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
- 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.