I spent the better part of my free time on Friday afternoon thinking through what has been going wrong in my linear algebra class this term. I had scattered notes in my little class journal with hints. Things like:

- “Too short. Needs a second act” (end of first day)
- “I’ve been stuffing too much into each day.” & “Not much linear algebra happened today.” (end of day three)
- “Need to tighten terms and their introduction.” (day four–Friday)

I had the good fortune to be at the Joint Mathematics Meetings over last weekend, and I got to talk a little with Ed Parker, who has been a mentor for me in IBL teaching. We didn’t have a long time to talk, but he reminded me of a really important concept to keep in mind when taking an IBL (Moore Method) mindset “down” to a course which is not proof-driven: **appropriate level of rigor**. I’ve heard him mention this before, but I really needed to face it now, and so it helped me a lot to hear it again.

At UNI, our linear algebra course is not a place where we expect students to write proofs. For most mathematics students it comes in the first term of the sophomore year, between Calc II and Calc III. And there are many students from majors other than mathematics. (I think I have plenty of scientists, and even a Music Ed major.) The department has decided to take out any expectation of proof-writing in this course, and instead focus on getting students to learn something about how a computer can help a working mathematician. So, if I am going to avoid introducing formal proof writing, how do I focus my IBL questions? What will count as a valid answer? Ed’s advice was that in such a course, he would count as “convincing” any explanation that a student could give, and then adapt if all the numbers were changed. (I think I have that right. I am going from memory here, and that is not exactly how he said it.) The point being that a student needs to find a principle behind her solution, and be able to use it and explain it flexibly. I like this a lot. That is a reasonable way to demonstrate understanding at the “pre-rigorous stage.” (I am stealing that phrase from Terry Tao.)

I feel pretty good about that. But what kind of questions should I ask? I have some conception of what kind of thing would be an acceptable answer, but what type of question will help focus students on a deep understanding of the material? It turns out I had an answer to that before the semester started, but somehow I never managed to do it. I was all focused on figuring out some sort of inverted classroom structure, but I was mangling the actual details of getting students working.

Now, for the few of you still reading (Hi Bret! Hi Dana! Maybe Vince? No. three readers is optimistic. I won’t pretend that there are any more.), here is the bit of reflective writing I did on Friday afternoon. It picks up right after the end of the comments I had above from day four.

No. It’s worse than that. I have some serious revision to do.

Things not working:

- Students can read the text and gain understanding
- Students work at interesting tasks that deepen their understanding.
- I support student growth by watching their zone of proximal development carefully.
- Students experience a balance between frustration and success.
- Work aims at the heart of our learning objectives.

Learning Objectives: :Students will demonstrate

- Increased use of precise language
- increased use of abstraction (as a verb)
- ability to create examples and non-examples of various phenomena — my appropriate level of rigor!!
- ability to choose when to use a computer to do a computation or make a visualization
- Content Goals:
- the three pictures: row, column, transformation
- Strang’s four subspaces
- Strang’s four central problems:
- Linear systems
- least squares
- eigenvalues
- singular values

And then I made a new plan, which goes like this:

For each class, students will

- read a section of Strang (and possibly Hefferon, too.) This part doesn’t change. except for Monday we are starting over.
- Do 10-12 exercises that I set. These will almost all be “create an example of [[something]], or explain why such an example is impossible.”
- Then we will take our class time to do IBL-style classroom presentations.

I will have to rejigger my syllabus and my grading scheme a bit. But *c’est la vie*. I feel better about this approach, and myself, already. I wrote the first assignment and sent it out yesterday. Everything will be in Sage .sagews format (so that students have to interact with the SageMathCloud–my chosen software package). You can go follow along on github if you like. If you are into github, you can find everything really easily by yourself, so I won’t post a link.

Back to work.

How does this relate to the Wiggins authentic assessment post you put on Google Plus today?

Also, your comment from Day 3 would sound ridiculous to me if I didn’t have any teaching experience. It is weird that we can do both too much and not enough at the same time.

Now I need to figure out GitHub. . .

The relationship? Figuring out “authentic assessment” is pretty close to “getting the right problem sequence” when you teach a Modified Moore Method sort of course, I think.

And, yes. I was pretty down on the whole of day three. I thought I could turn it around with a good day four. But that cratered worse. Or maybe we just hit the bottom of the hill that day. As for github, I’ll send you a message.

I really appreciate your ability to carefully reflect on your teaching. Ed Parker is one of my heroes. Also, we now have confirmation that at least two people read your post. Vince?

Yeah. The shout-outs are a cheap way to gin up engagement… But thanks.

I’m following as well 🙂

This semester I’m trying something new in Graph Theory and plan to write about it. I give the students five or so terms that I want the students to define. They need to give a precise definition, a lay-person definition, and give examples (and non-examples) of the definition. This happens before class and we can fine-tune understanding in class…

I would like to hear about that. As part of my euclidean Geometry course I ask students to come up with definitions of some terms. The students find it challenging, but it always gets them excited. It is a pretty good “leveller,” since anyone can participate at the level of checking out examples, but writing a rock solid definition is not so easy.