Here is a short report on the big experiment for this term, and a related note on a realization from today with wider applicability. I expect that this will start well, and then ramble on as I fiddle with some ideas.
I am taking part in a “Talking Teaching and Learning” group, and my homework this week was to think about the last few details of my new assessment structure for Euclidean Geometry. In particular, how will I handle the “regular, daily feedback” part of the process?
So, if I am to provide regular feedback to my students at “assessment opportunities” they take, how shall I do it? I want this to be meaningful and effective. And it would be nice if it didn’t consume my working time.
I think I will try a format beloved by politicians: I will ask and answer my own questions. If any of you wishes to play investigative journalist and ask other questions that I should be forced to answer, go hit the comment box. I would like to play.
What counts as an assessment opportunity?
Any student presentation, meaningful engagement in class discussion, a discussion with me outside of class where I learn something. Those things count as opportunities for me to assess student performance that don’t necessarily have written feedback attached to them. In each case there is plenty of verbal feedback from classmates–but I don’t always participate. In fact, I prefer to leave it to the students.
Why are written papers not on this list?
Students will get feedback in the form of a referee report on each paper. I am not as concerned about providing more structured feedback here because I feel it is adequately covered.
Why do you prefer to leave the process of verbal feedback to the students?
One of the skills I am trying to encourage is the ability to evaluate arguments critically and thoughtfully.
If there is a reason to leave the verbal feedback to the students, might written feedback from the instructor corrupt this process?
Oh, yes. That is my main worry.
How can you avoid this trouble?
um. uh. [blink. blink.]
I hope this writing will spur me to some ideas about that…
What are the goals for this written feedback?
I want to focus student attention on some aspects of what they did. Ideally, quality feedback should help speed up a student’s process of improvement by directing his or her attention to something concrete.
What kind of constraints are you going to impose upon yourself?
I am a constructivist at heart. The student must come to grips with the material and how to do it. Each one should do this on his or her own terms. One idea would be to give feedback by asking questions.
I am just not sure what kinds of questions I would ask that are detached from the process of running a class meeting. We handle lots of things in class, and I almost always do it by asking questions. Maybe I will just reiterate some of the unanswered questions. That doesn’t feel like a very good answer.
Another idea is to use a sandwich approach: mention something positive, make a suggestion for improvement, reiterate the positive outcomes. And be relentlessly optimistic.
Now I’ve run out of questions. So.
I think whatever I do will have to play to my strengths. I am at my best when I split my time as a cheerleader, mentor, and coach. Students are capable of amazing things, and sometimes they just need for me to believe in them and expect it out of them. Sometimes they just need a little bit of commiserating about how frustrating it is to do mathematics. Sometimes they need a concrete suggestion of what to do when they are stuck and at their wit’s end.
That was unsatisfying.
Here we are. 500+ words in, and no answers I feel wonderful about.
Never mind, time for some unbridled confidence.
When I got into IBL teaching, I recognized that a major asset I had was hubris. I just believed that I could do this. Usually it works.
What? So I will have to help each student in as individual a way as possible, thinking on
my feet and being careful about everyone’s feelings? Why should I worry? I can do that.
I’ll think about this some more, and just try to roll with it.
I don’t feel like I finished my homework.
…continuing from the previous post. Here are some other things I am thinking about.
- It is good to have a crowd. That is, it is good to have a group of like-minded people to lean upon. I missed that the last year or so.
- I need to refocus on intentionality in my teaching. I have made many sloppy, quick decisions lately. I want to be sure that the choices I make are being made for reasons.
- The trouble of finding a decent assessment method is killing me. Until I find some sort of partial resolution to this, I will not make any other progress as an educator.
- I want to refocus on mentoring students personally. I have gotten away from this. It has always been the case that students express fear about talking with me. I don’t quite know where this comes from, but it has always been so. In the last few semesters I haven’t done enough to counteract this. Students aren’t coming to visit me and discuss things one-on-one. This hampers my ability to help them.
- I need to sell what I am doing. I am not doing that enough.
With these in mind, I wrote some things I might consider saying to a hypothetical student.
Embrace Challenge. I am here to make sure you get stuck and help you get past it. This is the only way I know to foster true intellectual growth.
Math is hard. I am just laying that bare instead of hiding that behind slick lectures.
My intention is to give you many opportunities for growth and learning. I will try to put you in a situation where success is possible. But I will not put you in situations where success is guaranteed. There is no growth, no learning, no empowerment, in finishing a task you are sure of completing before you begin. So you will often feel unsure of how to succeed, at least at first, and maybe for a lot longer than you are accustomed. I am not going to guarantee your success, not on the small scale of a single task, nor on the larger scale of this whole course. But I am confident in your success. I believe deep in my bones that each student can eventually prosper here. My job is to provide you with guidance so you can grow and succeed in ways you were not capable of before.
