So many things I wanted to write about along the way, but it just wasn't a year where that was going to happen. Maybe posts someday? Who knows, but I wanted to reflect on what I could remember now. Mostly I teach math for preservice teachers, elementary, middle and secondary, and some our senior capstone, which I do as a bit of a history of mathematics. But last summer we needed someone to teach calculus online, and I was certified, and I find it hard to resist a teaching challenge. (Our university requires that you pass a little course before you're certified to teach online. Ironically an in-person course. And, of course, no training required to teach otherwise. )

The online course was hard. I knew I would miss the day to day, getting to know the students, and the learning community, but I had no idea how much I relied on the students to make the work problematic for each other. The interaction is sooo much of the thinking, and even good participation on a discussion board is not going to hold a candle. The lack of the day to day formative assessment made me feel like I am teaching in the dark. The course was asynchronous, so the disconnect was maximal. Some people finished the course by week 8 (of 12) and some did the entire thing from weeks 7-12. I required two video call/interviews, but not at any particular time. Now I think at least three, including one in the first week.

But when a teacher ed course got cancelled for low enrollment (we should be worried, I think), I got a chance to teach calculus 1 in person, and then in the summer eventually fell into calculus 2 for the first time in a decade. If you want to skip all the blah, blah, blah, and see the agendas and resources, here are the course pages (Google docs) for the Calculus 1 and Calculus 2. (Online calc 1 page, with more exposition as it was self-paced.) GVSU is a real hotbed of some calculus innovation, with Matt Boelkin's Active Calculus and flipping crazy Robert Talbert among others.

So what sticks out from the year of calculus? I think I'll try to write about (a) reordering calculus 1, (b) writing in calculus, (c) the interviews, and (d) some learner feedback. (I never got to (d)! Maybe next post.)

**Reverse It**

In discussions with #MTBoS folk about calculus (all of whom I cannot remember; Dan Anderson Paula Beardell Krieg, David Butler, Heather Johnson, Lana Pavlova...), the idea was floated to think about integration first. Why do we teach derivatives first? What are the consequences of that? I'm already in favor of moving limits to the end of calc 1. Hand waving was good enough for Newton and Leibniz, so it's good enough for me. What are the problems that limits solve, and do learners know enough to know they are problems when limits are taught? For me, it's about precision, and precision comes late in a learning trajectory. One of the problems with Derivatives First is that learners invariably think of them as antiderivatives, which makes the Fundamental Theorem of Calculus one big shoulder shrug. They see it as how to calculate definite integrals (first part) or never bother to parse it (second part).

The best thing about derivatives first to me is thinking about rate of change and the transition from velocity as an average to instantaneous velocity. But rate of change is difficult to visualize, I think, as it involves an abstraction even to get started. But accumulation feels more visual. Someone suggested penguins coming into a room... which led to this GeoGebra. It was a solid lead-in to Visual Patterns. That was a good concrete start, gave us reason to do some algebraic modeling, and good use of table representations which lead to thinking about differences (first, second and so on). We posed an accumulation question about the visual patterns, too: what if we wanted to build the current step and all the previous? You know, like we were just doing? That lets us get at the accumulation for the algebraic pattern we just found. And can we find a pattern for that? Always one degree higher, hmm. And the pattern for the first difference is one degree lower? And the number of differences to constant is the degree of the pattern? So much to notice. I tried a little, changing the patterns to bargraph, which also makes a nice connection to area under the curve. I want to explore this more.

One of the things about teaching calculus at college is that a fair number of students have had some calculus already. Not enough to test out, but enough to start a calc 1 class feeling like they've got all this already. Despite this really being algebra, it seemed familiar to no one. And their calculus connections helped intuition, but made them more surprised that it wasn't exactly the derivative and antiderivative.

The other thing that this helped with was keeping us away from rules to start. We started integrating for applications immediately, so we used tech to get answers. This helps put the emphasis where I want it, on the meaning of what is being added up or accumulated in the integral.

