(I apologize for getting this up a bit late. Almost two months after the initial post. I'm working on National Board Certification renewal and life got in the way. But what can you do.)

Given this diagram, what do you notice? What do you wonder? Annie Fetter (@MFAnnie) of the Math Forum tweeted that she wondered if segments BE and CD were parallel. This was not the first thing my students wondered about. They wondered if certain triangles were congruent and if certain triangles were isosceles. This was not surprising, because we had just completed a unit on congruent triangles and isosceles triangles.

So, what did they notice and wonder? I gave students about 2 or 3 minutes to create a list individually and then I put them in groups of 3-4 to compile their lists. Then, we went around the classroom with each group contributing one item and me creating a master list. This is something like brainstorming. All ideas were welcome. Once or twice we did have to move something from the "I Notice" column to the "I Wonder" column. For example, students said "I notice AGF is isosceles." However, since we were not given any segment lengths or angle measures, we had to move this to the "I Wonder" column. Most of the time students are given diagrams with angles and segment lengths given and the fact that this diagram did not have any measurements was slightly disconcerting to some of them.

Here is a sample of what they "Noticed and Wondered".

Next, I gave them some additional information about angle measurements. They were told: Angle 1 was congruent to Angle 2. Angle 3 was congruent to Angle 4 and Angle 5 was congruent to Angle 6.

This allowed us to move some of the ideas from "I Wonder" to "I Notice", and this lead to discussing

Here is a sample of what they "Noticed and Wondered".

Next, I gave them some additional information about angle measurements. They were told: Angle 1 was congruent to Angle 2. Angle 3 was congruent to Angle 4 and Angle 5 was congruent to Angle 6.

This allowed us to move some of the ideas from "I Wonder" to "I Notice", and this lead to discussing

__why__we could make the moves from "I Wonder" to "I Notice". Do you see the beginnings of a formal proof???
Class finished by having students work in their groups to write a formal proof focusing on congruent triangles. The class discussion was rich and the geometry ideas plentiful! Thanks, Annie Fetter for helping me and my students to "Notice and Wonder"!

Leigh,

ReplyDeleteThanks for your post. I've been teaching proofs and struggling to get students to "feel" the connection between statements and their justifications. I like the idea of everything being an "I wonder" until proven guilty.

twitter.com/mrcoleymath

I also think part of the problem with proofs is that you need to spend time with it. Students are used to math being a speed game - timed math fact tests, standardized tests, short questions with one right answer. The "Notice, Wonder" approach to a proof is new to students, because there is no one right answer and one "problem" can take an entire class period (or longer) to complete!

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