Talking to Reflect and Learn: Major Progress

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:

  1. developing an assessment method I am happy with using (focus on Math 3600 Euclidean Geometry)
  2. learning to teach Math 2500 Linear Algebra
  3. 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:

  1. I want some examples of correct proofs for students to see
  2. I want students to experience reading mathematical literature for understanding
  3. I want to be able to assume something or this class will never get anywhere
  4. I want to infuse the course with a sense of connection to history and wider mathematical culture
  5. I want the students to learn to critique everything, even Euclid (some of his arguments are wonky)
  6. 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.

The Big Unteaching Experiment: Rebooting the Machine

OK. Sorry for the confusion earlier today folks. I accidentally posted things meant for my classroom blog to this one. And they got automatically tweeted out. Awesome. So, now you know that I keep a simple blog for my students each semester. That way we all have a running account of what happens each day. I find it especially useful for my Moore Method geometry course, but it serves well if a student might happen to miss a meeting.

We are now past the half-way point in the semester. In fact, spring break starts right about…now. I have been worried for some time about the state of my experimental differential geometry course, and today I discussed with my students how we will go forward in a manner different from the first half of the term.

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The Big “Unteaching” Experiment First Major Report

My differential geometry course has been an interesting experiment so far. I decided that my aim for the course would be to put the students in a position to practice the skills to handle this situation:

You have a subject with a lot of literature. How do you learn it? How do you structure your own approach to the material? How can you be sure you have learned something?

This involves handing over the reins to the students completely. “Here is differential geometry of curves and surfaces. Go learn it.” To be fair, I knew it wouldn’t really work that way. So here is what I planned before the term began:

UNIT ONE: Differential Geometry of Curves in Space

  1. Assignment One: spend a week trying to get a big picture view. Make an outline of what seems to be essential, core material. Maybe note a few advanced topics for later study. At the deadline, we negotiated a common class syllabus.
  2. Assignment Two: Spend a month learning the basics. To enforce some standards, I asked them to produce the following things:
    1. a “Book Chapter” write up of the mathematics on our basic syllabus,
    2. a selection of “standard exercises” and their solutions to complement this chapter,
    3. a “demonstration piece”–this is something visually oriented that highlights some concept from the basic syllabus, and
    4. a proposal for an advanced topic to be worked out in the next three weeks.
  3. Assignment Three: complete the agreed upon advanced topic and share work with the class.

UNIT TWO: Differential Geometry of Surfaces in Space

…repeat all of the above for surfaces.

I knew this was a challenging thing to ask of my students, but I still wanted to do it. To help them a little, there was a time line for rough drafts of the pieces of this big project. no student met any of those. So it wasn’t too big a surprise when the final due date came along on Friday that things were in a sorry state. No amount of cajoling could convince them.

Attendance was terrible. When they had questions, we had wonderful discussions. But I don’t get the sense that they were making a reasonable schedule for themselves to keep pace.

So, things have to change. We will muddle through the first unit. I have given comments on the half completed projects, and they are due (again) on Friday. I hope this makes a difference.

In the second unit, I will structure things even more tightly. I hope that this helps them see how to pull this all off. If anyone has advice for what to tweak, I am all ears.

Early Semester Concern: Classroom Culture

Tomorrow will be the fifth meeting for each of my classes. At this point in the semester my only concern is setting up the right classroom culture.

Classroom Culture for IBL Courses

Inquiry-based learning environments can be wonderful because they are active, challenging places to spend time. The goal is to have students on the case at every moment. This means that they will be focused on what they don’t understand and why, and actively trying to shorten the list of things they don’t, yet, get.

Some fraction of the students relish this. They come to you brimming with a cautious confidence, a willingness to get involved, and some measure of ability to reflect on their own learning. These students make going to class easy. I suspect that such students will succeed no matter what kind of course they find themselves in.

But, at least where I work, most students are not in such a wonderful place for mathematics when I meet them. This fruitful attitude has to be demonstrated, encouraged, and sometimes preached. No matter how hard you sell this emotionally open way of working, it has no chance if you don’t set up the right atmosphere. Your classroom has to be a safe and supportive place before many students will take a risk in front of their peers.

All of this goes down in the first three weeks of class. It seems that after that the basic classroom culture is set and you have to live with it for the rest of the term. So I work very hard in the first few meetings to bend things in the right direction.

Today I find myself reflecting on my “culture setting progress” for my three courses. I have about four meetings left before my window starts to narrow.

