## Wednesday, October 30, 2013

### I Notice, I Wonder (Part 1)

Over the past month or so my Honors Geometry students have begun to have issues with my class. Not issues with me personally, but issues nonetheless. The bane of their recent mathematical experiences?  The thing that gives some of them nightmares of broken pencils and worn out erasers?  Yes, the very thing that I personally loved about Geometry in high school has become the new four-letter word in Geometry.  OK...so it has five letters.

Why don't students like proofs??  I think after 21 years of teaching I may have found the answer!  Well, at least part of the answer from attending a talk at ATMOPAV (Association of Teachers of Mathematics of Philadelphia and Vicinity) last Saturday.  The reason students don't like proofs...(insert drumroll here)...is because they don't understand them!  Thanks to Annie Fetter (@MFAnnie) from The Math Forum for helping me re-think this based on ideas from her talk "Ever Wonder What They'd Notice (If Only Someone Would Ask)?"

Gee - students don't like proofs, because they don't understand them.  Not really an earth-shattering piece of information there, Leigh.  I agree, but do we ask students to understand what is in front of them before they see the given information or what it is that they are to prove?  Do we require them to take some time to notice and wonder what is happening in a given diagram?  I know I had never done that before with my students.

So on Sunday night, I found a problem from our textbook that I would turn into a 2 day "Notice & Wonder" problem.  The image shown here is what my students got from me.  Notice that it doesn't have any information about angles or segments.  There is no "Given" information.

First, my students worked individually for a few minutes, completing what they noticed and wondered about the diagram.  Then, they were put into randomly assigned groups (Did I mention that I love playing cards for random assignment?) to discuss what they noticed and wondered.  Last, we came together as a class to compile a master list.

Before I share their lists with you, what do you notice and wonder?  Be sure to check back to see their lists and how we moved some items from the wonder column to the notice column.

1. I LOVE this problem! I notice all sorts of things that can be defines using the transitive property and the triangle sum theorem.

I wonder if BE and CD are parallel.
I wonder if AB and AE are congruent
I wonder if AC and AD are congruent.

2. I have trouble understanding the proofs myself. It's especially hard when in every diagram all our "i wonders" end up becoming "i notices". Without some ideas being false, it's hard to tell the students why they can't just assume all the things they see in the picture.

3. I think the important question is to ask them, "How do you know?" For example, if you think triangle ACD is isosceles, how do you know? What would be necessary to make the triangle isosceles. Ideally students would say that two segments (like AC and AD) would need to be the same length or that base angles would need to be congruent. If we aren't given either of these items, is there some way would could work with what we are given to get to one of these items (congruent segments or congruent angles)? This is essentially what constitutes a proof.

However, I think students are often overwhelmed, because they don't have a chance to list everything they think or wonder before diving into the problem. When students are given a prompt in an English class to write an essay, they don't start with the first sentence and then write the second sentence and it flows perfectly. They usually start by jotting down some ideas or an outline. For some reason, this is not modeled or encouraged with proofs. So, most proofs become very generic, basic and boring.

4. I like your idea of likening a proof to an essay. Perhaps diving directly into the steps of a proof is not the best approach. I think that brainstorming (noticing, wondering) would be the most effective if students had some incorrect ideas mixed in with the correct ones. Do you find this to be the case in your classes?

5. I agree that having students state incorrect ideas is part of the learning process. Part of value of incorrect ideas is not telling them that their idea is incorrect, but having them deduce that for themselves.

For example, maybe a student says "I notice that the pentagon in the diagram is a regular pentagon." Like any other "I notice" statement, I ask "what makes you say that?" or "how do you know? what makes a polygon a regular polygon?" At this piont the student starts to think aloud and at some point stops themselves and says "Nevermind." But then sometimes I go a bit further and ask them "Why did you say Nevermind? What are you thinking? We all can benefit from your ideas."

Validating student ideas (both correct ones and incorrect ones) is key to having students think like a mathematician and seeing math as more than just a list of rules/algorithms.