.

the point *is* to keep it simple.
"two-column" proofs *eliminate*
the need for prose style altogether.

let the english department grapple
with getting these fine young people
to know when they've written
a decent english sentence.
this is important work... far more
than writing proofs of any kind...
but it's *too hard*.

is this "statement" good code? (Y/N)
does this "reason" justify it? (Y/N)

lather, rinse, repeat.

math is here reduced to an
uncannily-computerlike
*binary logic*.

the "justifications" for the second column
will depend on *definitions* and *theorems*
(loosely; any previously proven piece
of code is here glorified with the name
of "theorem"); nothing else.
again, this is "keep it simple"
with a vengeance.

laying everything out with this
absurd-once-you-know-how
rigidity is precisely what makes
it *possible*... i say... for your
work with confidence; keep it
if you have any say in curriculum
(say i that never has had so far).

*this* proof is better than
*that* one because...
(of some very sharply
limited number of reasons;
big win for us until we're
sure everybody's on board
with "what's a theorem"
or 'what's a definition").

i'd rather do number theory myself
or for that matter "white to move
and mate in three" style stuff
in circular-tic-tac-toe or something.
but... the other compelling
i sure hope nobody comes along
and tries to *standardize*, e.g.,
(a,b)*[a,b] = ab
(gcd*lcm= product).

("standards are great; that's why
we need so *many* of them"...).

"reformers" know that getting students
*talking* about the math is the real win.
but the formality of two-column style
isn't a move *away* from talking
but simply (again!) the best-known
technology for becoming
darn good and sure
*the same things*
(as the text and the teacher
and our fellow students).
there's an awful lot of
baby-and-bathwater
going on in at least *some*
of the "down with geometry"
tendency...

benblumsmith said...

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.

Kate I want to hear more about these! What did kids say? What kinds of arguments did they find compelling?

Kate Nowak said...

Oh, Ben. You'd be so disappointed in my non math circly prostrate before the regents examsy self right now.

Remember how I yelled at those poor apple-cheeked Christian youth type kids disrupting lunch with their boisterous yay-God-ness at Notre Dame? That's how uncool I am in Geometry right now.

Basically I was happy that they were having discussions about how "SAS doesn't mean you are DONE. You might have missed something."

And I guess that's better than not having the discussion at all, but I know. I should have my credentials revoked. Bob Kaplan would be embarrassed to know me. Ellen would understand but be ashamed nonetheless.

Sue VanHattum said...

That's balderdash! At least I hope it is, because I am just as guilty.

I got 3 lovely students to come over to my house, with no credit or grades hanging over their heads, to be my unofficial advisory committee, and help me figure out how to teach beginning algebra better. And what do I do? I ask them those awful teachery leading questions, where you know there's a 'right' answer.

I just about wrecked the lovely magic pancake problem with my "What do you think would be a good next step?" nonsense.

But I copped to it, and we had fun, I think. They left my house wondering how to add 1 to 100. One of them (at least) will be thinking about it for the next two weeks and will call me, ecstatic, if she figures it out.

However, I don't know if I'll ever be able to retrain myself out of those terrible questions.

William Wallace said...

"let the english department grapple
with getting these fine young people
to know when they've written
a decent english sentence.
this is important work... far more
than writing proofs of any kind...
but it's *too hard*."

Actually, I think there should be more cross education. Especially for math. Math should be incorporated into history, political science, etc. And for fairness, math teachers should sometimes incorporate English, history, and political science.

How else can you get over the popularity of "I'm not good in math?"

benblumsmith said...

@Kate

Yo sorry I'm mad late. I never do the "subscribe to the comments of a post you commented on" thing, even when (as here) I actually ask a question. Need to get on that.

Anyway, so I'm self-conscious that I'm now being perceived as some sort of like authentic-mathematical-conversation priest. Anything the kiddies said that got you excited enough to write the words "amazing, and worth ten days..." was obviously amply mathematically rich for where they were at and the constraints you were under. No need for absolution from me. I was just curious.

Still curious, btw.

Kid 1: SAS doesn't mean you are DONE. You might have missed something.

Kid 2: ???

(And here's where I'm a dork for not asking you five months ago when there was any chance you'd remember.)

@William Wallace

I'm all for interdisciplinary things, but I'd like to point out a hidden assumption I'm perceiving in your rhetorical question "How else can you get over the popularity of 'I'm not good at math?'"

I may be misreading, but it sounds to me that you're implying that kids will not like / get good at / regard themselves as good at math unless a) math gets extra help by being additionally taught / shown to be relevant by being discussed in other subjects, or b) math gets enriched / spiced up / made relevant by other subjects being brought into it, or c) something similar.

If so, then the hidden assumption is that math itself is not compelling for students. Math class has failed to be compelling to many students but I believe in my gut that this has never been the fault of math itself.

unapologetic said...

@ben just a suggestion: