### Unqualified

In spite of…well, most of what I post here, I am confident that I have done a respectable job of working with my raw material (fundamentally uninteresting and inapplicable mathematics, students who should be in grade seven instead of university) this semester and last. When I emerged from a week-long isolation grading exams, I was able to step back and realize that a healthy majority of the papers I was grading belonged to students who had known nothing about the subject three months earlier, but who now had a working knowledge of it. Because of me. And the comments I got from students, for the most part, reflected that as well, not that I am above distributing course evaluations the day after a test, when my laziest and most obnoxious pupils can be counted on not to attend class.

My negative comments, unsurprisingly, came mostly from my precalculus students. Apparently, I am “too hard” and “too abstract”; I don’t “make the material interesting enough”, nor do I “relate it to real life.” (Of course, my attempts to make the material interesting and relate it to real life (see “word problems”, September 2004-present) meet with just as much hostility (see “too hard”, above).) Guilty, alas, on all charges; but I don’t see what I could have done differently while still teaching the material I was hired to teach. In any event, imagine my glee upon discovering, last week, that my successor is apparently a newly-minted Ph.D. whose area of expertise is - wait for it - *category theory*. Yes, the students who barely squeaked out of Precalculus 1 because they thought that equations were boring, abstract gibberish, have spent this past term learning Precalculus 2 under the tutelege of an individual who saw fit to devote his entire life to *actual* boring, abstract gibberish. *

Our resident category theorist (who I’ve previously referred to in this space as “Poor Sap”, a moniker that is actually more applicable now than ever before, for reasons I’ll get into as soon as I close these parentheses) will taking leave of Island U this month, after one brief term of employment. Reason: as it turns out, his category-theoretic ways were incompatible with the pedagogical needs of mathematical illiterates. Who’da thunk it? Perhaps both he and Island U would be happier if he were to find employment elsewhere, and we wish him the best of luck in his job search. Not that we’ll be providing glowing references.

Meanwhile, I also handed in my textbooks and keys ** this week. My time at Island U has also come to an end; but unlike Poor Sap, who didn’t make it through the mandatory probationary period that precedes an offer of permanent employment, I’d actually originally been hired for a single term, and managed on the basis of good work to get my contract extended for a second one. But that won’t turn into a permanent offer, because, well, I don’t have a Ph.D. in, for instance, category theory. Really - you can’t make this stuff up. Department Head and I parted on good terms: as far as he’s concerned, he told me, I’ve done Island U a huge favour this past year and he’d love it if I could stay on, but his hands are tied. He wishes me the best of luck in my job search, and I should know that he’ll be happy to provide a good reference whenever I ask for one.

And I know he’s sincere, and I feel badly for him, because now he has to go through the hiring process again. But feeling bad doesn’t get me a paycheque next month. Still, I reckon there aren’t droves of people out there who a) have Ph.D.’s, b) have the skills and the temperament to teach mathematical illiterates, and c) are willing to move to Island Town; so I wish Department Head the best of luck.

But I think that the most ironic thing to come out of all of this is that there’s actually a real-life use for a degree in category theory.

* Yes, I have studied algebraic topology and algebraic geometry, both of which make copious use of category theory, so I realize that I owe a debt to this bizarre field. I maintain, however, that category theory *talis qualis* is just so much abstraction piled upon abstraction, and I have no interest in entertaining arguments to the contrary in this space. That said, some of my best friends are category theorists.

** Oh, crap, forgot about the keys.

You need to move here and teach our mathematical illiterates. Among our current full-time teaching staff is at least one Canadian citizen (who is a dual citizen with some other non-US country) whose highest degree is an M.A.

California community colleges need competent math teachers with degrees equivalent to a master’s or above. If you have a Ph.D. in any math discipline, you could try for a position in the California State University (multiple campuses) or the University of California (ditto), except that at UC people interested in teaching end up in lectureships instead of professorships. Still, I know people who have been happy (well, not miserable) as lecturers at UC Davis for several years.

Good luck finding something soon.