**Writing Calculus**

Waaay back when I was first teaching calculus, early 90s?, there was a book from the MAA on student projects. They were really artificial problems, but fun, and the first I had seen the idea of getting students writing in math. This was long before standards based grading was even a twinkle in my gradebook.

What I found this year was that student writing was amazingly synergistic with what I was doing for SBAR. It is generally hard in a 14 week college course to get students up to speed on the idea of SBAR, and getting them to take responsibility for what they have and need to show is a current challenge. I use a holistic grading scale that boils down to an A means you have given evidence that you can solve problems like this, and a B means you can solve this problem. So how do you show you can solve a lot of problems with one solution? You explain what you're doing, why you're doing it and evaluate your answer to see if it makes sense. (Why is it so hard to get learners to check their answers?! Hmm, if I want them doing it, it should be part of my assessment. So...)

What I noticed this year is that the writing, which was explicitly about putting words together to form coherent thoughts, helped the learners to make the jump to more complete answers on SBAR problems. (Also, the writings can be SBAR evidence. Evidence is evidence.) I ask for 6 or 7 writings, they get credit for completion, get feedback on the 5 C's rubric, and then pick 3 at the end of the semester to be their exemplars, which can be revised based on feedback. Everything I've read about teaching writing requires an opportunity to revise based on feedback. They post writing to the discussion board, and then give someone else feedback. Partly to just get them reading other people's writing, partly to get them evaluating in a way that gets them thinking about their own writing. The writing assignments (specifics on the course pages) come in a variety. Writing prompts, specific problems, choose from a short menu, make a miniguide to a topic (thanks Paula Beardell Krieg - great assignment), or open to their choice. Everytime I surveyed learners as to their preference, there was a great diversity, so I feel constrained to variety. What's a little weird is that you can't just give those all as choices each time, some people want the structure of a specific assignment.

**Talking Calculus**

My colleague Esther Billings has been using conferencing really effectively for years. So I've always meant to... but it's hard finding the time was my excuse. Despite seeing Esther somehow work it out literally next door. For the online course, I required two video conferences, but many left it until the end of the semester, limiting their effectiveness. In in-person calculus, I kept the idea of two interviews, and made one about differentiation and one about integration. I thought then people would start getting them done when we were done with the topic. Silly rabbit. So this past semester, I did two interview periods, done during that time they counted for an A, after a B.

So much good happens in these conversations. For one, getting to know the learners better. For two, many more people came for office hours afterwards than beforehand. Three, it is definitely my most accurate form of assessment. Even with the writing that they are doing, it helped me know what they understand in a really specific way. Yes, that was an accurate mark in the grade book. But it was the best formative assessment that I have ever had.

Students came in, either with a problem they had worked on, a topic they wanted to talk about where I provided the problem, or something in between. Delightfully, a few students brought in something on which they were stuck - especially after the first experience. The questions I asked further gave me an opportunity to model the kinds of explanations I wanted on SBAR problems. I gave A/B if they need my support, but were able to recap in a reflection. If I asked a question that they didn't know where to start, I'd share why I asked and my answer.

I will not teach without these in some form again.

While we're talking talking calculus, a frequent comment from learners was how helpful whiteboarding was. #VNPS if you're lingooey. These were very high engagement lessons, usually with a list of problems from which groups chose a problem or two. I get to watch and assess, interact with individual groups, and ask questions to prompt more in group discussion. We concluded with one group sharing in depth, me sharing comments about what I saw in common challenges, or each group giving a quick summary.

**Next Time**

So much went on in these courses, but I was not good at writing along the way. I don't even know when the next time I'll get to teach calculus will be, but some of these lessons will come with me whatever is next. I'd love to hear your feedback, or questions, or how you think about these ideas. If I get to write more about this soon, I'll try to capture some of the learner feedback about these courses and features.

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