I'm teaching my college writing course again this Fall. You might think that's strange. As a math professor, why am I teaching writing?

My college (Harvey Mudd) is a school of science, engineering, and mathematics. Because writing is an important part of every discipline, the college shows its commitment to that idea by having every department contribute faculty to teach the college-wide writing course that every first-year student takes.

For me, teaching writing was a chance to think more deeply about how to improve my own writing. I taught this course for the first time in 2018, and as it happens, was also writing a book for the general public at the same time. The old adage "you teach what you want to learn" is really true. Every lesson I taught my students was also being put immediately into practice---the best kind of learning, for sure. The second time I taught the course in 2019, I compiled a list of parallels between the process of writing and the process of doing math.

Here's the list:

Both involve reasoning and making convincing arguments.

Both involve creative choices about what arguments are most compelling.

Both cause anxiety among some people.

Everyone can improve with practice and encouragement.

Both benefit from mastering technical skills as well as cultivating an overarching vision.

It is unreasonable to expect an argument to pop out beautifully the first time.

More often, getting good involves a process of repeated revision and creative choices to make arguments clearer and more compelling.

Doing them well can bring great personal satisfaction.

Clarity of communication is essential, and intertwined with clear thinking.

Both build virtues that will serve you well no matter what you do in life.

(You can see more ideas, contributed by others, by clicking the tweet above.)

I wondered: if there are so many parallels between math and writing, then how can teaching writing inform the way I teach math?

For instance, if I really believe point #1, then right answers are not as important as the right thinking that produces those answers. This means in a math class, I should devote time honing students' abilities to make convincing arguments to their peers, and giving them opportunities to do so through class discussion and peer feedback, as is normal in my writing course.

If I believe point #5, then I'd realize that helping students see the big picture is just as important as teaching them specific techniques/recipes/algorithms for doing things. This mirrors what happens in writing an essay, where writing, grammar and sentence structure are technical skills that should be mastered, but they are not sufficient for writing a great essay, which also needs to have an overall vision.

If I believed points #2, #4, #6, #7 and #8 then I'd give my students opportunities to revise their work, to improve the arguments they are making, so that they can ultimately be proud of the final product. In much the same way, when writing essays, a finished product doesn't pop out beautifully the first time. Instead, good writing is a process---it begins with a rough draft and in-progress ideas that grow through revisiting those ideas and revising them.

### Rough Draft Thinking

Amanda Jansen has a helpful new book, called Rough Draft Math, in which she makes and develops this idea very effectively. She's pushing back against the (unfortunately common) way of teaching math at the K-12 level that primarily expects students to memorize or compute things, and makes no effort to connect to the ways that students are beginning to make sense of the ideas. Rather than being affirmed for starting the journey of understanding, students are often shamed for wrong answers. Students who learn math this way grow inured to the idea that their thinking doesn't matter.

Of course, this is exactly the opposite of what learning math should be. Thinking is everything in mathematics. Thinking is where joy is to be found, when you truly grasp an idea and understand it. The process of learning is just that: a process. To grasp an idea is to start with an unfinished, possibly incomplete, understanding and, through exploring more and more examples gradually understand it better and better.

In fact, this is how mathematicians do math. When I am doing research, I start with an incomplete picture---a guess, a conjecture about something I've noticed. Then I explore more and more---doing examples, trying to reason about why something might be true, until I arrive at a more complete picture of an idea. Moreover, that picture is always being revised as I learn related ideas and place this idea in context. If this is really how math is done, then why isn't it taught that way at every level?

This is where I really appreciate Jansen's framing of encouraging what she calls rough draft thinking. As she defines it:

Rough draft thinking happens when students share their unfinished, in-progress ideas, and remain open to revising those ideas.

There's a lot packed in this. A math class should be an environment where students feel free to share their thinking, and feel no shame about brainstorming. They are encouraged, as Jansen says in her book, "to use language that makes sense to them...even if the language is not mathematically perfect." They are aware their understanding is limited, and look forward to revising their understanding as their thinking expands. Again, as I read this in Jansen's book, it reminded me of the way that research mathematicians work when they first start to work on a problem together. Often, when I work with a collaborator on a problem, I marvel at how imprecise our language is because the ideas are still fuzzy, and yet somehow we still communicate. Our language tightens and improves as our understanding grows. Doing math is a process.

