Here is a short report on the big experiment for this term, and a related note on a realization from today with wider applicability. I expect that this will start well, and then ramble on as I fiddle with some ideas.

This semester I have put a lot of effort into retooling *Math 1100: Mathematics in Decision Making*. This is a course which satisfies our Liberal Arts Core requirement in quantitative reasoning, and is typically a course for people outside of STEM majors. I have lots of arts and humanities majors.

The reason for reworking things is that the class size has doubled. In the last three semesters, I taught this using a group-work IBL style. All of the work was done in class, and the work was very much of the “guided discovery” variety. I found I could make this work with a class of about 30 students. It often required a lot of personality from me to help students overcome negative attitudes about mathematics or their perceptions about their ability to do mathematics.

When I found out that this semester there would be 68 students enrolled in my class, I knew I needed something different. It would be too easy for students to hide, because I can’t go get *all* of them and re-involve them in our work during a 50 minute period. There is a potential that there are just too many attitudes to shift even for your charming host.

I got some decent advice from my dear readers (Thanks, Bret! Thanks, Robert!) and hurried to put together a Peer Instruction & Guided Practice thing. I guess people would view this from the outside as a kind of flipped classroom setup. That is accurate, as far as it goes. For me, it is important that the reading and guided practice assignments are a structured mix of textual information transfer, embedded exercises **and selected inquiry-style questions** to prepare for class meetings. Then during class, I have used a mix of Peer Instruction polling (I use polleverywhere.com) and group-work IBL activities and investigations.

On the whole, I am really happy with this. It has given the class the structure it needs so that I can successfully keep so many students moving in a productive direction. I have a few reports from students who seem satisfied with it, too, but no real data about that.

I do miss the depth of the inquiry experience. In previous semesters, I could see students having giant “Aha!” moments during class. And I could watch the empowerment students take from it. I don’t know that this is not happening this term, but I have not seen them. I just miss that. I guess I will wait until the end of the term to say for sure, but I am about ready to pronounce this a successful experience.

During a conversation about teaching today, I realized something I need to do a little bit more for this class, and probably for all of my courses.

I need to sell the structure of class, and I need to describe in detail how to succeed in this environment. Those are actually two sides of the same coin. Both are part of convincing students that a different classroom setup will be a good thing for them.

I always make a point of doing this kind of thing when I teach an IBL class. I picked up early the mindset that an IBL class is just a structure for putting the students into the role of working mathematician and me into the role of research mentor. I talk with my students a lot about my motivations for this, about what to expect on a daily basis both in the mode of work and in the psychology of doing mathematics. I tell them I am trying to get them stuck in a place they need to grow, and then I will help them figure out how to get unstuck.

Now, I transferred only part of this lesson for my *Math in Decision Making* class. I have definitely shared with the students my motivation for the structure and a bit about the way it will feel. **But I have not discussed how to succeed.** At least, not with the same clarity or at the same level of detail.

The particular place I noticed this was in thinking about how the students may not be making the best use of discussion time. I do not necessarily fault them for this. I realize now, that I should not expect them to know how to have a productive conversation about mathematics. It is my job to tell them!

Here is a view of the problem. In mathematics, the closest thing we have to a work product is a well-articulated understanding of a challenging idea. Often our work involves idealized and abstract concepts, but no actual physical creation other than a communication of those ideas. So, where an artist can make a painting, a computer scientist may write some software, or an engineer may design and build a bridge, the most I can really hope for as a creation from my students is a well-made explanation, either written or oral.

Students have some trouble evaluating their own understanding. And this is natural! It is hard to test your own understanding. One can test software, or a bridge, by seeing if it performs its functions well and is free from defects.

But in mathematics, all we can expect is a communication of ideas and understanding. So, I think I will try to share this with my students. Further, I will tell them that this is how they should spend their time:

To get better at this, one must practice. But as each communication must have an audience, the students have to engage with each other. Explain your understanding to someone and let it be held up for criticism. Any criticism you receive is the key to improving your work. On the other hand, when performing the role of the audience, your job is to be critical so as to help your partner improve.

Well, now that I have written all of that, it seems pretty damn obvious. But I still feel like I have learned something new here. I have not quite put so fine a point on the “The work product of a mathematician is a communication of deeper understanding” before.

Though now that this sentence has hit the page, it makes me think of some of Bill Thurston’s writing. Maybe I’ll go reread *On proof and progress in mathematics* again. Yeah, I am pretty sure I have just re-discovered his main point from that. Ah, well. Now that main idea is mine, too.

Well, will you look at that. We are all out of time for tonight, folks. Until next week, I am your host Prof Noodlearms.

“In mathematics, the closest thing we have to a work product is a well-articulated understanding of a challenging idea.”

This seems obvious, yet I have never really thought about mathematics in this way. I should make sure that I get my students to start thinking in this way, too.

Thanks!

Bret

When I was writing, I started by thinking I had found something profound and important to me. That sentence was as close to the heart of it as I could get. And then it looked so plain I don’t know why I felt so excited a few minutes before.

Maybe its one of those things.

I liked that sentence, too. Just tweeted it out. I think it’s why a lot of calculus instructors include longer-term projects, for instance: it’s not that the content of the project is more inherently a part of the subject than understanding the Fundamental Theorem of Calculus, but it gives students and outsiders a tangible product from the class.