The thing is, I'm "bad" at math, but I'm good at doing things I'm "bad" at. I should probably explain what I mean here.

To be "good" at something is to fully internalize the fundamental mental tools of a discipline to the point where the conscious mind can simply take the tools for granted.

For example, I'm "good" at expository writing. Although I'm always making refinements and improvements, I have fully internalized the fundamental tools of grammar, punctuation, and spelling, as well as paragraph and larger-unit organization. I don't have to consciously

*think*about any of these elements; most of the major cognitive work has been moved to my subconscious. When I have an idea, it just "appears" in my conscious mind in properly constructed sentences and paragraphs. My subconscious is not perfect, and I do of course still have to think consciously about writing, but 90% of the work happens in my subconscious. Internalizing these low-level tools is not

*sufficient*to be "good" at writing, and it is possible to write well without these tools, but internalizing the tools makes writing well consistently and frequently easier and more enjoyable. I can spend almost all of my time thinking about the subject matter, rather than the presentation, and when I think about presentation, I can focus on "higher-level" tasks rather than struggling to make sure each sentence is grammatically correct. Similarly, I'm "good" at computer programming, and I've internalized the fundamental syntax and organization of computer programming, freeing my mind to think about "higher level" work.

(Note that my dry, abstract, and somewhat dense style is by design. I write what I like to read.)

My facility with writing is not a matter of "talent" or innate ability, except to the extent, I think, that I "innately" enjoy reading and writing. Basically, because I enjoy the subject matter, I enjoy practicing to gain these low-level skills. I don't have to "force" myself to read or write, and I don't have to "force" myself to write computer programs.

In contrast, I'm not "good" at math because I haven't internalized the fundamental mental tool of mathematics, which is ordinary algebra. I

*consciously*know algebra, but I haven't internalized it. I could, I suppose, but unlike writing and computer programming, I don't innately enjoy algebra. When I get a difficult algebraic problem, I have to force myself to solve it, and if I can use a crutch, like a computer-aided algebra system, I will do so without hesitation.

I don't fully agree with Doron Zeilberger; I don't think mathematicians should let computers do all of the algebra (including the "algebra" of integrals) precisely because so much of higher math seems to involve "creative" algebra:

*seeing*the "hidden" algebraic relationships necessary to solve complex problems; Seeing these "hidden" relationships requires internalizing the low-level algebraic mechanics. In writing, "seeing" how to express a complex thought requires internalizing the low-level mechanics of grammar. (Again, you can express a complex thought without internalizing the low-level mechanics of grammar, but it's much more difficult to do so: you can't just "see" the correct expression.)

I'm good at doing things I'm bad at because 90% of most interesting endeavors is seeing patterns, and I'm "good" at seeing patterns, precisely because I enjoy looking for patterns and practice a lot. I can do most anything that isn't pure "muscle memory," precisely because I'm good at picking up on the patterns within a field. But if I don't enjoy actually acquiring the muscle memory, my progress is limited: I won't practice. I'm not a big fan of "discipline," practicing things I don't enjoy doing for the sake of internalizing the fundamentals. I'd rather spend my time practicing things I actually do enjoy.

I'm just finished Calculus III (multivariate calculus), and I'm

*done*with math as math. Calc III, at least as I've been taught, is 1% generally interesting patterns, 2% patterns interesting to physicists, and 97% grinding out algebra. I'm not really complaining; Calc III is the gateway to a math degree (and most STEM degrees), and intensively practicing algebra enough to internalize it is absolutely necessary. You need to either really enjoy doing algebra, or have enough discipline to practice it anyway. But I have neither enjoyment nor sufficient discipline to continue.

I wouldn't change that math majors really should practice algebra continuously; they

*need*it. Still, there are a couple of things I wouldn't mind seeing in math instruction.

First, it would be awesome to have a math track for people who don't do math as math; focusing on the higher-level patterns in math which are (even if the underlying algebra is tedious) amazingly interesting, beautiful, and incredibly useful. A lot of different fields, including economics (and, to some extent, political science), can use a lot of higher math without having to actually grok the math as math.