Still, I reckon there aren’t droves of people out there who a) have Ph.D.’s, b) have the skills and the temperament to teach mathematical illiterates, and c) are willing to move to Island Town; so I wish Department Head the best of luckI don’t know. The NSF cut a lot of math funding this year and a lot of places lost VIGRE, so I’ve heard rumblings that not a lot of people are hiring postdocs so…you never know, canada could suck up all of our university-less, roaming new PhDs who aren’t getting jobs here…

Category theorist! No kidding! That is so funny!

(Am I the *only* non-mathematician reading here (and I have a math minor)? And, you know, searches which give results like “However, it is still evolving and the precise meaning of category theory, that is what it is in the end about, remains to be fully clarified” don’t help.)

wolfangel…I’m a waitress and I read, though I don’t know MS.

It seems this blog has attracted rather the large following.

The whole “you have to have a PhD to have a full-time teaching position” is probably the result of a couple things: accreditation standards, and intellectual snobbery. (The second is probably the underlying cause of the first.)

Now, I went to a bonafide research university for undergrad, so one understands why they might want to hire faculty with PhDs… However, they also had a couple tenured “instructor” positions. I’m not even sure they had Masters degrees. These tenured instructors were for the lowest level math classes around — some were lower level than that precalc class you taught. The dept. knew they had to get people who were good teachers for these classes, otherwise the students would never get through.

In math, it’s generally the case that the lowest-level classes are the hardest to teach. The grad classes are a snap to teach. Hell, you could just throw a book at the grad students, and you’re not about to hear them complain.

Anyway…. how about considering actuarial science? :)

I’m a geography grad student, but I like math. I’d never heard of category theory before you mentioned it.

wolfangel: My favorite definition comes from a nonmathematician: “Category theory is a ridiculously abstract framework that takes all the meaning out of mathematics.” (Quote from here: http://tinyurl.com/a63bz)

RH and TonyB - heh, thanks, but I’m staying in Canada. I’ve already sent out a bunch of applications, all in BC, but I haven’t head back from anyone yet. Which isn’t too bad, as I’d semi-planned to take this summer off…but I’d wanted to take this summer off and have something lined up for the fall.

Wolfangel and Mychelline - there’s no reason for people who like math to have heard of category theory (”unless they’re specifically studying it” I was about to add, but there’s no reason to do that, either.) More accurately, there’s only a reason to know what it is if you’re doing algebraic geometry or algebraic topology, as category provides a compact syntax for a lot of the ideas there. But, as far as I’m concerned, the whole subject is basically just glorified syntax, and the definition KimJ provides is spot-on. Category allows you to do things like study topological spaces via collections of groups they’re associated with, which help in classifying them. I can’t, however, for the life of me see what’s interesting about category theory apart from the fields it helps explain.

Meep - naw, I don’t think the actuarial life is the one for me. But I’m thinking that it may be time to put my original plan of, at some point, thinking about thinking about looking into doing some work for the military, into action. We’ll see what happens on the teaching front soon. I’d like to find more teaching work, but even if I find some, and it involves teaching university classes to innumerates, I know it’s only a matter of time before I burn out and need to find other work. (And while I agree that teaching lower-level math is harder than teaching higher-level math…the most challenging is teaching higher-ish-level math to students who can’t even do the low-level stuff. Man. That requires math skills (which I have), teaching skills (ditto), and the patience of Job (nope). A talented high school math teacher with a solid grasp of low-level undergrad math would do a better job than me at teaching precalc, though I still maintain it’s an impossible job.)

When I was in grad school (1977–1982), we students would occasionally sell books to each other we no longer needed. By longstanding tradition,

Categories for the Working Mathematician,by Saunders MacLane, was always sold for −$0.50.Would you consider teaching younger students? I wouldn’t have the patience.

I could suggest Cegep jobs, where you only need a MX, but you don’t seem inclined to move across the country.

Hmm, I know a few people who studied algebraic geometry in grad school. I will not bother to ask them about category theory! (Amusingly, one is now becoming an actuary while finishing up a PhD in linguistics, but hiding the actuary part from every single faculty member in the university; this person is also trying to convert other grad students to the actuarial life, because of those things called “jobs”.)

