I’m gonna ramble a bit before I get to the big thing. Just hang on. I wanna write one blog post today, not seven.

Now I got off to a better than average start by sending my students a questionnaire at the end of last term. It was a simple google form asking students about their backgrounds, what they are comfortable with from the list of prerequisites, and why they are taking the class. This turned out to be really useful. I learned these things:

- These are fairly advanced students: one junior, two seniors and five graduate students in our MA program. All have concentrations in mathematics, though one calls it their “second major, after computer science.”
- They are 50/50 split on why they are taking this course: for some it is required (the grad students), but some are there because it sounds interesting.
- Most of them are through our undergraduate analysis course, though a few are halfway through (its a year-long sequence), and one has not, yet. So I can almost-but-not-quite assume they know stuff about epsilons and deltas.
- About half of these students report wanting to continue their education as a next step (PhD programs in math), and the rest want to talk about getting involved in higher ed in some way (tutoring, community college teaching)
- They have widely differing comfort levels with using technology to do mathematics. This talked me down from some rather ambitious plans that would have made a mess.
- Despite having all passed the prerequisites, they report big gaps in their comfort level with some of the material that would be considered important before enrolling. I’ll have to manage this carefully. I think I’ll just drop “5 minute reminders” all the time.

I have a class that meets three days a week for 50 minutes each, in a room built for group discussion more than anything else. There is a chalkboard and presentation equipment, but it is all mashed together at the front of the room. If I want to present a lot, chalk is going to be awkward, but using the computer would be easier. There are lots of tables set with students chairs facing small groups of 4-6 people.

I have the university level learning goals, and the department level learning goals. Those are about as broad and vague as you would (should?) expect. Stuff about making arguments, solving problems, writing, speaking, and thinking critically, etc etc.

But what should I try to teach in my course? As an advanced course, I have the luxury of aiming more at content and a bit less at foundational skills. The course catalog is not as helpful as I would like:

https://catalog.uni.edu/collegeofhumanitiesartsandsciences/mathematics/#courseinventory

MATH 3630/5630 (800:155g). Differential Geometry — 3 hrs.

Analytic study of curves and surfaces in three-dimensional Euclidean space. Prerequisite(s): MATH 2422 (800:062); MATH 2500 (800:076); junior standing. (Odd Springs)

Right… I read that as, “Do whatever and have fun.” So… I had to come up with my own. I won’t pretend I am completely finished with this, but I DO have a good start, because I stole the idea of “Big Questions” from Kate Owens. My Big Questions are these:

- How can we use smooth functions to describe {curves, surfaces} in Euclidean 3-space?
- What is the difference between
*intrinsic geometry*and*extrinsic geometry*? - How do we understand the internal geometry of a {curve, surface}?
- How do we understand the external geometry of a {curve, surface}?
- What are some of the concepts of
*curvature*and how are they interpreted? - What is the difference between the geometry of a {curve, surface} and the topology of that object? How is the geometry of an object related to its topology? Can knowing something about curvature tell us something about topology?

We might not get to that last one, but I really hope to explore some interesting theorems in each of these regions for both curves and surfaces. Anyway, later today I hope to sketch out a calendar of study topics and see what I can reasonably pack into a term with 42 class meetings.

So, I started thinking about this: How do I assess student learning on those things? (yeah, they are a bit vague, but still.) Which are more important than others if I have to make hard choices?

The trouble is that I don’t really care. I mean, I want the students to do some math and learn some stuff. But I don’t particularly care exactly how they choose to do the math and I don’t care exactly which bits they learn.

That sounds terrible. But I mean it. I want to give the students some flexibility and autonomy. As long as they are doing work that involves the ideas of the course somehow, I will be happy.

So my stupid idea is this: a choose-your-own-adventure points collection assessment system. I am going to offer the following types of assessment options:

**Exams/Quizzes:**sit-down exams with no references aimed at small bundles of material coded to the big questions above. About 100 points each.**Weekly Homework:**regular assignments. But I’ll only collect*reports*(which I stole from Joshua Bowman) that are small, and students may turn in their “best two problems of the week” for real comment, credit and revision. About 25 points each, 5 for the basic report, 10 each for good written solutions.**Projects:**Independent Projects on related material or topics just off the main thread of ideas in the course. delivery of these to be negotiated, but I will definitely accept short papers and oral reports. point values will vary, depending on the project, but from 20 to 100 points each. I have a list of about 10 ideas for this. I’ll need to think up some more. I will also invite students to suggest project ideas.

Then I will assign grades based on some sort of stupid table I write up. Collecting X points will mean a grade of Y. Each particular thing will be assessed as pass or fail. (for exams, that will be question-by-question. Also, I’ll have to write some rubrics and make them public.) If you pass you add the points to your total. If not, you have the chance to revise your work, or try a different assessment. I will have to make two columns to this table, because the undergraduates and graduates have to have different expectations. I am going to try to set things so that a student can expect to pass the course in a variety of ways. In particular, an undergraduate should be able to get an A in the course if they `do well’ on three exams and weekly homework. In particular, if a senior who plans to go to industry just wants a C so they can graduate, they can pick their level of commitment, learn some stuff, and call it a day.

So, oddly enough, my adventures with alternative grading systems has led me here. I am inspired by standards based grading and mastery grading and specifications grading, but I have no strong feeling about exactly which standards are important. And NONE need be. I just want students to learn some (relevant) stuff and show me that they did it.

Now is the part where you help me, assuming you read so far. (Hi, Mom! — I’m kidding. My mom would not have read this after seeing the title.)

- Why is this a bad idea?
- What would you want to know about this after it is done? Say I wanted to write a PRIMUS paper about it, what would you want that paper to address?

Set up: I have taught this course nearly 20 times from essentially the same set of notes. I wrote the notes myself, so I know them pretty well.