I wrote a while back about attending the latest Legacy of RL Moore conference in Austin, Texas. On one of my plane trips after the conference, I did a little reflective writing about the event… on a pad of paper! Now that is just not fair to you, or to history. Really, if what I say wasn’t so important for everyone, would I have been given this blog? So, I will write out that reflection again, and we’ll see if I still believe it.
Ed Parker’s Opening Address
This was a good meeting for me. I needed the chance to reconnect with some basic principles, and get back to thinking about basic issues of how to be an effective IBL instructor. I think I might have gotten a bit lazy, and hence less effective, in the last two years. I finally felt comfortable enough not to think about my instruction every free moment, and I let it go a little too much.
The opening talk was by Ed Parker. Ed ran the first New Users Workshop I attended in summer 2008. He was also kind enough to answer emails when I was getting started. In short, Ed is awesome, and I was very interested in what he might have to say.
The big points that Ed discussed were these:
- An appropriate level of rigor
- Assessment Opportunities
- Possible Social Challenges
The first two stick with me the most.
What does it mean to choose an appropriate level of rigor for a course? Well, Moore Method courses are about making sound arguments and defending them. You have to decide what will count. Ed’s point was that you have to take into account your “input audience.” This doesn’t mean you give up doing mathematics, but it should influence which parts of the subject you use as a playground. And I didn’t take that to mean exclusively which material you work with. Instead, the lesson I take away is that you can be careful about what part of the process of doing mathematics you use as a focus.
As an example, Ed mentioned that at the “intro to argument” phase of development, some of his problems take the form of correctly parsing a definition, and some of them are explicitly about separating semantics from grammar. These are skills that he finds his students lack, and so he asks students to engage with these things directly.
The second point about assessment is something I have been thinking about a lot lately. In fact, Ed provoked me into a thoughtful mood, and I don’t much recall anything about his third point. First, the fact that he framed the discussion by choosing to discuss “assessment opportunities” really sat well with me. That is how I think of my course, too. Each class meeting is full of students taking the opportunity to be assessed. When they don’t take these opportunities, I can’t really help them. I think I will be more explicit about this with my students and use this kind of language when we have “dead air” time in class.
I wrote down two main points from this portion of the talk.
- Model correct use of language, and praise any correct, or near-correct, use of language.
- Give outlandish rewards for solving problems.
The second scheme is meant to counteract the psychological difficulty students encounter when participating in an IBL course. Ed shared a model for grading, which I will restate in my own terms.
- C — earned by displaying understanding of the course material.
- B — for such a display and solving some problems
- A — for the above and “awesomeness”
I generally like this scheme. I don’t know how I will adopt it, but I like it all the same.
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.
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.
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.
Originally Posted 05-22-2012
I had taught dynamical systems once before, in the spring of my first year at UNI. Then I used a textbook (Alligood, Sauer and Yorke) which turned out to be too advanced for my students, and I asked them to do too much. The class was lecture/homework with a big final project that involved reading a scientific paper from when Chaos was just being recognized everywhere. Mostly, the class was too hard, but the project bit went pretty well.
This time was going to be different! I planned an IBL approach and I wrote my own notes. Also, as an experiment, I went with a structure described to me at an IBL workshop at UT Austin run by Mike Starbird and Carol Schumacher: each meeting the class was responsible for a small snippet of tasks (usually 4 or 5 of them), and we would spend the first 25–30 minutes of class working in groups to solidify our ideas and refine arguments, and then discuss/present to the whole class.
I won’t try that particular structure again. It didn’t work for me. That is, students didn’t seem to take the personal responsibility to get all of the work done each meeting. Many used the group meetings as a way to avoid coming to grips with the material on their own. Etc, etc. All of this made it hard to evaluate some of my students, as they really did their best to hide and nothing in my written expectations said they couldn’t. In retrospect, one thing I changed from the structure as it was described to me likely made a big difference: I did not require them to turn in written work every class period for evaluation. I don’t have a graduate TA for this type of class, so the workload wasn’t feasible for me. But I see now that this might be a key part of making that particular IBL structure work.