Eric J, is that

negativefifty cents? That’s a riot.I’ve actually taught younger students, but in a very different context - gifted high school-aged math geeks at an academic summer camp. Best damned teaching gig out there: marvellous kids who love the subject, crave knowledge, and are thinking about math even when they don’t “have” to. I’ve also mentored the coolest 12 year old kid in the world, and I’ve done some one-on-one with kids aged four to eighteen, of various levels of ability, and I’ve enjoyed it. But I don’t think I’m cut out to teach younger students in large groups - that requires someone more sensitive and more extraverted than I; I also don’t feel any overwhelming need to be part of the socialization process, which is part of the job description when you’re working with the younger set. (A high school English teacher I’m acquainted with once lamented to me that he spends 10% of his class teaching English, and the other 90% being a therapist to his students. Of course, working with adults is no guarantee that you won’t have to do that, but the subject-teaching:socialization ratio is a lot higher.) I’ve also found that my dry sense of humour, which is a subtle but important part of my teaching persona, falls flat with younger kids, so I don’t think I’d be that good at keeping their attention.

Speaking of jobs, I have a friend (from undergrad) who started a PhD here at the same time as me but bailed out after a year with an SM. Then he bummed around for a couple of years, teaching a couple of CEGEP and university courses, before starting law school two years ago (why law school? intellectual curiosity). But now he wants to get back into math starting in September…

This is awful news.

I hope that you get a job for the fall real soon.

Would you consider teaching high school math in California? We have plenty of mathematic illiterates to go around, and with your experience, you would start at $40,000 (US) plus benefits.

The law might be changed soon, but currently you would be tenured in 2 years.

Can’t blame you for wanting to stay in BC. The entire Wonk family (WifeWonk, TeenWonk, and myself) love Victoria (Vancouver Island, really) and have visited several times.

Do keep us posted on the job search and on your adventures during your summer off. This is, after all, the only place I can turn to on the net for relentless abstraction, curmudgeonship*, a strange, dark humor, and seeminly almost a good-natured tolerance of my own strange, dark comments. (Not to mention that most people can’t write half as well as you.)

*I made up that form of the word.

Wolfangel — I had a math minor, too. I use lots of math in my current career as a permanent graduate student, but it’s all applied stuff. So, like you, I’m not really a mathematician.

I’m pretty confident that I will be able to make a good living in my own province, and quite certain I can do same in my country of citizenship. All due respect to the US - lovely place to visit, but I don’t want to live there.

I reckon I’ll have plenty of free time this summer to keep blogging, wes. Don’t know about what, mind you…hopefully I can spend much of the summer near good hiking and a pottery studio that has decent hours. And there are the other eight major Gulf Islands that I haven’t visited yet. Seems unwise to go into a lot of details about the job search while it’s active, though; but when I find employment, y’all’ll know.

The feeling is pretty much the same from here in NYC — love to vacation various places in Canada (Quebec City, Vancouver), but I wouldn’t want to live there.

Unless I had about 5 homes. Maybe when I become CEO…

I think I’ll just echo the general sentiment here; best of luck on your job searches. Keep us posted, I guess.

Sheesh. Even *category theorists* refer to category theory as “abstract nonsense”. Poor Sap.

Best of luck hunting for a job. May you (finally) get the students you deserve!

I have a PhD in mathematics, and my dissertation was in algebraic topology — VERY heavy on the algebra side. The bulk of it had to do with generalized homology theory, which is primarily category theory.

The reason category theory is appealing to some (like me, at the time) is that it’s the ultimate plug-and-chug experience. There are no nettlesome questions about “what does this mean”; you just set up these equations with unbelievably bizarre notation and start chugging away. Although I knew (at the time) enough about category theory and homological algebra to know that my dissertation was a solid piece of work, I am convinced that one of the reasons my defense went so smoothly was that my dissertation committee was completely pummelled into submission by all that notation and it only partially had to do with the quality of the content. When I started my dissertation, I didn’t have to understand even a word of what my subject actually MEANT… I just had to be really good at keeping track of symbols as they passed through one equation after the next.

