When students are smarter than an answer key

When I first saw this image, I had so many thoughts that went through my mind. Although I could write about any of them, I want to take time to look at the student’s thoughts.

As explained in a comment on George Couros blog, when making small talk with people, the question usually comes up about what we do for a living. When they find out that I am a teacher, the next question they always ask is, “What do you teach.”

My typical answer, “Students.” Granted, there is a lot of math that gets discussed, but my focus tries to be on developing the student. After all, other than a high school math class, how often will an average person ever use the “CPCTC” (or “Corresponding Parts of Congruent Triangles are Congruent”) theorem? What about factoring? I think most real-world polynomials of interest cannot be factored.

So why even both having math classes if we are teaching students skills they will never use outside of the classroom? For me, it is because I am trying to teach so much more than math. I am teaching how to follow directions and proceedures. I am teaching logical thinking. I am teaching how to explain their complex thinking to others. I am teaching how to persist, especially in light of difficult problems.

I am also trying to get them to think outside the box. (Also one of the reasons I have participated in Odyssey of the Mind for so long, especially at the regional and state levels.)

I love this student’s answer for this problem. I think the writers of this problem (and consequently the answer key) just assumed that we would assume the pizzas were identical. The purpose of this question was to help students realize that (if the pizzas were the same size), Marty could not have eaten more pizza that Luis, and so the statement that Marty ate more is incorrect.

Marilyn Burns recently wrote a blog post about algorithms and shared the work of a student she called Omar. Part of his work subtracted 98 from 100 and got 88 instead of almost nothing. He also took 120 and subtracted 50 to get an answer of 420 — more than he started with!


Problem like number 8 can hopefully help students learn to self-correct and realize their own errors.

But when I look at this student’s answer, I love how they thought outside the box! They didn’t confine themselves to equally sized pizzas. If the statement made is correct (and why wouldn’t it be; the teacher gave it to us), then there has to be something about the problem that may not be obvious, but allows the given problem to be true. Having pizzas of different sizes would certainly qualify.

If my pre-teen son and I shared the same pizza, we both might eat “half” of the pizza, but I can assure you that when I cut the pizza, the “halves” will not be equal size so I will still eat more of the pizza. I’ve also cut a pizza in quarters and then cut the quarters in half (to make 8ths) or in thirds (to make 12ths) so that my wife and I can have the same number of slices as our children, but still get more in our bellies. If I can get creative serving pizza, why can’t this student?

Granted, my examples were using the same pizza and this problem refers to different pizzas (it must be different pizzas since 4/6 + 5/6 of the same pizza makes more than one pie; also the references are to “his pizza” and not to “the pizza”).

I worked with a Geometry student last week who mathematically got a correct answer, but rationally, she didn’t think the answer made sense. The problem asked her to take a pool (shaped as a quarter sphere) of water, and to find the volume of a model that shrunk each dimension by 6. Although I didn’t know the numbers she used, I knew that the answer would be 216 times smaller than the larger tank. She thought that answer would be too small, so though she made a mistake.

We talked about the reasoning for the difference being so large. As a parting problem, I asked her to calculate the area of pizza next time she ordered it, to see what kind of value she was getting. I gave the example, “Would a 6″ pizza have half the area of a 12″ pizza since 6 is half of 12?” and as a follow-up, “Would we want to charge half as much for a 6″ pizza as the 12″ inch?”

What do you think?


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