Hewitt response

Hewitt does an excellent job of putting into words certain ideas that I've only ever thought of on a surface level. I appreciate the distinctions he makes between aspects of math curriculum and the ways in which students interact with those aspects. My understanding is that "arbitrary" refers to terminology, notations, conventions, and other choices about how math is actually physically done. These need not be the way they are, but are mostly propagated due to convenience and perhaps some long forgotten historical basis. I really like the idea of addressing this aspect of math with students. For starters, I think math is more interesting when it seems less like a set of seemingly pointless rules and viewed more as a creative endeavor. In practice, the arbitrary must be taught by teachers for students to memorize, so they're able to actually communicate and interact with the standard mathematical landscape. In contrast, what is “necessary” is more so the mechanics, processes and results of mathematical reasoning. These ideas can also be taught in such a way as to be simplified to seemingly arbitrary methods and algorithms, but they are more so necessary in the sense that they represent a set of idealistic learning goals which focus on students' relational understanding of the material. On that note, there is definitely some similarities between the dichotomies of arbitrary vs. necessary and instrumental vs. relational. If I were to try and combine these ideas (as I understand them), I would start by saying that necessary content can be taught either relationally or instrumentally. However, the ideal situation, at least in my mind, is that necessary content is taught relationally. When it is taught instrumentally, the necessary is reduced to the arbitrary in the eyes of the students, making demands on their memory instead of engaging any kind of deeper awareness. Again, I think the perfect-world image of a math class involves students deriving necessary ideas from their own relational understanding of the curricular content, perhaps making use of their own personal set of arbitrary conventions. In practice, I think it’s understandable to sometimes focus more on instrumental teaching, and blur the lines between what Hewitt might consider either arbitrary or necessary. He raises a good point about how students may or may not be willing to engage with the more creative aspects of mathematics. For example, they may have no interest in distinguishing between the necessary and the arbitrary and may just consider the subject as entirely arbitrary, only valuing an instrumental understanding. To be honest, I think that’s also probably the prevailing student opinion, which is a failing of math education. As it stands, for practical reasons such as covering a course-load’s worth of content and maintaining student engagement, teachers definitely need to balance their lessons in terms of the degree to which the necessary and arbitrary are separated, oscillating between relational and instrumental teaching. It also occurs to me that what is considered necessary or arbitrary is at least slightly subjective and may differ slightly from one teacher to the next. I would also add that even the arbitrary can be taught somewhat relationally, providing some rationale and historical context for the choices which have been made about math over the years. I think part of an excellent math course involves equipping students with the tools to do math, but also giving them ideas about how to develop their own mathematical tools. I would argue then, that the idea of the arbitrary existing within any mathematical framework, is in itself a necessary idea, and worth not only pointing out to students, but also celebrating.