This week I have started to new ongoing conversations that I am really excited about.

### A Formal Discussion Group

First, I joined a small “Talking Teaching and Learning” group on campus. This is a multidisciplinary group of people who wish to have a small community for working on issues related to being an educator. One of the ground rules of this group is that the conversations are confidential, so I will just say that I hope to use the group as an accountability mechanism for me. I shared during our first meeting that I will be working on three things in the near future:

- developing an assessment method I am happy with using (focus on Math 3600 Euclidean Geometry)
- learning to teach Math 2500 Linear Algebra
- refining my approach to Math 1100 Math in Decision Making

I will probably talk about some of my thinking here, as I noodle through things, but this will be the last mention of the group. I think this is a fun idea, and I am looking forward to participating.

### An Informal Chat over Tea

Today I had a longish discussion with my colleague Scott Peters. Scott teaches political science at UNI, and we sometimes play soccer together. He was curious about what IBL might mean for a social science course! I am so glad I did this today.

Sometimes you just need to start talking and see what comes out of your mouth. Then you can evaluate it and decide if you really mean it. [Hell, that is why this blog exists. Just replace “talking” with “typing.”] The conversation with Scott was nice because he was very thoughtful and because he comes from a very different discipline. This meant we had to talk about and navigate through to the important commonalities that are really about teaching and learning from an inquiry based learning viewpoint *without reference to mathematics*. I learned some very important lessons from our talk today. They are important enough that I want to write them down, so I can find them again later and feel guilty when I realize I haven’t internalized them well enough. (*Hey, look at that. I wrote exactly that thing down two years ago. What was I thinking?*)

##### Lesson One: Introducing Students to asking their own questions should be done in a narrowly focused context

Scott floated the idea of having students pick their own questions to work on–essentially he wanted them to develop their own small program of study. He has tried something like this lately, and was unsatisfied. I reflected upon my EG experiences and shared that I invite the students into the process of asking questions and making conjectures, but I do it in a very deliberate way.

At the beginning of the course, I set all of the questions, and I model making new questions and conjectures during class when the opportunity arises. Sometimes a presentation doesn’t quite hit the mark the student wants, and then an easy way to “salvage” things is to state a theorem encapsulating exactly what the argument **does** prove and then make a conjecture that covers the gap in the argument. Sometimes an argument looks juicy enough that I just ask the presenter if they can think of any conjectures that come to mind in the context of their work.

Later in the semester, I explicitly ask students to find and prove unstated theorems that are analogous to ones already in our records. For example, after we have proved a bunch of things about rhombi, I introduce the notion of a kite and set them loose. Even later I can work in a more open-ended way. By mid-semester, some of the students have their own observations to share and they ask permission to make conjectures. (Iowa students are so polite and deferential.)

Anyway, the main point is that without realizing it, I have things structured to slowly acculturate the students into doing mathematics *including what it means to ask a question and what kind of question we might have a chance of answering.* Importantly, the question-asking is also done in very narrow, specific contexts. That allows the students the freedom to practice asking their own questions, but only gives them a big enough sandbox to do so in an appropriate fashion. I get high-quality work out of them because they come to understand what that means first. Apparently, I am so awesome that even I didn’t recognize it.

Scott pointed out that one of the things going on is that I am only allowing students the opportunity to ask their own questions after they have begun to get a sense of what a proper mathematical process of finding answers is. He felt that this was lacking in his approach. I can’t say for certain if that is true for him, but **[expletive deleted]** that is definitely true for me! This is a big part of why my Differential Geometry course failed this term. I gave them all the freedom in the world, which is waaaaay too much. What an eye-opener.

##### Lesson Two: Replicating the Success of Euclidean Geometry might require more faithful replication of the format

Scott asked me about my textbook choices. This gave me a chance to talk about my rationale for using Euclid’s *Elements* as a text. For all of you, the abridged version is this:

- I want some examples of correct proofs for students to see
- I want students to experience reading mathematical literature for understanding
- I want to be able to assume
*something*or this class will never get anywhere - I want to infuse the course with a sense of connection to history and wider mathematical culture
- I want the students to learn to critique everything, even Euclid (some of his arguments are wonky)
- I don’t want to pretend the students don’t know anything, even if they really don’t. (They have all had a geometry course in high school. But mostly they don’t have anything but vague memories.)
*The Elements*acts as a convenient bandaid. The facts we need that they are likely to recall from previous schooling are in there.

I tend to think of the course as a mini research community: I am the grand mathematical guru, and my students are new graduate students who wish to be mentored into the professional mathematics community—but all about a millenium ago, when aspects of this planar geometry stuff is still cutting edge. *The Elements* is our full suite of reference literature, and then I set a research program for the group around it and extending it.

Scott latched onto the idea of using historical sources as a way to structure the development of his material. I have always liked this idea, but I haven’t done strictly that. And this is another thing that hit me! This is missing from my other courses. I don’t have the “extant research literature” for students to grapple with and use as a foundation. But maybe I need to make that. So, for Differential Geometry or Linear Algebra I could make some synthetic replacement for Euclid’s *Elements* by looking in the historical record. At this point, I don’t expect to find such a convenient piece of ancient scholarship for other courses that plugs into just the right spot, so I’ll have to create something.

This sounds like a lot of work, but it might be just the thing.

Though right now I have my doubts about using such an approach for Math in Decision Making. I don’t know why. I hope it is not a prejudice on my part. More reflection required.

Hi,

A couple of comments here:

“Introducing Students to asking their own questions should be done in a narrowly focused context:” I struggle with this. I seem to wildly oscillate between giving the students too much of a context (and having them fail because of it—too much freedom) and giving the students too narrow of a context (thereby limiting the students’ creativity—too little freedom). If you are doing this well, you are awesome.

“Replicating the Success of Euclidean Geometry might require more faithful replication of the format:” are you suggesting that for , say, linear algebra if you simply gave them a textbook, and then devised a pseudo-research program for them. So the idea is that they would learn the stuff in the textbook by working on problems beyond the textbook? My description is overly simplistic (the students would need more structure), but I think that this idea could be brilliant (or awful—one of the two). I am kind of thinking of doing something like this in abstract algebra in the spring now. I have no plan yet, though.

Bret

The point is that I was doing this bit of “invite students to ask questions” successfully without understanding it. In one class where I just let it happen in context, it works. In any place where I try to do it explicitly, it has failed miserably. the difference definitely seems to be the scope of allowed context. “come up with three conjectures about kites using our work on rhombuses as a guide” always works. (And later those students are prepared to just ask questions without prompting.) But “find an advanced topic and figure something out that interests you” always bombs.

and for replication: yes. I am suggesting that I need some incomplete version of a linear algebra book, and we will prove things and solve problems to extend it.

I don’t really know how to do that at the moment for linear algebra. For Euclidean Geometry I am using the obvious choice. For Differential Geometry, I would have to go back to something like Monge, Darboux and Gauss.

I find this exhilarating and terrifying. Linear algebra is not scheduled until spring, so I have some time to work it out.