Teaching is not something someone is mostly born to do but actually requires you to do a lot of unnatural things, [and not do things] that are hard to learn not to do, things you do in everyday life that you just can’t do as a teacher.” – Deborah Ball
I’ve been thinking more about the ways in which teaching is not intuitive, and in particular about the ways I need to work deliberately to train myself to avoid certain types of thinking that might seem to be reasonable at first glance. My intuition is powerful, but that is not a natural thing — it is intuition that I need to build by being deliberate about my practice, constantly working to get better, and avoiding some of the traps that my brain falls into as I try to do the work of teaching. Here are a few of the traps I have been working to train myself to avoid.
I have beliefs — about the nature of teaching, about what my students know and can do, about whether a certain instructional activity is going to be engaging or promote learning. As a human being, when I see an ambiguous situation, I tend to interpret it optimistically with respect to my prior beliefs. That’s the way my brain works, and most of what happens in a math classroom is in some way ambiguous. If I really want an activity to be engaging, I will tend to focus on the subset of students who are engaged — potentially forgetting that another significant subset was not engaged, and maybe that activity I was so proud of was actually not the best choice for that lesson. On a more basic level, when I want to know whether my students understand a given concept, invariably some will understand it, some will not, and many will have some understanding but have work left to do. My natural bias is to focus on the students who “get it” (who are likely to be the vocal students in any case), and to interpret ambiguous understanding in my favor — that a correct answer means they have a strong grasp of the concept, or that an inarticulate response in which I hear some good ideas means they’ve reached the mathematical goal. Or maybe I am really proud of this example I came up with to help a student understand exponential growth, and after that example they get a question right. Maybe that example was the one they needed. Or maybe they just needed some time to organize their thoughts, or they needed more examples but that example alone was not more or less useful than another. Or maybe I saw they were confused but had to help someone else before giving my prize example, and another student explained it to them better than I did. If I do the natural thing and accept my assumptions, I am going to go on believing incorrect things about what is happening in my classroom.
I teach students, not classes. “They understand it” is my natural instinct after teaching a lesson — I want them to have understood it, I am looking for evidence that they have understood it, and I gravitate toward the simple, concrete answer. The answer instead, almost always, is that some students understood it, some didn’t, and a great deal are in the middle. Rather than generalizing about whether or not “they” “got it”, I would be much better off looking at some student thinking, really critiquing myself, and thinking about where I am going to go next that builds off of the very specific and nuanced things that each students knows or does not know.
I remember most new math I learn because I already know a great deal of math, knew math fits well into my prior knowledge, and I use math all the time. That is not the case for my students. Some things they forget quickly. Even things they understand well during class will be forgotten eventually if they do not practice it and continue to deepen their understanding. The intuitive thing to do is to say, “I taught it, they just forgot it” — and as a consequence, to review the day before the test, then move on forever. If I want to make sure my students retain and continue to build on their knowledge, I need to do the counterintuitive thing to constantly give students distributed practice and chances to rehearse and consolidate their knowledge in order to build effectively upon it in the long term, and check for understanding, no matter how well they understood it the first time.
Teachers are experts in math. We have a great deal knowledge chunked in ways that reflect deep mathematical structure but are not logical starting points for novice students, and we have mathematical skills practiced to a point of automaticity without conscious thought. This helps me solve math problems, but it does not help me help students solve problems.
In that multiplication problem, the 2 is carried above the tens place — but it actually represents 200, not 20. Have you ever noticed that? An English teacher friend who was helping a student with math homework saw it when a student kept making the same (fairly logical) mistake carrying. But math teachers don’t naturally notice these features — it’s not necessary for us in the process of actually doing math, and we have to work constantly to put ourselves in someone else’s shoes and consider where their perspective will take them.
This is not an exhaustive list, and I could give many more examples, particularly from my first year. These also aren’t insurmountable obstacles, and I have good tools to work against them. But doing this work takes constant vigilance, in both looking for and noticing what is happening in my classroom in all of its messiness, and constantly interrogating my assumptions, asking myself whether what I think is happening in my classroom is actually happening.