I know this is a math ed blog, but we need to talk about writing. Humans communicate through language. Language involves receptive processes (listening and reading) and productive processes (speaking and writing). We math teachers tend to focus on speaking to communicate thinking over writing. I’m not talking about communicating answers, I’m talking about communicating our thinking. As I visit classrooms, I hear teachers asking students to explain a process or their reasoning, but I seldom see requests for written explanations. If you want to hear teenagers groan, ask them to write in math class (seriously try it out).
As a math major, I was surprised at just how much writing there was in college mathematics. Where was all the writing in high school math? Why was written communication so much more prevalent in college math? Didn’t I say I was a math major because I didn’t want to write? It may then come as little surprise that as students enter their first “proof course” in college, many students struggle to communicate their thinking. How can we help support students in expanding their comfort and skill in writing about mathematics?
One thing that helps is to explicitly discuss the purpose of writing in math and what a strong mathematical explanation entails. For example, I have told students that we are writing to share our problem solving processes, convince me (mathematically) that that process has resulted in a solution, and that you understand why it is indeed a solution. I don’t stop there, I also talk about what a good explanation entails. This might look very different from class to class or grade to grade. For example, in kindergarten (yes we write in kindergarten) we show our thinking with pictures and numerals as best we can. In high school, we might use complete sentences that include well labeled diagrams to further evidence our thinking. I also ask students to avoid pronouns because I often don’t know what it refers to and I want them to be precise in discussing their ideas.
As we develop writing, something that is helpful is to allow students to share their writing and follow along with it to see if you can recreate their strategy and solution. Doing so usually uncovers aspects that are not clear or not logically consistent. For a recent elementary PD session, we made a list of expectations for math writing so that students could begin thinking about what constitutes an adequate explanation as they wrote them. The hope was students would start thinking about their writing and understand what makes a clear explanation. We adapted down to kindergarten and up to fifth grade (see below for the general idea). We didn’t just give the list to the students, we practiced writing and then pulled out things that were clear and less clear and talked about what made them clear (or not). We then talked with the students about each of these in the context of their writing. It takes a lot of time, but we hope that over time if we continue to use it then they get used to communicating their thinking in writing.
Some of my favorite prompts for writing in math involve things such as write a letter to your family to explain how you solved this problem. Pretend you have a pen pal and explain your thinking to them. Somebody said they don’t believe your answer is correct, write a letter to prove you are right. Write about an error you think someone could have made on this task and explain why you think people might make that error. There are lots of ideas for prompts, but you could also simply ask students to explain their thinking.
So, as you kick off the school year and discuss expectations, make sure to develop and discuss your expectations for writing and help students understand them. Also, make sure not to shy away from writing; if your students aren’t good at something that is a pretty good clue you probably need to do more of it, not less. If you have some routines or ideas or resources for addressing writing in math I’d love to hear them!