## Math teaching needs overhaul

My friend Tara sent me a message:

The President’s Council of Advisors on Science and Technology submitted a report on the challenge to producing more college graduates with STEM degrees. In particular, they point out mathematics as a bottleneck, and recommend (on p. 29) that “teaching and curricula [be] developed and taught by faculty from mathematics-intensive disciplines other than mathematics, including physics, engineering, and computer science.” Of course, there are physicists, engineers, and computer scientists on the Council, whereas there is no mathematician.

On some level, they do have a point. They seem to say that we (as a nation) are not doing a good job of teaching K12 mathematics. I strongly disagree with their conclusion that we should therefore take the college-level teaching of mathematics away from the experts in mathematics.

Hmm. I don’t know. I’ve been sounding a warning for a while now that math departments are way too complacent about the way they teach undergrads. I try to make people think of a math department as, to some extent, a brand, and that we should be trying to attract good majors and we should be trying to get more people psyched about math. To that end I am constantly trying to get people to care about the calculus curriculum, which is always at risk of being taken over by the physics, engineering, and economics departments, and I’ve consistently introduced “introduction to higher math” courses which explicitly teach proof techniques.

But there’s a major problem, at least in the very top research departments. Namely, the professors actually think math *should* be a hard and elite major, and that gives them an excuse to not care about the quality of the undergrad classes. That’s not how they say it, of course, but my experience is that’s how it works.

The other reason I think it makes sense to be a bit concerned about the brand is that if we mathematicians don’t start doing it, then someone else will start doing it for us. This President’s Council of Advisors report is exactly saying that. On the one hand it could be the kick in the ass that math departments need, but on the other hand considering how much reporting they are asking for, it could mean a tremendous amount of paperwork as well as a loss of independence of the math community.

I say mathematicians respond to this by admitting there’s a problem and coming up with a good plan that they organize and control. Otherwise I do think something else will and should be done.

Interestingly, there also seems to be a call in this report for more good math tutors. It reminds me of a commenter from yesterday who wants to start something called “Tutor for America”, which I think is an excellent idea.

I’m quite pessimistic about the situation. And, by now, my view is the problem doesn’t start with calculus or the undergraduate program. It starts with the *graduate* program. I don’t believe we even train most of our Ph.D.’s properly, and it only gets worse as we work our way to earlier years. And I maintain that if we start by training our graduate students properly (remembering that most of them will *not* become research mathematicians but will work either in industry or teaching positions), this will already improve the situation you describe. But I’ve never been able to get anybody to take my views seriously.

When I was at Princeton the department was very concerned that the math major was too small and elite and contained only those kids who were planning to go to math grad school, so they formed a “Broadening the Math Major” committee, which resulted in me developing …. “introduction to higher math” courses which explicitly teach proof techniques. So I don’t think top departments are blind to the need for this sort of thing!

Not completely blind, no. But on the other hand, they had a post-doc develop one course. Not earth-shattering.

Not that I think it’s all their fault, I don’t. I don’t even think they have *that* much power over the underlying cultural problem. But they could be doing more.

Our department is small enough that we cannot teach sections of our service courses tailored to disciplinary needs, so I recently tried to drum up interest/support in applying for a grant in which we would develop disciplinary-specific modules in collaboration with other departments. I could get no traction, partially because my chair is concerned that I get more papers before I go up for tenure, but also because our curriculum committee is concerned about a loss of control of our courses if we engage in this dialogue. We are lacking in manpower, so would need real substantive involvement of engineers, etc.

I also know universities where the engineering departments have taken back all of the math courses to teach them internally because the math department was not giving students the tools to apply the theory. In other words, this “experiment” that the letter proposes is really not much of an experiment at all! I am not sure why they don’t just study the programs where this is already the case and determine whether there is a change attrition rates stemming from the mathematics courses before throwing more millions of dollars down the math ed grant rabbit hole. I suspect that there isn’t, but perhaps I am wrong.

I don’t personally see a problem with people from other disciplines teaching the math courses if we are doing it badly. The problem that I do see is those courses being watered down so that more people will take them. A friend of mine in the business school regularly complains that her students cannot understand basic rate problems. Our business school removed the math bottleneck to the degree by dropping calculus as a requirement!

Really good points! I think you are right to ask why they haven’t already seen what the STEM rates are for those schools.

