For all students in my class there exists a chance to learn. True or false? This week the topic was universal and existential statements. How to prove them true; how to prove them false. How to negate them. How common they are in advanced mathematics. I also returned two assessments: one quiz and one homework–the results were bimodal: students doing well or not well at all–this bimodal distribution of assessment seems to have replaced the normal curve–if it ever existed in graded assessment.
In informal assessment I found the normal distribution, or I imagined it. From responses to my questions during lectures a small number of students appeared to be fully engaged. From posture and distraction another small number of students seemed bored. The majority of students seemed to be listening and watching and taking notes–I don’t know if they were transcribing or interpreting–it might be nice to have an observer in the room.
In preparing the lesson I personally gained a deeper understanding of the universal quantifier as a short-hand for a conjunction of multiple predicates and the existential quantifier as a short-hand for a disjunction of multiple predicates. It may be possible to do more with this understanding in my next round of teaching discrete.
Universal instantiation and existential instantiation–the foundational ideas of general rules and particular instances. These concepts seemed excessively formal when I reviewed them, but as I saw them on the board they gave the impression of great stones of support for the mathematics we use every day–solid and necessary, but best left submerged from view.
Several students dropped my class this week–the drop-without-grade date is Sunday. They sent no messages to say why. Almost all of them took the first two quizzes and submitted homework. Most did well enough. Perhaps the amount of work required may have discouraged them–I collect and grade homework weekly; I quiz weekly. Did some of them dislike my method of teaching? I had to force myself to write that last sentence.
Some students came to office hours to have me solve their harder homework problems. There is a tension in teaching between scaffolding and challenge. Some faculty see too few failures as low standards. Some faculty see too many failures as improper support. My current thought is that a teaching moment is an experiment that has no control. All lessons learned about teaching are anecdotal. If there are universal rules can they ever really be true for all particulars?