Teaching Math a Third Way

I was reading Harry Webb’s advice to a new secondary teacher, describing his usual classroom procedure for “senior maths”, as an addendum to his earlier post on classroom management. And I thought hey, I could use this to fully demonstrate the difference in math instruction philosophies.

Harry’s lesson is a starting activity, a classroom discussion/lecture, and classwork.

So here’s what I did on Friday for a trig class, which is certainly “senior maths”: brief classroom discussion, class activity (what Harry would call “group work”), brief classroom discussion. And I think it’s worth showing that difference.

The kids walked in, sat in assigned seats grouped in fours—strong kids in back, weakest in front. I often forget and start before the tardy bell, just laying out what we’ll do that day. I never check homework—the kids take pictures and send it to me, and I eventually get it into the gradebook. I don’t really care if kids do homework or not. They take pictures of it and text or email me. I eventually check. If kids have homework questions, they’re to let me know during the tardy pause and I’ll review them on an as-needed basis. But yesterday, the kids hadn’t had homework, so not an issue.

When the tardy bell rang, I had just finished sketching this:

(this next bit is what I think Harry would call classroom discussion):

“Can anyone tell me the relationship these triangles have?”

I got a good, solid chorus of “similar” from the room—not everyone, but more than a smattering. I picked on Patti, up front, and asked her to explain her answer.

“They have two congruent angles.”

“Good. Dennis, why do I only need to know about two of the angles?”

Dennis did the wait out game, but I’m better. After a while, he said, “I don’t know.”

“Do you know how many degrees are in a triangle?”

“180. Oh. OK. If they add up to 180, and two of them are equal, the third one has to be the same amount to get to 180.”

“See, you did know. Jeb, if two triangles are similar, what else do I know?”

Jeb, in the back corner, said “The sides have a constant ratio.”

“More completely, the corresponding sides of the triangle have a constant ratio. Good. How many people remember this from geometry?” All the hands are up. “If you had me for geometry, and about eight of you did, you may even remember me saying that in high school math, similarity is much more important than congruence, for high school math, anyway. Trigonometry will prove me right once again. So while I hand out the activity, everyone work the problem.”

When I got back up front, I confirmed everyone knew how to solve that, then I went on to this:

“I don’t want everyone to answer right away, okay? I’ll call on someone. Give people a chance to think. Which one of these variables can be solved without a proportion? Olin?”

Olin, very cautiously: “x?”

“Because…”

“I can just…see what I add to 8 to get 12?”

“Right. Now, that probably seems painfully obvious, but I want to emphasize—always look at the sketch to see what you know. Don’t assume all variables take some massive equation and brain work. Now, how can I find the length of the other side? Alex?”

“I’m just trying to figure that out.”

“You’re assuming the triangles are similar? Can she do that, Jamie?”

“Yes, because the lines are parallel.”

“Hey, great. Why does that help, Mickey?”

“I don’t know.”

“Cast your mind back to geometry. Which you took with me, Mickey, so don’t make me look bad. What did we know about parallel lines and transversals?”

“Oh. Oh, okay. Yeah. the left angles are congruent to each other, and the right ones, too.”

“Because….”

“Corresponding angles,” said Andy. I marked them in.

“Okay. So back to Alex. Got an equation yet?”

“I don’t know what I should match with what.”

“Okay. So this, guys, is the challenge of proportions. What will give me the common ratio that Jeb mentioned? I need a valid relationship. It can be two parts of the same shape, or corresponding parts from different shapes. Valicia?”

“Can I match up 8 and 6?”

“Can she?”

“Yes,” said Ali. “They are corresponding. But we don’t know what the short leg is.”

“We don’t need to,” says Patti. “6 over 8 is equal to y over 12.”

After finishing up on that problem, I turned to the handout.

“I stole this group of common similar triangle configurations, just as a way to remember when they might show up. But we’re going to focus on the sixth configuration. Can anyone tell me what’s distinctive about it?”

“It’s a right triangle with an altitude drawn,” offered Hank.

“True. Anything unusual?”

“No. All triangles have altitudes.” He looked momentarily doubtful. “Don’t they?”

“They do. So take a look at this” and I draw a right triangle in “upright” position. “Where do I draw an altitude?”

“You don’t need to….Oh!” I hear talking from all points in the room, and pick someone up front. “Oscar?”

“That’s the altitude,” he points. I wait. “The—not the hypotenuse.”

“Melissa? Can you give me a pattern?”

Melissa, in back, quite bright but never volunteers. “If the leg is a base, then a leg is the altitude.”

“True for all triangles?”

“No. Just for rights. Because the legs are perpendicular.”

“Right. So back to Oscar, what’s different about this?”

“The hypotenuse is the base.”

