Was thinking about an exercise in a textbook (@emilyriehl's Cat Thy in Context). It referred to a def given in a previous exercise, which I had "already done." Nonetheless, my understanding of that def was clearly insufficient to the present purpose. 1/
After being stuck for a while, I finally asked myself to build some sort of visualization of the previous definition. Once I found a mental image that worked for me, the answer to the exercise was clear. But it struck me that nothing had prompted me to do this beyond being 2/
stuck for a while and having a (frankly, inconsistent) habit to search for an appropriate visualization when stuck. Neither the metacognitive skill of remembering to search, nor the concrete skill of how to go about building one's own appropriate visualization, were ever 3/
the explicit object of my (20+ years of) math education, but they were both critical to my success here. This was only visible to me because I'm me. If you were trying to help me, how would you have any idea what I needed or how to help me find it? 4/
When a student "isn't getting" something, we often have very little info about what's actually happening in their mind. Often they are reasoning totally coherently but about something different than we think we're talking about. Or sometimes, like the above, 5/
they need (I needed) a mental image of their own, different from anything we've said (or drawn, etc), but don't know they need it. How many times have we accidentally made a student feel dumb for being in this position? #iteachmath 6/
A takeaway: teaching is curiosity about what's in students' heads. (Note to self: go beyond asking students to explain themselves. Probe their mental representations and images. "Can you draw a picture of what you're thinking?") 7/
Another: math accomplishment is as much or more metacognitive and learning skills (when stuck, shift gears in X way; how to direct your mind to do X) as content skills. (NB: this does *not* mean "teach problem-solving skills instead of content." Not going there.) 8/8 #iteachmath
