Tag Archives: Generalizations

Math Reasoning Stages

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Tracy Zager (@TracyZager) asked for some thoughts/pushback on these stages of math reasoning imagined as a flow, so here are my thoughts based on my experience in the classroom…

Pattern Sniffing –  After students see a pattern I find they continue, using that pattern, for a while before thinking about a generalization. So, maybe “Extending Pattern Using the Pattern” comes after this in my mind?

Wondering – When they wonder, they definitely look at relationships, but I am not sure they wonder if it will always be true at this point? Now that I just wrote that, I am thinking maybe “Extending Pattern Using the Pattern” comes after this one?

Articulating – “Can I communicate what I am seeing happening in a precise way?” I don’t know if they are thinking too much about it at this point but more seeing it happening? Could

I don’t know where this fits necessarily, if it is embedded in one of them, or if it really fits at all:), but there is a point where mathematically students prove a generalization works with certain number and not others because “the numbers have to work that way” (structure) without the conceptual proof of why that is. For example, “Even dimensions of a rectangle will give you an even area.” Students can make the statement that it has to work every time because when you multiply even numbers it is always an even product….true, but isn’t there proof to that. So, it is like a string of proof by depth?

Investigating and Explaining Why – I feel the relationships and patterns question to themselves comes back up here too.

I love thinking about this process for students and the teacher implications between each step. What questions and/or feedback do we give as students go through this that isn’t too helpful or leading, but not too vague that leave them in one spot spinning wheels? Paging @MPershan…

*Chatting with Tracy after I wrote this, she was focused on mathematicians, not students. I find some holds true in both cases to different sophistications. 

Hope that helps a bit Tracy! Hopping on a plane but as soon as I have wifi I will add a couple more question I have to think about around this!

-Kristin

Is the generalization ever too much?

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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