“Right. So it turns out that the altitude to the hypotenuse of a right triangle is….interesting. Turn over the handout.”

The above conversation, which takes a while to write out, took about 15 minutes, give or take. I would expect Harry Webb has similar stories.

The next part of my lesson is the “group work” that Harry and other traditionalist think leads to “social loafing” and wasted time.

The kids are in ability groups of four; they go to whiteboards spaced all around the room: two 5X10s, 3 4x4s, and self-stick on bulletin boards that works great—I even have graphs attached.

And I just give them instructions and say, “Go.”

Is this discovery math? Hell, no. I give them all sorts of instructions. I don’t want open-ended exploration. What I want for them is to do for themselves and understand what I would have otherwise explained.

In the next 50 minutes, using my instructions, each group had identified the three triangles:

There’s always a surprise. In this case, more of the kids had trouble proving the similarity (that is, all angles were congruent) than with the geometric mean. I actually stopped the activity between steps 1 and 2 to ensure everyone understood that the altitude creates two acute angles congruent to the original two–which I frankly think is pretty awesome.

Even before they’d quite figured out the point of the angles, they’d gotten the ratios:

Each of the nine groups found the second step, proving the altitude (h) is the geometric mean of the segments (x & y) on their own; I confirmed with each group. Once they’d established that, I reminded them that the third step was to prove the Pythagorean theorem and to look for algebra that would get them there. Four of the groups had identified the essential ratios, identifying that a² = xc and b² = yc.

At that point, I brought it back “up front” and finished the proof, which requires three non-obvious steps.

a² + b² = xc + yc (reminding them about adding equations)

Then I waited a bit, because I wanted to see if the stronger kids pick up on the next step.

“Just think, a minute. Remember back in algebra II, when you were solving for inverses.”

“…Factor?” says Andy.

“Oh, I see it,” Melissa. “factor out the c.”

“Right. So then we have a² + b² = c(x+y)”

“Holy sh**.” from Mickey.

“Watch the language.”

“That is so cool.” says Ronnie, who is UP FRONT!

“if you don’t know what they’re saying, everyone, look at the diagram and tell me what x+y is equal to.”

And then there were a lot of “Holy sh*–crap” as the kids got it. Fun day.

I wrapped it up by reminding them that we were just doing some preliminary work getting warmed up to enter trig, but that they want to remember some key facts about the geometric mean, the altitude to the hypotenuse of a right triangle. Then I go into my spiel on the essential nature of triangles and we’re all done. Homework: Kuta Software worksheet on similar right triangles, just to give them some practice.

This lesson would rarely be included in a typical trig class, whether reform or traditional. I described the thinking that led to the sequence. But it’s a good example of what I do. (Also, as many bloggers have pointed out, my attention to detail is dismal, both in blogging about math and teaching it. Kids usually pick up on stuff I miss, and if it’s something big, I go back and cover it.)

I vary this up. Sometimes I go straight to an activity they do in groups (Negative 16s and Exponential Functions), other times I do a brief classroom discussion/lecture first (modeling linear equations and inequalities). Sometimes I have an all practice day or two—I’ve covered a lot of material, now it’s time to work problems and gain fluency (that’s when the tunes come out).

I originally had more but somehow the length got away from me, so I’ve chopped this down.

I have developed this method because I was never happy with traditional math, whether lecture or class discussion. The difference is not solely about the method of delivery; my method requires more time, and thus the pace is considerably slower.

The jury’s in on reform math: it doesn’t work well in the best of cases, and is devastatingly damaging to low ability kids. Paul Bruno refers to reform math as the pedagogy of privilege, and I agree. But it’s worth remembering that reform math evolved as a means of helping poor and black/Hispanic kids. Why? Because they weren’t interested in traditional math methods, and were failing in droves.

Ideally, we would stop forcing all kids into advanced math. But since that’s not an option, I think we need to do better than the carnage of high school math as we see it today: high failure rates, kids forced to repeat classes two or three times Given the ridiculous expectations, traditional math is due for some scrutiny, particularly in its ability to leave behind kids without the interest or high ability to carry them through. Let’s accept that most kids can’t really master advanced math. We can still do better. This is how I try for “better”.

I still have problems with students forgetting the material. I still teach kids who aren’t cognitively able to master higher level math. I’m not pretending the problems go away. But the students are willing to try. They don’t feel hopeless. They aren’t bored. I don’t often get the “what will we use this for” question—not because my math is more practical, but because the students aren’t looking for an argument. (And when they do give me the question, I tell them they won’t. Use it.) However, as I mentioned in the last post, I now have had students two or three years in a row. They were able to pass subsequent classes with different teachers, but they haven’t lost the ability to launch into an activity and work it, having faith that I’m not wasting their time. That tells me I’m not doing harm, anyway.

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