Today I had a really interesting conversation with my Euclidean Geometry students. It was a new wrinkle on a thing I had planned for, and it shows how you can still be surprised by an IBL class even when you have run the same notes more than a dozen times.

# Differential Equations Progress

I have made a lot of progress in planning differential equations for this fall. I still have a lot of work to do, but I think I have made all of the big decisions. I finished the syllabus and turned it in to the department, so all of those decisions are “final.” Run-down after the jump.

# Differential Equations Outline

I have been thinking about my differential equations course all afternoon and evenings. I have lots to do to put it together, but here are my big decisions:

#### Stuff that Stays the Same:

- Build around Hirsch, Smale, and Devaney’s
*Differential Equations, Dynamical Systems and an Introduction to Chaos, 3rd Ed.*This is for the focus on modern dynamical and qualitative techniques (which I like). Basic coverage is chapters 1-10 (core material), 11 (basic biology models like Lotka-Volterra and SIR), 14 (Lorenz System), and then some lighter coverage of chapters 13 and 16 (I wanna tell the story of the 3-body problem and Poincaré’s discovery of homoclinic orbits). - Some Lab Explorations: use several of the activities in HSD as assignments. I think 5 of them will do, with one due every three weeks. We’ll use CoCalc as the main tech, augmented with Field Play. Lab assignments are to be peer graded.
- Week Two is all about using that technology. We spend the week in a computer lab learning the tools and getting the first lab written up.
- Weekly “homework reports.” Instead of asking for a homework assignment that has to be marked seriously for correctness and returned, I’ll just ask for a report on what kind of work that the student did during the previous week on the assigned reading/exercises/etc. This course is full of more advanced students, so I feel it is appropriate to move this responsibility onto them. (I took this idea from Joshua Bowman.)
- An extra assignment for the grad student(s) to read and report on the proof of the Fundamental Existence and Uniqueness Theorem.

#### Stuff that is New:

- A (nearly) weekly “technique” assignment. The last iteration of this course didn’t have quite enough work on important, well-known techniques for solving systems by hand. So, I’m going to add some. These won’t happen every week, because at some points we will have new linear algebra to learn, etc etc. Anyway, the new topics are: separable equations; homogeneous equations; exact equations; integrating factors; linear equations; numerical methods (Euler and RK45); Picard’s method; power series methods (easy ones); Laplace Transform; Fourier Transform.
- A weekly “techniques quiz.” This will be a one item, 5-7 minute quiz on each Monday to check up on the techniques in a simple way.
- Split the Final Exam into an “in-class” and a “take-home” portion. The in-class version is on Monday of finals and will ask big-picture questions. The take-home part will be like one of the lab experiments, aiming for summation of ideas.
- Maybe I’ll switch to using Jupyter notebooks rather than Sage worksheets? I have to think about it. Maybe that is too much for this iteration.

If I have all of this right, the cycle of work will be something like:

Monday – technique quiz, TJ preview of material

Wednesday – homework report, class discussion

Friday – labs due (sometimes), class discussion

Tomorrow I’ll revise last year’s syllabus to reflect the changes. Then I’ll start writing the assignments and assessments out, update the course web page, and set up CoCalc for the term. (It might take till Monday or Tuesday to get all of that done.)

# Simple Progress: web site & geometry

Well, I made some good progress. It took a bit longer than I hoped (what doesn’t?), but I have now made general updates to my UNI web page and I have reset things for my Euclidean Geometry course. [Everything! all the assignments, the syllabus, etc]

There is nothing truly *new* in my planning for that course, except that I have decided to add a little extra communication with students about their progress via an email every three weeks. I didn’t write this into the syllabus, but I plan to do it. These emails shouldn’t be too big a burden because the class only has 10 students enrolled. (I hope no one drops, because that is the minimum required before the Dean and Provost start making noises.)

Oh, and I reread Tim McNicholl’s *The Extreme Moore Method*. It is just as a I remember it. Still feels like home. One of my challenges will be making linear algebra and differential equations feel as comfortable.

Now for a little break, and then move on to differential equations. I hope to have that one done by tomorrow afternoon.

# Getting Ready for Fall 2018

I have started getting ready for fall term, and I feel like I have some *thinking* to do. So, I return here to use this as my write-to-think space. Writing to think has always been a good teaching practice for me, and I am thankful for the amount of feedback I get from you, my only reader. (Hi, Mom!)

This term I am teaching three courses:

- Differential Equations
- Euclidean Geometry
- Linear Algebra

Classes start on August 20th, which is less than two weeks away. I hope to have plans made and syllabi constructed by the end of this week. That is unlikely, of course, but I bet I can have two done and make a good start on the third by Friday afternoon.

