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.

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?

Things I Need to Run Differential Geometry

Time to get down to brass tacks. What things do I want to have in hand at the first meeting of my differential geometry course this semester? What things must I create before I meet the students so that everything can run in an orderly fashion? (Given the course structure, I am using the word “orderly” in a rather loose sense, here.)

  • An Official Syllabus: this is the document required by state law. It has to have certain things on it.

  • An IT structure for dealing with recording student achievement and demonstrations of competence. (Somewhere between “scattered pieces of paper” and “My own web-app grade book database with customized reporting options.”)

  • The first assignment: this is the one that gets students rolling towards making an overview and choosing their learning goals.

  • My own outline of how to handle phase one of the course: the theory of curves in the plane and in space.

  • A set of ideas for projects (Josh Bowman helps me here)

  • A list of examples to fill out my “example a day” plan.

  • A tightly controlled structure for a warm-up project to help students get over the disorienting freedom I will give them. (With great freedom comes great responsibility, or something like that.) (Patrick Honner helps me here.)

  • An annotated bibliography to share as a guide to the available literature students might use to get started. This will lean on my book review project, which is languishing right now. I will do a few more in the next week.

  • Something to guide my students through the “advanced calculus gap.” I don’t need real analysis, I don’t need full-blown differential topology, but I do need more than the standard multivariable calculus and linear algebra courses require. Spivak’s Calculus on Manifolds is probably aiming too high. I have a copy of Shifrin’s Multivariable Mathematics, so I will look through that. I might have to make my own thing here. I hope not.

  • Speaking of Shifrin, I need to arrange the latest of edition of his undergraduate diff geom book as a course pack for the students.

  • Some way of carting my stack of books to class each day. Seriously. I need to figure out a mechanism for getting my undergraduate level differential geometry library from my office down to the classroom. I need a cart, or a bunch of sturdy square bags or boxes with handles.

  • A killer opening. I want to grab attention with the first twenty minutes. This stuff is seriously cool, and I want to build a little excitement for the material.

  • Some serious spine-stiffening. I have to present this craziness, and I have to do it in a self-assured and encouraging manner.

Have I missed anything? I think I can have half of these squared away by the end of the day on Friday, with good progress on the others.

Objectives for Differential Geometry

Regular readers (all three of you) are aware that

  1. I am scheduled to teach differential geometry this coming term.
  2. I am very excited about this.
A minimal surface

a minimal surface

Less well known is that I am also terrified of this experience. I taught the course in the Spring of 2010 and the experience was not all warm and fuzzy. The biggest problem was the disconnect between what my students knew coming in, and what I thought they would know. Suffice it to say that I have a much better understanding of what it means to have a student who passed linear algebra and multivariable calculus at UNI.

Let’s take it as given that this iteration of the course will also contain surprises, but I hope they will be much smaller ones. And I am at a place now as an instructor where I am much more alert for this kind of trouble, so that can only help. (Oh, please, please let that be a true statement.)

So, now I have spent a fair amount of time in the last two weeks worrying about how this course will all work out when I should have been properly enjoying time with my family. I started by trying to write a set of learning goals for the students.

Continue reading

Project Based Differential Geometry

Next semester I am assigned to teach some of my favorite material: differential geometry. This is only the second time I have this assignment, so I still have a lot to do to think through the issues and plan a course of study.

I expect to have a class full of well-motivated and interested students. so this gives me the opportunity to do something fun: a project based class! What does that mean? I think it means the following things: I will give the students a clear list of goals to meet, and negotiate with them over projects they can work on to display their mastery of the material. There will have to be some structure to ensure that all of the students meet goals on core material. There will have to be some idea of “timed check-ups” to make sure that students are on a reasonable pace. (No course can be mastered all in the last two weeks.)

I will assign a textbook to make sure that the students all have one decent reference (and have it in common). Last time I used a “not exactly published’’ book by Ted Shifrin from the University of Georgia. I liked the text, so I will recommend it again. I won’t use it, though. I don’t intend to lecture unless students ask for it. I have a long list of other books to give them, and I think I will put a host of them on course reserve at the UNI library, too. (This will be the first time I have done that.) My list of suggested reading will need a bit of an update, too, as there have been a few new books at the undergraduate level in the last few years. I might have to write to some publishers to build a new list.

Another thing I would like to do is to give “an example a day.” I love the subject for all of its generality, but it is full of beautiful examples. I want the students to see lots of them. I am not sure if I will give all of these, or if I will assign them to students. Maybe I will start and then pass them off.

What I need to prepare:

  1. an updated copy of Shiffrin’s text to sell as a course pack
  2. an updated reference list
  3. a list of content outcomes for the students to demonstrate.
  4. Some real ideas for structure. You know, something I can explain to my students and defend at a disciplinary hearing.
  5. BIG DEAL: a sense of what the course is really about.

I’m not sure about that last one. I need it bad. I know what the material is about, of course, but I need to figure out what skills I am trying to teach. I am starting to think I am trying to teach the process of structuring your own learning.

Items 3 and 4 are intertwined. I know that I can split the semester into roughly two parts: curves and surfaces. And in each part I must develop a list of content objectives that I consider necessary, and longer list of “optional’’ projects where students can demonstrate deeper understanding of more challenging material.

Well, more when I figure it out.