The second thing I'd like to change is how math is taught in economics. Just as a political scientist, not to mention a ratical revolutionary communist, the pretense that economics is not a

*normative*discipline is ridiculous. To uphold the pretense, economics has retreated into math; "economists" just prove mathematical theorems that they suspect might have some tenuous relevance to how people produce, distribute, and consume goods and services. I have been advised many times that if I want to get a Ph.D. in economics, it's nearly useless to study undergraduate economics; top grad schools would rather have candidates who are great mathematicians who know little to nothing about economics than people with a deep understanding of economics with less than the most excellent mathematical skill. I have just enough discipline to master enough math to get a Master's in economics, but I can think of few endeavors I would find more boring and pointless than to do what passes for economics at the Ph.D. level. If I'm offending any of my current or future professors or advisors, oh well; they will have to console themselves that they are at least ensuring that a future political scientist and theorist with not be completely ignorant of the structure of capitalist economics.

I'm not saying that economics should not use a lot of math. Math is an extremely useful language for talking about the world. I am saying, however, that unlike mathematicians, and perhaps unlike physical scientists, economists do not need to be "good" at math. Being "good" at math is, I think, useful for purely

*descriptive*fields, but economics is normative (and anyone who tells you differently is trying to sell you something). All the problems in economics that I find interesting are not about finding new ways of describing the world in rigorous mathematics, they are about looking at how our social relations interact and intersect with real economic behavior (producing, distributing, and consuming real goods and services). (Hence I'm more-or-less a "Marxist.") Math is useful, but not fundamental. It's more important to be "good" at economics, to internalize thinking about real economic behavior, than to be "good" at math.

But the world isn't as I wish it to be. Fortunately, there's enough wiggle room in the system that I can educate myself in what

*I*want to learn while still doing what academia wants me to do to gain the credentials that I need.

**ETA:**I've since improved my math.

I'm bad at writing for much the same reasons you are bad at math. I don't like the idea of studying grammar one bit but I recognise that my eduction in it is almost non-existant.

ReplyDeleteI don't recall ever being taught it in school but I will confess that I might just not recall it as I was spectacularly disinterested in school.

At this point, I don't really know where to begin. This might just be an excuse because I don't want to begin!

One might not be good at the "procedure" (things we take for granted) but I think there is a "deeper" understanding of a topic. Deeper meaning understanding what is actually going on in the process. What are the proofs. What is the historical context of mathematical discoveries. Is it discoveries or invention by the way? Classic question. I've been thinking lately that math may be as close as we (humans/people) can get to coherently describing a theory of knowledge; as in, math is the foundation of human knowledge. I don't think that math is "transcendental", but I'm curious as to what your perspective might be on the role of math in knowledge and truth (or Truth^TM). Its an idealization but it seems (to me) to be foundational for science and philosophy (philosophy that I like that is). I'm thinking that maybe math is a system that relies on axioms that is the idealization of our limits as limited beings and not something that exists in its own right.

ReplyDeleteAnyway; I hope that this post is of enough substance for this blog. I don't doubt that you are a math teacher at heart; I'm not implying that a non-response would equate to the negation of that perspective by the way. Maybe I'm too reliant on this blog for informing my own opinion and I apologize for taking advantage of you that way (as I do it). eergh, I really want to post questions though...

Sorry for the late reply, Mon.

ReplyDeleteOne might not be good at the "procedure" (things we take for granted) but I think there is a "deeper" understanding of a topic.I'm not so sure those two things are cleanly separable. They are different: one can certainly have a mechanical grasp of a procedure without understanding what's really going on. However, I think that the procedural knowledge, which is really just about seeing patterns in the math, is required for a deeper understanding; more importantly, a deeper understanding doesn't advance one's knowledge until the procedures are well understood.

Classic question. I've been thinking lately that math may be as close as we (humans/people) can get to coherently describing a theory of knowledge; as in, math is the foundation of human knowledge.I dunno. I don't know that math is any kind of knowledge. It seems like a useful tool when gaining real knowledge, i.e. empirical science, but I'm very post-modernistically skeptical about math as knowledge per se.

I hope that this post is of enough substance for this blog.But of course,

mon ami.Maybe I'm too reliant on this blog for informing my own opinion and I apologize for taking advantage of you that way (as I do it). eergh, I really want to post questions though...What do I write for

butto try to help inform people's opinions!? You have nothing to apologize for, and please feel free to post any questions you might have.