Since building early successes is important for one’s mindset, I am going to start with easy tasks and work toward the harder one. Here is what I have to do:

- Re-read Tim McNicholl’s paper on the Extreme Moore Method, which was an important part of my conversion experience to IBL teaching. I want to get re-energized to make decisions based on my core beliefs. Since I last read the paper about 10 years ago, I might find out that some of my core beliefs have shifted. That’s okay, and it will be important to find out.
- Plan Euclidean Geometry and reset it for this term. I am making no major changes, so this should “go fast.” But I should at least think it through again.
- Plan Differential Equations. I don’t have time to make big changes in this one, though I would like to. At the moment, I am thinking about adding a “technique of the week” component. Currently the course is focused on numerical & qualitative work, and I want to put in just a little bit more of “this is how you find a solution to an ODE by hand.” I also hope to switch the technological components to using Jupyter notebooks instead of Sage Worksheets. (I am still using CoCalc as a platform.)
- Plan Linear Algebra. This is the big one. I have so much to rethink and redo. I’ll say more when it is time.

So, if you get excited about the details of trying to teach college math courses, follow along in the next few days.

# A Failure: Indifference to Building Student Anxiety

I had an up-and-down day in classes today. I am still thinking about how linear algebra went early this afternoon.

We are now in the fifth week of the semester here at UNI, and my linear algebra class has been doing lots of tasks about the geometry of **R**^{2} and **R**^{3}, especially those things that help us understand how to think about vectors, lines, planes, and the dot product in situations where we have a chance to draw the picture. Things have been getting steadily more challenging, and the tasks for this week are hard for them. For example, a task that made everyone unhappy today was this:

Consider the line in

R^{3}given by the parametric equation

t → (-6,-2,1) + t(3, -1/2, 1).

Find the point on this line which is closest to T = (1,1,1).

That uses everything we have learned so far, and requires a little bit of thinking about the geometry. It is a great task. It was one of seven I asked them to do for today. (It was probably the hardest one.)

But Friday is the first exam.

I should have thought a little more carefully about this. I have lots of young students, and I have actually set things up in a way to make them more anxious than need be just before the exam. I set a trap for all of us, and then led the whole class into it with a big grin on my face.

Also, I am writing all of the materials myself. This means that if the students feel anxious and are having trouble with the text, that is my fault, too. There is no faceless textbook author to be grouchy at. I can’t be the friendly guy who saves them from the confusing writing.

We had a short chat in class, and I sent them a longer message by email after I had a chance to reflect. I HOPE that I have helped calm their fears a bit. Time will tell.

I am going to have to think more carefully about how I structure things when I edit and revise for next term.

# About Linear Algebra This Term

I am having trouble writing this morning, so I hope that blogging here will loosen my brain and thoughts will start to spill out properly.

I am working on my linear algebra books. “What’s that?”, you say, “You are writing books, plural, for a course?” Yes. I have taught linear algebra many times in the last few years, and I finally felt dissatisfied enough with my old materials and prepared enough with my ideas about the course to write everything from scratch. So, I am writing these:

- A primer, or reader, which discusses the basics of linear algebra but sticks entirely to the plane and 3-space.
- A workbook, which has a sequence of tasks I can use to run an IBL course.

My basic model for this is how I run my Euclidean Geometry course. I have the students read Euclid’s *The Elements* Books I-IV, and I have designed a sequence of tasks to go with that. The idea is to treat *The Elements* as the existing research literature, and then pose ‘research questions’ which drive the students to understand the material and learn to do math on their own. In geometry, we are working on proof-writing, definition-making, and conjecturing. This works great.

I want to copy the model for linear algebra, so I need something to replace *The Elements*. So I am writing it. I am trying to use the old-fashioned, discursive style that you would find in math texts pre-WWII, too. The text is much more a narrative, and avoids the now-standard “Defn-Thm-Proof” setup. Definitions and results pop up as we find them.

So, now I am about 75% of the way through writing chapter one, and I am past deadline for getting this to my students. They have the first half of the chapter, which got us through yesterday. I had hoped to give them the second half to read last night… and I failed.

My current sticking point is making a transition from talking about norms, angles, and the dot product in the plane, to talking about the equation of a line through the origin. I have written the start of three different versions today, and been happy with none of them. So, I am going to take a break and start again in about an hour.

Maybe I’ll read the copy of Math Horizons that just arrived.

ps. If you want to follow along with my crazy experiment, I use github. The files are here.