So I can see how Poor Sap could crash and burn in a precalculus course. If all you see when you look at mathematics is notation and number- and symbol-crunching, then that’s how you’re going to teach, and this is precisely what students at that level are worst at. (Come to think of it, the majority of students at ANY level of math short of PhD studies have a hard time with abstraction and using notation in a proper sort of linguistic way.) I’m very glad that I got out of the algebraic topology business shortly after getting my PhD and got into more pedagogical and applied areas; and I am in a good position to eschew the idea of mathematics as symbol manipulation.

The employment procedures of universities are hard to understand. The policy of hiring PhD’s makes sense in some situations and not in others. For a limited-term employment contract, it seems like the policy might be to accept all applicants and then hire the best teachers rather than the most highly educated (in terms of diploma count) person. We have two non-PhD’s in our department who specialize in working with the lower-level math students and with preservice high school teachers — and they do it far better than any of us PhD’s could. So I guess if could offer some encouragement, it would be that at least in the US, there are plenty of schools out there with more realistic policies for hiring well-qualified teachers on the college faculty, and you’re bound to run into one eventually.

Robert, your comments about not having to know “what [your] subject actually meant” captures exactly why category theory fails utterly to appeal to me. It also captures, alas, a big part of why I ultimately left of grad school: eventually it got to the point that math felt like so much symbol-pushing, completely removed from anything that ever made me love the subject in the first place. I’m a very intuitive thinker, and the idea that math ought to be taught in the linear axiom-lemma-proposition-theorem-proof progression offends me. To read most math papers, you’d think that mathematicians received the Holy Axioms from God Himself, rather than coming up with them

because the problems they were working on demanded new approaches.Back in grad school, I finally switched advisors in large part because my old one DID tend to approach math in the ALPTP order, and didn’t see why I was asking pesky questions such as “what is the significance of this theory” instead of cranking out some formulas. Consequently, I take issue with the idea that “abstraction and using notation in a proper sort of linguistic way” somehow represents the pinnacle of mathematical achievement: I know plenty of people who can crank out proofs from first principles, but who are also woefully banal thinkers and will hence never publish any mathematics of lasting value.

That said, Poor Sap’s precalculus students didn’t crash and burn because they’re not yet proficient in the higher-order cognitive function of pushing symbols around - that’s not what they’re “worst at”. They’re a lot worse at relating those symbols to Real LifeTM. Ask them to find the ages of two children if they sum to 10 and have a difference of 2, and most can’t.

MS, I teach a course called “Methods of Problem Solving” (MOPS) in which the very thesis of the class is that problem solving is NEVER done in a linear way– that problem-solving demands creativity, the ability to understand the meaning and significance of problems in multiple ways, and curiosity. I designed this class and have taught it for four years and it’s largely an effort of mine to PREVENT people from going down the same sort of symbol-crunching path that I went down and, mercifully, got off of. This is actually the approach I take in ALL my courses, even the “liberal arts math” course I teach (math for poets) because I am convinced that math refuses to be locked inside the linear progression that you mentioned.

One thing I always stress in MOPS and other courses is that the way you read math in a textbook is NOT the way the math actually came about — there, it’s stripped of all the sweat and failed experiments and discoveries that went into it. The real process of mathematical discovery is organic, unpredictable, and uncertain. But this is what also makes it exciting and worth studying. I was fortunate enough to have professors in grad school who understood this and did what they could to get me to question the meaning of what I was doing and undertake math as a means of discovery. It was ME who wanted to just push symbols around. Years later, I found they were right and I was wrong, and I’ve been a lot happier since then.

So I do not hold that symbol manipulation is the pinnacle of anything — but I do think that being able to take a problem, find the mathematical concepts in it, and then translate those into symbolic form is a sign of very good mathematical thinking. And using notation properly is important. But not the most important thing.

When I say that symbol manipulation is what my students are worst at, I do mean that they’re not very good at it, but I also mean that almost every other problem or deficiency they have can be traced back to the idea of “mathematics = symbol manipulation”. They have problems thinking conceptually and applying what they learn to new situations because the first thing they want to do is throw algebra at the problem without thinking the problem through first. They’re not necessarily really good at algebra or symbol manipulation, they don’t enjoy it, and yet they refuse to let go of the algebra and think of problems in more creative ways. I think this is because occasionally, throwing algebra at the problem *works*. It’s just enough reinforcement to keep them coming back to algebra, despite their lack of skill or desire for it. Don’t we usually call behavior like this an “addiction”?

