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I’m probably going to get a lot of flack from my physics friends for saying this, but teaching math is a crazy amount of fun, and it may (gasp) even be more fun than teaching physics. When I first heard I would be teaching Algebra II honors this this summer, I was worried. An entirely new prep, in a different subject I hadn’t really taught before—it was a big push outside my comfort zone and it seemed like a lot more work. But after today, I’m really thrilled that that I got pushed in this direction. The newness of what I’m doing and unexpected-ness of what our class finds is so fulfilling.

Let me tell you what happened today that made me a true believer in the joy of math teaching. My colleague, whom I co-teaching this course with, and I decided to start our first unit on linear functions with the cup stacking activity. I cribbed dan’s notes and put together a simple 3 act presentation with a photo of me next to a stack of cups, and then when students told me they’d need my height, I gave it to them (200 cm—more on that later).

Distressing part here—not more than 5 minutes into our discussion, a student blurts out and basically gives away the problem to the entire group. I gulped and felt a pit in my stomach—”okay,” I said, “time to start working in small groups on these problems.” As students began to find what they thought was the answer pretty quickly, I asked them to start to complicate the problem and think of their own questions. Now things started to get interesting very quickly.

Here are some of the cool tangents we followed from this cup stacking problem:

• Pyramid stacking: one group decided they wanted to figure out how many cups you would need to make a pyramid stack of cups equal to my height. Knowing that each cup was 12 cm tall, they able to figure out that they would need 17 rows of cups to equal my 200cm height. Then they realized that starting from the top cup the next row down had one additional cup, and so they just needed to add all the integers from 1 to 17 to figure out how many cups they needed. The super cool part is that, rather than pouch this into their calculators, the students wanted to find a pattern, and with a tiny hint from me saw that 1+17=18, and 2+16=18, etc, which they quickly generalized into the formula $\frac{n}{2}(n+1)$. But they didn’t stop there. They decided that they really wanted a formula for the number of cups, $x$ you’d need to reach a given height, $h$, rather than the number of rows you’d need. And so they used their previous work to express the number of rows as the height divided by 12, and came away with this formula for the number of cups in terms of the height:

$\left(\frac{h}{12}+1 \right)\left(\frac{\frac{h}{12}}{2}\right)$

Needless to say, I was speechless.

• Undoing the mistake game: When we presented our work, I asked students to follow the Mistake Game as described in this outstanding post by Kelly O’Shea, and one group of students latched right on to an excellent mistake of not fully seeing that the base of the cup is different from the full height of the cup, and so they created a mistaken formula, $\frac{h-12.2}{2}=c$, as shown below. Along the way, the student was able to explain the meaning of each of the numbers (12.2 is the full height of the cup, 2 is the height of the brim of the cup).Bu rather than putting in the original formula the student had created that was correct $\frac{h-10.2}{2}=c$, she tried to take the wrong formula and fix it another way, by subtracting 1 from the total number of cups, as shown in the whiteboard:

And this led to a great conversation about how we know these two formulas are equivalent because they are the same when we simplify them, but how the unsimplified form is probably easier for someone to understand, since the individual terms in the equations are related to the actual measurements of the cup.

• Why units matter: another group seized the idea that there are 3 measurements for the cup—the cup height (12cm) the base height (10 cm) and the brim height (2cm), and these form a system of measurement not unlike inches, feet and yards. Then they found that I’m 94.85 cups high, and so I asked them to figure out how much they’d need to cut from a cup to equal my height—.15cm? This led to a great discussion of the full story of a number and the value of units.
• Mixed stacking: another group wanted to figure out the height of the stack if you mixed stacking cups end to end with interleaving them, and so they came up with the formula: $z=10x+2y$, where z is the height of the stack in cm, x is the number of bases visible, and y is the number of brims visible.
• Inverse functions: one group took to graphing the height of the stack vs the number of cups, and another graphed the number of cups vs the height of the stack. This led to many interesting conversations about whether the graphs should be points or a line, and the meaning of the intercepts with the horizontal and vertical axes. We were then able to compare the equations and see that they were really telling us the same thing when we solved for the other variable.
• Making equal stacks: A group got to wondering about how to make a stack of interleaved cups the same height as a stack of end-to-end stacked cups, and they realized there are an infinite number of pairs of nend to end stacked cups and minterleaved cups that can have the same height.
• I’m not 200 cm tall: One astute group realized there is no way that I’m 200 cm tall, which led to a conversation about how we need to check our answers and the numbers we are working with for plausibility.

