I’m planning for Spring 2019, which is MONDAY, and I have a lot of work to do still on *Math 3630/5630: Differential Geometry*. I am behaving like a professional and trying to plan things the way one should, start with (1) figuring out your context, then (2) write student learning outcomes (SLO), then (3) plan assessment of those, finally (4) settle on teaching and learning activities to prepare students, and when all that is all done write the syllabus. (Thanks to Dee Fink for showing my how to organize this correctly.)

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.

## Step One: Context

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.

## Step Two: SLO

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:

**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)

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

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.

## Step Three: Assessment Plan

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.

## Questions for Readers

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?

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