What follows are my notes on the Hyman Bass article from 2011, “A Vignette of Doing Mathematics…”, about the slices of cakes.
1. His description of his internal process of working on a problem comports with my experience of my internal process of working on a problem. IT’S LIKE HE IS IN MY BRAIN.
2. He seems to imply a salient difference between an expert practicing their discipline and a novice learning a discipline. (Especially when he says “In doing mathematical research, unlike school work, we don’t want to expend great effort trying to solve a problem that has already been solved, (unless our intention is to find a simpler solution or proof). So, once a question we confront resists our first serious efforts, it is wise to consult the literature, or expert colleagues, to find out what is already known about the problem. This is also appropriate in school mathematics if working on an open-ended and long-term mathematical project. Mathematics is a hierarchical subject [NB; hierarchical!], and we don’t want to constantly reinvent the wheel. But of course this means learning to interrogate and learn from the expert knowledge of others.“). This view (expert vs. novice) resonates with me. There exist activities that are appropriate for trained mathematicians to engage in while doing mathematics, while those same activities may or may not be appropriate for a novice in the service of learning mathematics. I’m thinking in particular of the definition sorting activity (of course).
3. The initial problem, “If c cakes are to be equally shared by s students, what is the smallest number, call it p = p(c, s), of cake pieces needed to make this distribution?”, initially seems a good candidate for a middle or high school complex instruction task. Not sure about groupworthy-ness, just because I think I would want to do this problem on my own… at least until I got stuck. Once Bass started talking induction though, I became doubtful that this problem might be effectively used in HS, as induction generally isn’t covered in HS.
4. When Bass says, “Consulting the literature in pursuit of the questions above was the occasion for learning some very interesting mathematics”, I am struck that this is the first time I remember him mentioning that he learned mathematics. He talks here about the “learning” of mathematics to occur when he consulted the literature, not before… not in the “doing” of the mathematics… he doesn’t “learn” math by exploring and proving his conjectures, connecting representations, etc… he “learns” it by reading about what others have done. I searched the article for the word “learn” and Bass mentions mathematicians learn mathematics by doing a google search! When he’s doing his proof thing, he does refer to assertions proven in the paper as “what we have learned”. Interestingly, not “what *I* have learned” but that could very well be a result of the quirky style in which mathematicians tend to write.
5. I thought about our task debrief questions, (what math might students learn, what math do they need to know), for the problem quoted in 3.
What math would they learn by doing this problem?
Unclear… they prove a theorem, so they’d learn that theorem. They might practice GCD, LCM, fraction arithmetic, Euclidean algorithm, depending on solution path.
What math do they need to know?
GCD, LCM, fraction arithmetic, Euclidean algorithm
There’s been this ongoing tension for me about what constitutes math (obvs), what we might mean when we say “learn math”, and how that might differ, if at all, from “know math.” Learning is the process by which we come to know? If I know it, then I don’t need to learn it… the desired result of learning has been achieved if I know it, I know longer need to continue to learn it.
6. What CCSSM content standards might the problem in 3 address? I have concerns that precisely because this is such a rich problem, that its utility for assessment or teaching is limited. (Whatever the heck we mean by teaching, and whatever relationship that might have to learning and knowing. Oh bother). Here’s the sentence frame I want to fill with math *content*, but have been unable to… By completing this problem, students will be able to <blank>. I can put any number of the mathematical practices in there, but I want my students to know the content, that’s priority one. What’s the point of a math activity if we aren’t learning math content?
7. Also, definitions! Bass needs a thing, he defines a thing. No correct or incorrect definition, just choices, then proofs.