A Fun Problem, and Critiquing the Reasoning of Others

So I’ve got this new thing going where, once a week, we spend class solving a few worthwhile problems. Problems totally independent of whatever unit we’re in, just math that isn’t trivial, but is accessible for as many students as possible, and pushes students to think carefully and reason clearly.

The first problem we did is pretty straightforward. Check this out:

It’s from Robert Kaplinsky’s awesome site, and is an image of the cash seized in a drug bust — his site has a thorough lesson plan, go check it out! I dispensed with the noticing and wondering for this one because the question I was looking for really jumped out at my students and they didn’t want to be slowed down on the way to finding out how much money was in that pile.

The math: I had students estimate first, but didn’t spend too much time on their estimates. I love how visual this problem is, but there are a few pretty abstract obstacles to estimating the amount of cash — students didn’t want to estimate without getting a few pieces of information out of me, and I found that estimates anchor pretty heavily onto what they hear other students saying. Maybe something worthwhile there, but not worth the time this for me.

Anyway, I gave students a few minutes on their own to lay out a plan for how they might count the money, then share with each other. I then had students name everything they needed to know. Each class settled more or less on the dimensions (length and width in bills, and the height of the stacks) and the denominations. We discussed the fact that it might be ok to ignore the smaller stacks out front because they were so small. The length and width aren’t too hard to estimate, and I showed them this image
to establish that they all seemed to be $100 bills. Finally, we used a ream of paper (500 pieces)
to estimate the height. At that point, I set them loose. They had to decide on estimates for each dimension on their own, decide how to account for the shorter stacks, and explain how they found their answer on their own. I told them I was looking for three things — neat, well organized work, an answer that makes sense, and a clear, logical explanation.

The sequel: Students who finished early answered three questions — did they think their calculation was too high or too low? If they were in charge of actually counting all of the money, how would they do it? How long do you think your method would take?

The feedback: This is something new I’m trying. I had students trade papers — without showing any exemplars or answers — and critique each others’ work. I’m still working on a polished rubric, but they were looking for three things — 1, is their work clear, 2, does the answer make sense given the problem and the work shown, and 3, did they write a clear and logical explanation.

This was tough for some kids — giving feedback without right/wrong to jump off of. A few kids got a bit lost and needed some help, but overall I was really happy with how they did. I think it was especially effective because this problem kindof begs for a great explanation. When we make kids write complete sentence explanations for boring old normal word problems, we cheapen the value of a great explanation — and you wouldn’t want to solve this problem without explaining your work, because you want to prove that your answer is the best, and how you estimated the different dimensions in the problem.

Anyway, after the round of feedback, I had planned on giving students a chance to revise, but there wasn’t a ton of time left, and I wanted to get through another fun problem that maybe I’ll write about later. I grabbed a few pieces of great work to show off on the document camera to give students an idea of what that might look like, showed the youtube clip with the answer, and we moved on.

I don’t know if this was anything crazy special. The mathematics behind it is well below the Algebra I level my students should be at, but it felt accessible yet still challenging, and is a good jumping off point as we move into more problem solving over the next few weeks — especially since some of the big things I wanted to get out of it — the explanations and feedback — are separate from the content. I’m looking forward to refining some of these systems and going for it again with some new problems in a few days!

2 thoughts on “A Fun Problem, and Critiquing the Reasoning of Others

  1. robertkaplinsky

    As I read through this I was reflecting on the differences between our implementation approaches. My favorite difference was your approach to having students give each other feedback on their explanations. I really want to try this out and read more about how your technique grows in implementing this.

    Without having tried it, I wonder how it would go if you said to kids, “Pretend that you have never done this problem and you know nothing about it. How well would you understand the student’s reasoning based only on their explanation?” I wonder if a simple rubric like one I use would work. A 2-point version is 1 point for correct answer and 1 point for sufficient reasoning to justify the answer. A 10-point version is 5 points for correct answer and 0 to 5 points for sufficient reasoning.

    This is great stuff and I look forward to reading updates.

  2. dkane47 Post author

    I like your idea about simplifying the rubric — one of my worries about this protocol is that it will work well for students who are confident in their mathematical skills, but students with less confidence won’t engage in the reasoning. Any way I can lower the barrier to entry for those students is definitely a big win here. Will definitely post as I keep working on this.


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