Also, I was trying to get them to use Sage for the numerical work required. Basic dynamical systems has a nice experimental feel, where you can get the computer to do some basic things for you and then poke around the data looking for patterns. What I learned is that even fairly advanced mathematics students who otherwise seem comfortable with technology will balk at having to do something that requires a simple “for” loop. I’ll have to write code snippets for them next time. I had a few computer science double majors, and they got the class limping through that portion of the course. But really, that is when things hit a wall. The class never really recovered. I realize that I have to work on how to teach how to use the computer to do mathematical investigation. Computers are an incredibly powerful tool, but only if you have some basic way of using them. (Like Sage/Python confidence.)
In the end, we did not get through all that I think we could have, but we did “enough”. I certainly want to do more next time.
What will change?
First Item: More structure to the learning of how to use the computer.
Second Item: Pre-written scripts to hand out so that the computer is not the obstacle.
Third Item: Back to a structure I am more comfortable with–one that involves more personal responsibility on the part of the students.
Originally Posted 05-21-2012
More Reflections: Linear Algebra
Another new class for me this past semester was linear algebra. At UNI this is a Sophomore level course, it is titled “Linear Algebra and Applications” but is sometimes taught as a transition to proof course. Basically, this course is written into our curriculum so vaguely that it can be whatever you want it to be. We have had some conversations about this lately, and I hope that we come to a more focused common understanding of what should be happening in linear algebra.
I tried to run a blended environment with lots of features. I think I got a bit of a Frankenstein’s monster problem as a result.
- Do at least a few serious applications of linear algebra (Google’s PageRank algorithm, curve fitting)
- cover some basics of the geometry of lines and planes in the Euclidean plane and in Euclidean 3-space for intuition
- Discuss matrix algebra as an entry point to modern algebra ideas
- get into the nitty gritty of solving systems enough to “see” the proofs of most of the major theorems (rank-nullity, invertible matrix theorem, etc)
- See linear transformations as a way of organizing the work of linear algebra, and use them as a way of expanding the idea of a function
- introdue the mathematical software package Sage to help handle visualizations and tedious computations. I am a pretty novice user of this software, but I am enthusiastic.
And that is just the list of things I come up with off the top of my head. I am sure I was trying to do a couple of other things at different points of the semester.
To make things more interesting, I also chose to use the open source textbook by Woodruff and Grout. Jason Grout is a Sage developer and helped get me involved in a grant project about using Sage. His book has some nice features, but it is still a work in progress, and he was a week or so behind me in class days.
I didn’t exactly set myself up for a roaring success.
What I Learned
Students are terrible with technology. Sure they can use their phones, but only because they care about that. They are absolutely inept at watching things like syntax and grammar for interactions with Sage (or anything similar).
So prepare for the least technologically inclined student you can imagine. Then go back and prepare for one who makes that student look like Guido van Rossum and Linus Torvalds put together.
If you expect the students to use the software, you have to force them to do it. Make the homework assignments only available as Sage worksheets, for example. Even then some will try to avoid it.
Make sure that the technology instruction is in very small bits. Learn one command or process at a time. Do one each day. Again, force them to become literate by immersion, but do it very gently and very slowly.
You have to stay true to your own strengths. I am sure that the Woodruff-Grout book is wonderful for them. It is their baby. It was fine for me, but it always made me uncomfortable. It just wasn’t my perspective. I haven’t used a textbook in a long time, and now it chafes.
What Will I do Next Time?
I am teaching linear algebra again in the fall. Here is what I will do differently:
- Write my own IBL notes
- distribute those notes as Sage worksheets
- Add explicit instruction on proper Sage usage for linear algebra exploration. A little bit at a time. No, I mean really little bits.
More importantly, my own IBL notes won’t be just a set of tasks. I’ve been thinking about what makes my Eucliean Geometry Course so successful, and one feature is this: I use Euclid’s Elements as a text. That is, students have a reference work, and then I structure problems around it. My task sequence is about coming to grips with the ideas in Euclid by doing related things. I bet I can make something like that work for linear algebra, too. It means basically writing my own supplement to a Schaumm’s outline. I get to propose the interesting tasks with real depth. The routine stuff is in the book. My job is to blend them together. This is going to take a lot of work.
I think the switch to a more pure IBL structure is the only way I can be true to myself and have a successful result. The more I talk in class, the worse things get. All that ever happens then is that I prove that I can talk intelligently about linear algebra. That is not really the point of class, is it?
So, what will I write? Well, I have ordered the latest Schaumm’s outline for the official textbook. I think I will also transcribe my own linear algebra notes (from a class offered at Williams College in 2006–2007) into my own book. And as of right now I am on the lookout for interesting linear algebra problems. I think I might spend some time learning some more functional analysis to find interesting ideas for challenges.
Full Steam Ahead!