This post is a bit lacking in focus, since it mentions the problem of K12 math education, the problem of college math service courses for other disciplines (i.e. the calculus curriculum), and the issue of attracting math majors. To me these are extremely disparate issues.

For example, you mention “intro to higher math” courses, which I think are great, but they are probably the exact opposite of what other disciplines want from us.

The calculus curriculum is really the biggest educational challenge facing the mathematics community. I think that most mathematicians (including myself) think that we’re doing it wrong, but what should we replace it with? The reform calculus movement was supposed to answer this question, but a lot of mathematicians hate it even more than traditional calculus. It’s easier to stick with the status quo when it’s so hard to build consensus for any specific change. And of course, the scope of any potential changes are limited by the broken K12 curriculum.

As for the idea that majoring in math is a little too “elite” at top schools, I’m not sure that this is such a big problem. In fact, it probably leads to higher quality classes at those schools, which is a tradeoff that shouldn’t be totally ignored. In any case, this problem has only minor significance to the big problem of nationwide STEM education.

Final thought: I think the only real solution to the problem of increasing the number of STEM graduates in this country is the economist’s answer: higher salaries for STEM graduates.

When “pure” math seems to stagnate into a state in which grad profs are writing textbooks aimed only at each other rather than at undergrads, then “applied” math departments tend to dominate the novel and ground-breaking teaching and techniques.

Such incestuous textbooks have actually affected the originality and utility of the bleeding edge, e.g. it took years of popular use in electrical engineering for the Kroniker delta to be accepted by purists only when they could finally explain it with new proof approaches, and more recently only computer techniques have been accepted to “prove” long intuitively-accepted theorems.

Also more recently, incestuous textbooks have seriously degraded the viewpoint and use of probabilist statistics, encouraging the downright wrong-headed acceptance of bad (if not fraudulent) use of medical, financial, and business “probability theory” in corresponding academia departments (especially at the “prestigious” Ivy League level) that are desperately trying to establish hard science credibility.

It seems to me that the pedagogical use of hypertext textbooks could make incestuous texts quaint if not obsolete. The same digital book that first introduces basic math concepts could take a student all the way through graduate research level disputations without cluttering a paper version with distracting, boring if not maddening footnotes to satisfy the purists.

You shouldn’t treat ‘math departments’ as a whole, since there is some incredible variation from school to school – some departments care more than others, some departments see their mission as support of engineering and the sciences rather than producing graduate students (or even mathematics majors) of their own. Some departments have a focus on the teaching of mathematics (as opposed to mathematics itself) – it really depends on the school.

On the other hand, I (speaking as someone who’s taught undergraduate mathematics for several years now) notice that the primary indicator of success (of my students) is their ability to handle basic algebra, a weakness in trigonometry comes second. I also find that those students weakest in algebra are often weak in basic arithmetic as well (what do we do about the problem of students who are unable to perform the subraction 5-25 correctly?)

This kind of discussion often moves in the direction of just what it means to have learned mathematics – and I have had three faculty tell me that I should not expect any student to be able to do basic arithmetic, that they couldn’t (which I had trouble believing) and it was unimportant in the teaching of mathematics. (Clearly, I disagree.)

As for your friend’s note – I have anecdotal evidence that supports the Council’s report, one of my students told me of a friend of hers that decided against a STEM major specifically due to the difficulty perceived with learning mathematics. I think that *this* is a major issue, and the response (which I see across the scientific disciplines) of trying to pretend that research is a game, that the day-to-day work of scientists and mathematicians is somehow ‘fun’ and ‘exciting’ does a disservice to both our profession and to the students. *I* find mathematics fun, I enjoy fiddling with a problem and figuring out the answer – not everyone does, nor should they. On the other hand, I firmly believe that, fun or not, both science and mathematics as problem-solving methods *work* – and furthermore, they work for *anyone*. I make an effort to point out to my students that the bulk of the difficulty they have in mathematics is an over-reliance on calculators to do the work, and a lack of attention to detail coupled with weakness in the basic skills of algebra and trigonometry (which are built on weak foundations of basic geometry and arithmetic).

I belong to several linkedin groups frequented by maths educators, for example http://www.linkedin.com/groups/Math-Connection-1872005 The ‘broken K12 curriculum’ in the US and ‘maths phobia’ in students, parents, elementary school teachers are recurrent themes. Also the emphasis in schools (not university/college – I am talking about children) on calculus and ‘Maths for understanding physics’ rather than statistics and ‘Maths for understanding decision making’ is of interest to maths teachers, who feel that maths should be for everyone, not just future STEM majors.

