## Trying to Understand what my Students Understand

Starting to think about school again and this question has been clanging around in my brain. On my last test for my AP Calculus BC kiddos I included the following question:

My BC gang absolutely nailed this question. Almost every single one cited concavity for part b noting that a function with positive slope AND positive concavity will increase at an increasing rate while the tangent line increases at a constant rate. So, moving to the right of the point of tangency means that the function has pulled away from the tangent line. They almost uniformly used the language I just used with slight tweaks and maybe a little less detail since they were operating under time constraints. I was proud of them for such detailed answers to an important principle of graph analysis. However, after the happiness faded there was a nagging concern that arose. I worry that they are SO good at citing this language that perhaps they are simply responding to a familiar prompt. I am not here claiming that these talented students do not understand this principle. I am here claiming that I am concerned that I have ‘trained’ them too well in responding to certain prompts, that I have enabled them to simply repeat a claim that I have made convincingly in their presence. I want to do some deep thinking about how I can circle back to this idea and ask this question in a form that is similar enough that it is clear what I am asking, but different enough that my students will have to say something different to betray their understanding. I would love any advice on how to continue to poke at/probe how deeply my students understand this concept. Any clever ideas out there? Drop a line into the comments section or tweet me over @mrdardy

## My Students are Making Some Smart Guesses

This conversation was a wonderful way to end our day on Friday. I am delighted that my students are comfortable enough to make these guesses out loud and even more delighted that they are making such good guesses right now. I pointed out how helpful it is to play with GeoGebra to check these guesses and I hope (I hope hope hope!) that some of my students are making a habit of this.

## A Delightful Conversation

Last week in my Geometry class we had a fantastic conversation about a homework problem. Here is the problem in question –

I wish that I could take credit for having written this, but I am certain that I ‘borrowed’ it from somewhere. Likely from the fantastic resources shared with me by Carmel Schettino (@SchettinoPBL)

If you recognize the above problem as your own, feel free to claim it and let me know. Know in advance that I am very grateful for such a rich problem to tie together ideas of distances, slopes, line equations, properties of squares, and triangle congruencies all into one tidy package!

## The Decisions We Make…

I have two sections of Discrete Math this year, one in the morning and one right after lunch. During the fall term, each of these sections had 7 students. We all sat at a single group of desks together and had some great conversations. A number of the students have spoken to me about how much they enjoy this atmosphere. It does not work for everyone of course, some students prefer not to have the expectation of participation, they would prefer to quietly observe and have more time to think before speaking. Our school is on a trimester schedule and this Discrete course is set up as a trimester course where students can move in or out and not have the demands of previous knowledge from this course. So, I have done some thinking about how to make this course modular. One of my sections expanded from 7 students to 16 this term and we are in the process of figuring each other out and how this new group will mesh. One of the students who has been in the class the whole year commented that class seems more quiet this past week. Interesting that more than doubling the size of the class has resulted in a quieter atmosphere…