A Big Class

Update on that big IBL class:

A little over a week ago, I posted a plea for help on Google+, and a note here. I will soon be running a mathematics for liberal arts students (“math for those who might not wish to be there”) course. In the past I have run the class as an IBL experience, using group work heavily. This was working at an acceptable level for sections of 30-40 students. This semester I have 68 enrolled. And my friend Doug Shaw has two sections of 72 each.
After reading and thinking it through, I will take the combined advice of Bret Benesh, Robert Talbert and Vincent Knight. I can’t quite count on my audience to be as self-directed as Vince’s, but I am happy to stay within the family of student-centered, active, social-constructivist teaching techniques and use a form of peer instruction/guided practice. (Is that your term Robert? Or did you borrow it?)

Poll Everywhere

As a practical matter, I will be using www.polleverywhere.com as a student response system to help run classes. UNI has a site license which will make it possible to use polls with more than 40 respondents. The advantage of PollEverywhere is that it allows the use of any web enabled device or any cell phone with a text messaging plan to post a response. That will bring the number of students who don’t already have a useful piece of technology down near zero. I hope it is zero. I am working on a back-up plan in case the number is not exactly zero.

The downside to Poll Everywhere is that questions are only really allowed to be short strings of text. At least, that is what fits in their web app naturally. I can imagine times I want to ask questions based on a picture or a graph. fortunately, they allow you to embed a poll into any web page by generating a little snippet of javascript. I will be investigating this tomorrow to see if it is useable without destroying all of my prep time.

Other Materials

The other big hiccup is that I was planning on using an IBL script. This isn’t appropriate for my new course structure. But it is far too late to order a textbook as a reference. So it looks like I will be writing a different style of course notes this term. I think I want to keep the “discovery” feel. (I doubt I can get all the way to “inquiry” with this many students.) So, I shall be looking through the materials on the Discovering the Art of Mathematics site for inspiration, but not outright plagiarism.

When I get moving, these materials will start showing up in my github repository for course notes. Feel free to follow along.

At the moment, I still plan to discuss Cantor’s theory of the infinite, something significant about probability and statistics, and something topological. I usually lead a unit on classifying surfaces, but I might switch that up for something about knots or tangles. Frankly, anything past Monday feels so far away, I am unqualified to talk about it.

Here goes nothing.

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.

Book Review: Pressley’s Elementary Differential Geometry, 2nd Ed

Pressley's Elementary Differential Geometry

Pressley’s Elementary Differential Geometry


It is time to return to the book reviews! Our next book is Elementary Differential Geometry, 2nd Ed by Andrew Pressley. This is a pretty recent text. The first edition is from 2002, with the update published in 2010. The book has an attractive price point from Springer, and you can get it from Amazon.com for even cheaper.

Pressley’s desired approach is to make the subject as accessible as possible. In the preface, he writes:

Thus, for virtually all of the book, the only prerequisites are a good working knowledge of Calculus (including partial differentiation), Vectors and Linear Algebra (including matrices and determinants).

The tone of the writing bears this out, as does the author’s care to explain basic material. This text is definitely aimed at the modern student, and it conforms to the standard expectations for what a recent textbook on an advanced subject should look like.

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Book Review: Struik’s Lectures on Classical Differential Geometry

Dirk Struik’s Lectures on Classical Differential Geometry

Next up: Lectures on Classical Differential Geometry, 2nd Ed by Dirk J. Struik. This is an oldie and a goodie. The text is currently available from Dover Publications, and you can pick it up from Amazon.com, too.

The original version of this book was published in 1950, and some material was added for a second edition in the 1980’s. This is a truly wonderful little book, but I can’t recommend it as a text for today’s students.

I bought my copy back in 1994 when I took an undergraduate differential geometry course at Ohio State from Neil Falkner. I felt like I couldn’t really absorb the book then, and I was one of the more successful students in that particular course. I am pretty sure that means it is an inappropriate choice for the average undergraduate. Looking at it again, I can’t see what caused me so much trouble, except that the language is old-fashioned. This add an extra layer between the student and the material.

I also know that I learned most by going to lecture and that I didn’t really study my textbooks. (I hadn’t learned how to do that, yet.) So maybe more of this is my fault than Struik’s, but I bet lots of students don’t really study their textbooks effectively. Absent taking time for direct instruction on how to read a math book, it falls on the instructor to pick something easier to digest.

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Book Review: Banchoff and Lovett’s Differential Geometry of Curves and Surfaces

Banchoff and Lovett’s Differential Geometry of Curves and Surfaces

So far, all of the books I have reviewed have been old ones. That is, the “newest” book was do Carmo, from the 1970’s. Now it is time for a recent book, Differential Geometry of Curves and Surfaces by Thomas Banchoff and Stephen Lovett (2010). The book is published by A K Peters, Ltd. which is an imprint of CRC Press. The text is also available from Amazon.com. The Amazon.com page also gives you the option to buy electronic versions!

Being a more recent book, several features of more modern textbooks are readily evident: the now-standard Definition-Theorem-Proof style, many examples and exercises, and some Java applets available from a publisher-hosted web site. That same page has a link to the list of errata.

This book appears to be written for a current undergraduate audience, and it comes the closest to meeting my idea of the right set of prerequisites: multivariable calculus and linear algebra, and maybe some rudiments of ordinary differential equations. If you are choosing a new text, you really should have a look at this one.

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Thanksgiving Book Review: do Carmo’s Differential Geometry of Curves and Surfaces

do Carmo’s Differential Geometry of Curves and Surfaces

It is time to look at a classic: Manfredo do Carmo’s Differential Geometry of Curves and Surfaces. This text is very popular with professional mathematicians, and it has been a standard for a long time. I know many people who were taught from it as undergraduates.
The course I took did not use this, but I used do Carmo’s Riemannian Geometry as a text during my first year of graduate studies. (The two books are sometimes referred to as “baby do Carmo” and “grown up do Carmo” when it is not otherwise clear which one is being discussed.)

The text is published by Prentice Hall, which these days is an imprint of Pearson. You can also find it through Amazon.com. A simple Google search also yields a lovingly compiled list of errata.

Because I used this books brother at a formative stage of my mathematical training, reading it is very comfortable. For me, it is a bit like going home for Thanksgiving.

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Book Review: O’Neill’s Elementary Differential Geometry, 2nd Ed

Barrett O’Neill’s Elementary Differential Geometry, 2nd Ed

The next book in the stack is Barrett O’Neill’s Elementary Differential Geometry, Revised 2nd Ed. I read a portion of this book in the car today when it was not my turn to drive. Eastern Iowa and western Illinois are visually uninteresting this time of year, so the only distractions were from the kids in the back of the minivan.

This book is published by Academic Press, which is an imprint of Elsevier. Given the ongoing boycott of Elsevier products, I am reluctant to give this a recommendation. If you care about the state of journal publishing (and you should), then you have to think about how much you care about this and whether you want to extend your boycott to products besides journals.

None of that really has much to do with the quality of the book under review, so I’ll soldier on. Oh, you can also get a copy from Amazon.com.

This book is the closest thing I have found to an appropriate text so far. The only concern is the choice to use Cartan’s method of moving frames throughout.

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