I have intentionally set things up to force my students to grapple with the ideas of what a mathematical definition is. This comes from using Euclid’s *Elements*, which has some crappy definitions in it and lots of other important but undefined terms, and also because I give them slightly messy definitions to use. At some point, they become unhappy and we end up having really productive and thoughtful conversations about what definitions are for, how we make them, how we use them, etc etc. All of this is centered around things like “quadrilateral”, “polygon”, “convex”, “interior” and other important terms from classical planar geometry.

For context:

A

quadrilateralis a figure consisting of four points, no three of which are collinear, in a given order and the four line segments joining points next to each other in the list.Usually, we specify only the four points, so quadrilateral ABCD consists of the points A, B, C and D, called

verticesof the quadrilateral and the line segments AB, BC, CD and DA, called thesides.

and

Let n be a natural number. An

n-gonis a figure consisting of n points A_{1}, A_{2}, …, A_{n}, prescribed in order and calledvertices, and the n line segments, calledsides, A_{1}A_{2}, A_{2}A_{3}, …, A_{n-1}A_{n}, A_{n}A_{1}.A

polygonis an n-gon where n has not been specified.Note: Commonly used terminology includes the following: 3-gon = triangle, 4-gon = quadrilateral, 5-gon = pentagon, 6-gon = hexagon.

There is all sorts of juicy good stuff here (like: 4-gon and quadrilateral are actually not exactly the same, and both allow for non-simple figures). I have come to expect lots of these confusions mostly because I **designed** the course to make them come up.

And today was the big day. It has been building for weeks. The students are unhappy with my definitions and want to fix them. Fine. Then they have to fix them and live with their choices. We had the big conversation.

The part that got me is that they were all under the impression that I had not even defined the words *vertex* and *side*. In fact, they told me, “You wrote the words down and put them in italics, but that’s not a real definition.” They all insisted that I hadn’t given them definitions of these terms. So they wanted to redefine the word vertex.

This was surprising! It was also awesome. Wow. Anyway, I had fun, and we have more to discuss on Monday.

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We will read Hirsch, Smale, and Devaney’s 3rd edition. I really like the focus on qualitative and numerical work, the choices of lab explorations (which are tech agnostic), and the variety of motivating applications in the exercises. We’ll cover chapters 1-11, which is qualitative technique for ODEs in dimensions one, two, and three up to the Poincaré-Bendixson theorem. Then we’ll hit applications from population dynamics (ecology and epidemiology) and the Lorenz equation. I’ll spend the last week sharing the story of Poincaré’s discovery of the homoclinic tangle and chaos in the restricted 3-body problem because it is such an awesome story and such beautiful math. I’ve added a “technique of the week” component to bring in some classical material that I learned last millennium.

From the student point of view, a typical week will feel like this:

- Start Monday with a quick technique quiz (from last week), then get an overview of the main ideas of the coming material. Start reading.
- during the week, read and work exercises from the text, also read a short intro to a particular “solve by hand” technique and try some samples
- Wednesday turn in a “homework report” on the work you did in the previous week. Don’t turn in the actual work, just your report on hours worked, difficulties encountered, successes, etc. Have a class discussion about material you are finding challenging.
- Friday have more conversations about material. Every three weeks or so, turn in a lab on this day.

There are a couple of things worth noting:

- I have chosen a Mastery Grading kind of set up: Students will be allowed reassessment on the quizzes; most things will be marked for “good faith effort toward completion” and heavily commented on as preparation for the final exam.
- Everything will be pretty low stakes — leading up to a comprehensive final exam, split into an in-class portion (big picture ideas) and a take home lab assignment.
- I will have to write a really good set of specifications for that exam.
- Week Two will be a little different as we spend it in the lab playing with the tech we will use throughout the term: CoCalc, Jupyter notebooks, SageMath & Python programming, an online visualization tool for planar vector fields
- Week 15 will be story time. I don’t wanna pile new stuff on them as the try to finish up requirements and prep for the exam.

I have previously used CoCalc in some of my courses (with SageMath and Python), so those parts aren’t too scary. The new part is switching to Jupyter notebooks, and I think that is a pretty easy transition from SageWorksheets. It is very easy if I use the SageMath kernels available in CoCalc. Plus Robert Talbert has made a really nice 3-part YouTube tutorial for his students about the basics of Jupyter notebooks. (It is an easy search. You can do it.)

From my point of view, each week will have to look like this:

- Monday: Give short technique quiz. Give introduction to the main ideas of the week. Announce (remind students of) the next week’s set of assignments.
- Tuesday: Mark and record quizzes
- Wednesday: Return quizzes. Collect homework reports. Maybe work an example?
- Thursday: Read, comment on, and record homework reports.
- Friday: Return homework reports. Lead class discussion. Collect lab assignments electronically.
- On those weeks with lab assignments, I’ll be commenting on them from home over the weekend.

Of course, I have to write up all of the assignments. If I were to do it in one go, it would probably be a week or so of work. I don’t have all of that time right now, since I have to plan linear algebra, too, so I’ll just try to get the first month ready by the end of the day tomorrow. That means: 4 homework report assignments, the lab week materials, the first lab and its solution, the first 4 techniques and quizzes to assess them. If I have a little more time, I’ll write up the second lab assignment and solution.

One silly thing I haven’t thought through, yet, is that my assigned classroom is terrible. I have 8 students in a room with tiered seating (narrow fixed tables) for 65. And there is too much furniture at the front of the room: the computer equipment takes up at least half of the front of the room in a huge cabinet. I have lots of chalkboard space, but it is “student unfriendly”. I don’t know if I can move the class, but I might have to ask.

Last step before bed is resetting the course web page.

]]>- 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.

- 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.)

]]>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.

]]>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.

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