### Encouraging Rough Draft Thinking

How does one encourage rough draft thinking? For teachers (or parents), there's work to be done to create a welcoming environment where such thinking can take place. Rough Draft Math is filled with ideas about how to do this at the K-12 level: establishing a culture, and designing rich tasks that allow for communication and revision. Teachers of mathematics will find a lot they can use in this book, and I highly recommend it.

As a college professor, I'm challenged by Jansen's ideas to think about what rough draft thinking would look like in a college math course. The traditional non-interactive lecture---in which a professor recites polished proofs of important theorems and students take notes---doesn't allow space for rough draft thinking. It is the college-level equivalent of asking students to memorize things---in this case, Some Famous Person's Polished Proof. I remember seeing many such Proofs presented when I was in college. Although I'd be awed by the elegance of such a Proof, I often felt like: how would I have ever thought of that myself? That can be demoralizing.

So, what can a college math professor do? Here are some ideas. Many of these can be done in almost any kind of class, but the more interactive your class is, the easier it will be to implement some of these strategies.

Award partial credit on homework problems when students suggest strategies, even if they are not ultimately able to solve the problem.

Publicly affirm any student that offers an idea in class, and build upon it. In almost any 'wrong' idea there is some right idea that you can build upon to solve the problem. Or, you can pursue the idea, and then discuss what you learn by going down that path---perhaps it is an insight that affirms that this approach will not work. You are inviting students to see how you take a 'rough draft' idea and learn from it, inviting conversation about it. This may mean abandoning what you planned to do in class in favor of going down a path that a student has suggested.

Encourage students to openly discuss with other students strategies they tried to solve a problem that did not work. In one class, I made this a requirement: to earn participation points in a class, students would, over the couse of the semester, have to present 5 'productive failures'---things they tried that didn't work and what they learned from it.

Discuss your own rough draft ideas with your students. For instance, you might openly discuss in class your own journey to understand a concept: "when I first learned concept X, I used to think that it meant y. After a while I began to realize that Y was not sufficient to capture the idea, because of this example." Or, you might explain a research problem you are working on, and how some of the ideas you are thinking about are still in rough draft form.

Before presenting the proof of a theorem or a solution to a problem, invite student discussion to generate ideas about how it could be proved. Here's an example. In one class I teach (Galois Theory) there is a deep theorem that is hard, and the proof is quite unmotivated. Every time I've lectured on this theorem, I've just presented the Proof of Famous Theorem. And I've never been satisfied with that. So last time I taught it, I decided I'd ask students to brainstorm strategies for addressing this hard proof. Two students volunteered their ideas. Neither would work, but we talked about why---and then I showed them Famous Proof, and what was neat was how both of these strategies actually appear in some form that proof. It made everyone pay more attention to the proof because the class had come up with the strategies themselves.

Give students opportunities to revise their work for a higher grade. For instance, in my analysis course, on the first 3 assignments, I allow students to revise and resubmit after getting graded feedback. This incentivizes students to pay attention to the grader comments and improve their work. It also tends to level the playing field, since students enter my course with vastly different backgrounds, and revision allows students some time to adjust to my Guidelines for Good Mathematical Writing.

Incorporate peer feedback in your course. For instance, have students swap homework assignments, and offer helpful critique on how to improve each other's writing. You should coach students on what good feedback looks like. They could use the Guidelines (linked above) as a start, but you should also explain how to offer feedback in constructive ways.

Design a final project in which a student takes some bit of work they've done earlier in the semester and revises at the end of the semester. For instance, they could choose a solution to one of a selection of difficult homework exercises. In addition, ask them to offer a reflection on how they've grown in their thinking and communication.

So much of doing mathematics is really a process: of starting with an idea and improving it. As Jansen reminds us: we can make sure our teaching reflects that.

Wishing you all the best in your pandemic teaching adventures,