Is the generalization ever too much?

We love having students make generalizations in math class. Is this always true? Will it work for every number? If students can answer those questions, we feel we have created a successful learning experience for students, right?

Well, after attending a session today on supporting teacher learning in the CCSS, it led me to question if there is a time when the generalization hinders a learning experience? For example, we sat down to this problem: “Find all possible dimensions of a rectangle where the area equals the perimeter.” We worked through the problem individually and then together as a group. After coming up with 6×3 and 4×4 by guessing and checking, we started forming some ideas towards a generalization that would push students past guess and check. After some discussion, we concluded that the dimensions couldn’t be two odd numbers and there was a time when the area grew more rapidly than the perimeter so those larger dimensions would not work. After trying to set up an algebraic equation to formulate a generalization, we stopped to share as a group.

Long story short, we were told the generalization to find all possible dimensions with equal area and perimeter was that if a rectangle with sides a and b, a = 2b/(b-2). Now my question is this, does this generalization alienate a large group of students? I know as adults, we persevered and created viable arguments; however at a certain point we saw no entry point for many 6th grade students to answer this question. As adults, we were even at a loss after a certain point of working. Attentions started to stray and side conversations began. On the flip side, if i had left without the generalization, I would have left frustrated. But did that generalization help me make connections between length of sides and area and perimeter? I would argue not.

I feel that if we are going to have students make generalizations, there needs to be connections among entry points and when there is not a visible connection, I am at a loss.

Any general thoughts ;)?
– Kristin

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