“What hidden obstacle could be causing my failures?”
My mental shorthand for this is the following experience: I try to pull open the silverware drawer. It jams at an inch open. I push it shut and try again, same result. I pull harder, it opens a tiny bit more before stopping.
Reflection: Some physical object is getting in the way of the motion. Something could be on the drawer track, but more likely it is inside the drawer. It is a rigid object, because I always stop at the same place, although slightly squashable because I was able to yank and pull a little harder. It is probably striking the inner wall of the cabinet in which the drawer is mounted. It is on an angle because I can’t see it when I look through the inch gap. There is a fork or knife angled up and poking against the inner wall. Digging around with my finger quickly finds a fork.
Since then, I’ve brought up this question by asking myself “what is the fork in the drawer”?
For example, my linear algebra students generally seem smart and attentive, but they become confused whenever I do a detailed computation with inner products. After some thought about which computations confuse them, hypothesize that whoever taught them basic matrix manipulations didn’t teach the “transpose” operator, and particularly didn’t teach the rule (AB)^T = B^T A^T. Fixed very quickly. (Of course, I also try to encourage them to ask questions about what confuses them, but I think that it is impossible to ever get a class comfortable enough questioning you to not need to think on your own about what is the underlying difficulty causing confusion.)
“What hidden obstacle could be causing my failures?”
My mental shorthand for this is the following experience: I try to pull open the silverware drawer. It jams at an inch open. I push it shut and try again, same result. I pull harder, it opens a tiny bit more before stopping.
Reflection: Some physical object is getting in the way of the motion. Something could be on the drawer track, but more likely it is inside the drawer. It is a rigid object, because I always stop at the same place, although slightly squashable because I was able to yank and pull a little harder. It is probably striking the inner wall of the cabinet in which the drawer is mounted. It is on an angle because I can’t see it when I look through the inch gap. There is a fork or knife angled up and poking against the inner wall. Digging around with my finger quickly finds a fork.
Since then, I’ve brought up this question by asking myself “what is the fork in the drawer”?
For example, my linear algebra students generally seem smart and attentive, but they become confused whenever I do a detailed computation with inner products. After some thought about which computations confuse them, hypothesize that whoever taught them basic matrix manipulations didn’t teach the “transpose” operator, and particularly didn’t teach the rule (AB)^T = B^T A^T. Fixed very quickly. (Of course, I also try to encourage them to ask questions about what confuses them, but I think that it is impossible to ever get a class comfortable enough questioning you to not need to think on your own about what is the underlying difficulty causing confusion.)