Differential Geometry

I think this is not going to be a problem. I have four graduate students, two seniors, and two juniors. I worry most about the juniors, but they have each had two previous classes with me. Things are going to be fine, I’ll just keep an eye out for them.

Math in Decision Making

We had a stellar first meeting. Our conversation was fun and engaging. The students started figuring stuff out, and the brave souls who were my first presenters handled themselves well. At the end of the day I thought I might have already sealed the deal on a good class culture.

At the next meeting, I had seven new students. Apparently a colleagues class was under-enrolled, so it got cancelled and most of the affected students switched to my section. Seven out of thirty is a rather large portion, so we have to start over. Things are going slowly now, but I have lots of hope. The typical student in this course is “mathematically bruised,” so I am treading pretty carefully. Time will tell.

Euclidean Geometry

This is going pretty well. I have taught this class so many times that I can almost do it with my eyes closed. The thing to watch out for this term is that the class is larger than usual (25 students). I have to work a little harder to get everyone involved.

One Other Thing

It is time I learned all of my students’ names. I am really terrible at this in general, so tomorrow I will resort to taking photos of my students and making a seating chart that matches the photos. Then I will make myself a screensaver out of all the pictures with names written under all of the smiling faces. A few days of studying and quizzes usually gets me most of the way there.

Opening Day: Spring Semester Begins

Yesterday was the first day of the spring semester. (At this point of the year, it feels awkward to call it the spring semester.) I had a wonderful week at the Joint Mathematics Meetings last week. I got home on Saturday pretty late and very tired. Sunday, I lounged. Seriously. I didn’t do any work at all. I slept in a little. Then I took a nap with my preschooler cause she needed a cuddle. Before I knew it, it was 3pm. So, it was a good thing that I had taught each of my courses before and had spent serious time getting mentally ready for them over the break. I didn’t have every piece of paper in place by Monday morning (I still don’t), but it was not a disaster. So, how did it go?

Differential Geometry

Regular readers (hi, mom!) know that I have been thinking a lot about this, and that I have an unconventional plan. I have decided to refer to this structure as “unteaching” a course. I explained the basic idea and the reason for it, and described the first assignment. Not ten minutes had passed. Then things got a little awkward. I think the students were just so surprised that I didn’t have a whole fifty minutes worth of activity planned for them, that they didn’t know what to do. Most set themselves the task of starting to look through the books I brought with me. I still think this will work out, but I wonder what is going to happen tomorrow.

Euclidean Geometry

I followed my “usual plan” for the first day, and things went well. (I am on iteration nine of this particular course, so I have some things figured out.) This is a Moore Method course, so we practiced the structure of presenting and critiquing. This went well. We got a theorem. I explained all of the basic structure and rules (and a little of why I chose them) and then sent them off to do some mathematics. I am a little concerned that there are so many students–25 is a lot for a Moore Method structure. Not too many, just a lot.

Math in Decision Making

This is a IBL course, done as a “group work modified Moore method.” I only have 25 or so students and it is a relief. Apparently the last minute check on placement test scores forced six students out of the course. This is inconvenient for those students, but convenient for the rest of us. I seem to have a good group. They dealt well with being asked to participate, and we had a good, light-hearted conversation about the mathematics. It will take time to set out the right classroom culture, but yesterday was a good first step. I have decided to have semi-regular homework so I can give them feedback on their writing and they can practice on some of the more important ideas outside of class. I think I will do homework about once a week, and it will be short and focused.

All in all, a good first day. Now I have to go finish polishing things for day two.

My Classroom Blog

My Classroom Blog

This semester I am keeping another blog. (I know, terribly unfaithful of me.) The one you are reading right now is for “reflective practice writing.” The one I just linked to is for communicating with my students. It will have links to class documents, assignments, short recaps of what happens in meetings…

This semester I have three courses that will use some sort of IBL, and so both blogs should be quite busy.

Examples for Differential Geometry

So, I must be some sort of ancient fuddy-duddy. It is true that I went to college before the internet was really a pervasive thing like it is today, but there is not excuse for not executing some simple searches of likely places.

I was goofing off and looking through my Google+ stream, and I came across this nice link in a post by Alexander Kruel:

Of course, on it there is a link to their list of surfaces:

Now I just have to sift through and pick some examples to show off to the students. I have a few to add of my own that don’t appear on those lists exactly. Hey, that gives me an idea!

Students could improve some wikipedia pages as a way of sharing their work. I have never done that. Has anyone tried that? Is it plausible as an assessment?