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 ;)?