Anyhow, I can tell that you’re the kind of person who wants to inject the ideas of wonder and creativity and thinking into your math classes, and those sorts of teachers are rare and great. Students may not realize it, but they are getting quite a good education under a teacher like that.

“…eventually it got to the point that math felt like so much symbol-pushing, completely removed from anything that ever made me love the subject in the first place.”

I, too, am unfortunately largely feeling the same way (as I complete my masters level degree at Cambridge)…

You know what I have trouble understanding? The kind of people who become academic administrators. What kind of person would want a job in which he wants to keep a valued employee, but can’t, because his “hands are tied” by irrelevant policies? What possible satisfaction could there be in such a role?

Yeah, there’s stuff like this in business also; way too much of it; but at least people who are managers usually have *some* opportunity of making actual decisions as opposed to merely being human policy-following engines.

David,

If the administrator is a department head, as it was in this case, it may not be a matter of “wanting” the job so much as just ending up with it. The role of department head in many places is just a regularly-rotating chore that each faculty has to do at some point.

Robert…good point. “Chore” would certainly appear to be the correct term if the job has so little authority…

Not only does Department Head have to follow inane rules that he doesn’t agree with, he also has to do all of the work that no one else wants to do. Guess who’s teaching precalculus in the fall?

MS: I wonder if you’d have been happier if you’d tried to do a math Ph.D. in…any other department than Math. In Engineering or Bio or Statistics or Economics or Theoretical Computer Science (my field), you can focus on cutting-edge math in an environment where most people are only interested in the math if it has “significance,” and only a few freaks are interested in symbol-pushing and summitting the heights of abstraction.

I love to spend my days proving theorems, but I don’t see any reason I ‘d want to do it in a math department.

Hunh, never thought of that, though I’ve thought of finding work outside of academia that required me to do real math. Might go back for a Ph.D. one of these years, but not in the near future and not where I did my Master’s, that’s for sure.

There are interdisciplinary PhD programs popping up all over the place. University of Chicago, for one, has been offering a PhD in “financial engineering” which is sort of math/cs/economics hybrid. Not up my alley, but certainly has appeal to some.

I have a former undergrad advisee who got double degrees in Mathematics and Biblical Literature, who is now finishing up a PhD in “computational linguistics” at Indiana University. He had to become fluent in Hebrew, Greek, and Aramaic for his Bible degree, so he’s an expert at language; and about half of what he’s doing now turns out to be either stats or semigroup theory. He loves it.

Math people also tend to do extremely well in law school and med school. Speaking of the law, I recently supervised one of our math majors doing an independent study in technology law, and I found out that intellectual property law is very demanding from a science and technology standpoint, and it’s a wide open and extremely interesting field of law. Somebody with advanced mathematical training would be perfect for that kind of law, and since there appears to be hardly anybody out there with advanced math training AND a law degree, the supply of IP lawyers is low and the demand is high. (My student is going to work for a while after graduation and then try to get into the IP law field.)

So yeah, there are a lot of interesting opportunities out there that aren’t just sitting around proving theorems about contravariant functors all day long.

Carnival of Education, Week 13It’s here, the latest of Carnival of Education. Once again, the contributions remind me how many great ideas and great writers there are in the Edusphere. These submissions are elegant, excellent, touching, and enlightening. Enough of me blathering, …

In a strange twist of events, Perl 6 development has benefited greatly from Haskell and it’s use of monads.

http://www.perl.com/pub/a/2005/03/03/pugs_interview.html

http://www.engr.mun.ca/~theo/Misc/haskell_and_monads.htm

I’ve never seen the ultra practical camp benefit so much from the ultra theoretical. It’s like one of those ‘the day the universe changed’ type of things.

Oh, man. That’s nuts. What are all the category theorists going to do now that they know that the nonsense they’re dealing with is not so abstract? Why go on?