## Why this was aweosme

During all of this time my brain was racing—students were coming up with ideas I hadn’t thought of at all. This just doesn’t really happen as often in physics, where I think I can honestly say I’ve seen every thing a constant velocity cart scan do. Everyone was working to extend their ideas further, and when we paused class to write out quick summary tweets of what they were learning, the tweets were all about seeing multiple solutions, learning new ideas and #awesome (more about that in a future post). Somewhere along the way, students stopped just writing down math symbols and pushing them around the paper, and instead were deeply focused on the meaning of the equations they were writing and finding new ways to check their reasoning. Not one did a student ask me “Is this right?”

## Why I still have a ways to go

At times this class felt frenetic for me with ideas flying around all over the place, and if it felt that way for me, I bet there were moments for students where it was just a sea of confusion. Also, on the next day when some students came back to their work, I noticed that a bunch of them didn’t have anything written down from the previous day, they’d done all their work on scratch paper. Both of these things tell me we need a much better way of organizing our work and pulling out the key ideas from discussion so that students don’t get completely lost and think that I’m going to be asking them to determine the formula for pyramidal cup stacking when we’ve only been working with linear functions.

I tried to give a bit more structure at the end by just looking a the two cases of the interleaved cups and the end stacked cups, and I think this discussion was able to pull out some bigger ideas about graphing and linearity but I would have liked to help students come to more of these ideas on their own, rather than taking over the last 10 minutes of class.

So I think I’m starting to see my desire to teach a math class focused around problem solving that vies students the room to explore problems of their of design as well. I’m also deeply interested in developing more structures like keeping a problem solving notebook that will help students who might be feeling lost when ideas start flying about.

I’d love any feedback or suggestions you may have to help me to address these concerns.

September 8, 2012 8:43 am

very cool!

2. September 8, 2012 10:32 am

Brilliant class, John. I chuckled at your claim to be Dan M’s height and your students’ (apparently) blind and unquestioning faith in whatever comes out of their teacher’s mouth.

Letting go of a lesson is the ONLY way to get those thoughts from students that we teachers were “trained” long ago to not think to ask. It is the very inexperience of the student that allows him/her to ask what we wouldn’t think. Thankfully, they don’t know any better.

I love your props to math teaching, but in my mind, math & science have always been two versions of the same attempt to distill meant out of the patterns we find around us–both in our minds and in the world around us. It’s especially cool when what we learn in one of those domains informs what we can know in the other.

3. September 8, 2012 2:50 pm

Sounds like you’re in for an awesome year. A question I’d love to hear from you on (maybe in extended format, not just a quick answer here): what experiences did your students have before they got to your class that developed their abilities to model with mathematics in this way?

• September 8, 2012 9:16 pm

Really good question.

• September 8, 2012 9:19 pm

Dan,
That’s a great question. I think I’ll get some insight into this when students turn in their automathographies in a couple of days, which I’m hoping to be able to blog about. Almost all of the students in my class are sophomores, who took an Exeter style problem solving course that used the Exeter Course 2 materials, so I think that probably has a lot to do with where they are mathematically.

• September 11, 2012 8:31 am

Yep, that would do it. 🙂

4. September 8, 2012 8:14 pm

I concur on the #awesome. 🙂 I have the same “frenetic, students with no notes” experience a lot. One strategy I’m going to try this year is insisting on a whiteboard template (I’m going to start with Jason B.’s Evidence, Claim, Reasoning approach). At the end of class, I’ll document scan the whiteboards and print a hard copy for every student. That way, the whiteboards end up in their notebooks for future reference. It’s a lot of paper, but since they’re saving so much paper by not taking notes, I don’t feel so bad 😉

• September 8, 2012 9:21 pm

I like this idea. I definitely am planning on photographing all of our whiteboards, and I like the idea of having students approach whiteboarding with a greater emphasis on making sure that their boards tell the complete story of their problem solving.