I do think that improving maths teaching in early the early years is the place to start; I suspect that if, in general, the undergraduates were more ‘maths savvy’, undergraduate maths teaching would up its game to keep up, almost by default.

Speaking as another math educator with a number of years at the college level, I strongly agree that this is the fundamental problem. It’s not sexy, it’s not glamorous – but there’s a reason for all that soul-deadening drill, drill, and drill some more in the basics that used to be de rigueur in K12. Just as in music teachers insist their students practices even so humble a scale as the C over and over and over again, contrary to the legions of kids protesting (rightly) that they already “know” it.[1]

As it is, my students brush aside any, say, trig review in the calculus series as being for dummies or at any rate, beneath them. And then – of course – they are unable to solve integrals by the technique of trig substitution in any but the most mechanical and stereotypical fashion.

[1]In fact, I blame the parents of the children. As math-oriented people, we all know that the “secret” to being good at math is through practice, practice, and yet more practice. Pretty uncontroversial, right? Yet parents these days seem to shrink from forcing their children in grades K12 to do this sort of out-of-class drill. They apparently think we should be burning up our one allotted hour a day with their kids doing this so that they don’t have to – never mind that one hour a day three days a week just doesn’t cut it in the first place.

Sign me off as “disgruntled math teacher”.

I was a math major in high school (1970) because my father said that’s what I should do. I had the great luck of having great teachers, and of having text books that were elegantly and intelligently laid out (as opposed to the books my kids had, which were a total mess of distraction). The problems I see with math teaching in k – 12, right now is

1. Teaching stuff too early. The capacity for abstract thought starts in mid teens or later, but it’s now all the rage to teach as much math as possible as early as possible, which tends to hurt boys (who mature later) and everyone who decides they’re not smart enough to do math. Calculus in 11th grade is quite normal now. In my generation, Calculus in 12th grade was exceptional. My dad took it in college, and he studied engineering.

They teach geometry so early now (9th grade), that they no longer teach proofs — cause kids are too young to handle them. A great, great loss.

2. Math is presented as a hoop to jump through to prove how smart/ambitious you are rather than a subject with a history, wide ranging applications, and a place for the imagination.

3. For some reason Calculus is a sine qua non. I’m not sure why. I mean, it’s just a technique. Why can’t it be taught when it’s needed? What I wanted to learn more than anything else was how to think like a mathematician, how to have fun with problems, but nobody taught that. Not even the good math teachers. Why does it have to be Algebra, Geometry, Trig, Pre-Calc, Calc? Why can’t there be logic? and topology? and number theory? and set theory, and groups…..?

4. The only thing worse than k-12 math teaching is college math teaching. I took vector calculus and linear algebra at UCLA and nearly died of boredom. No teacher or TA took the slightest bit of trouble to explain why any of this stuff might be interesting/useful/curious/beautiful….etc.

I gave up on math in my second year in college. Then I got a Ph.D. in English, and now I write programming books.

I did see that math was beautiful, but I wish I had met one teacher, just one who could have shown me why.

I think that the “calculus in 11th” grade thing is produced by pressure for entry into elite universities, not from the populations’ generally increased mathematical ability (so that is related to your point #2). But like you I’m a bit baffled by the calculus emphasis. The majority of the population might be better served by studying up to algebra 2, but such that they actually understood it. Right now there are kids studying calculus in 11th grade that probably don’t have solid grasp of algebra 2 concepts.

Point #4 appears to be a side effect of the structure of undergraduate studies in the US – in this case the “service course” phenomenon. It tends to be that all STEM students get put through some or all of the same dry cookbook Calc I-III + Linear Algebra (+ ODEs) sequence. In some other countries (such as the UK), where students may be forced to declare an area of study early on (a trade-off to be sure), mathematics students don’t necessarily sit in those courses with engineers, computer scientists, etc.. They get less cookbook versions.

Some other thoughts on the HS curriculum inspired by your post:

1. There’s too much repetition (and students still fail to grasp concepts after all that repetition)

2. Euclidean geometry has always struck me as a weird choice, because it’s not well connected to the other classes. Hardly anyone does “axiomatic” geometry any more, it’s very artificial. We might be better off teaching proofs by interweaving them into other subjects, and perhaps some elementary number theory, set theory, etc (like the “Discrete Math” courses many comp sci majors have to do).

3. There is too much focus on trivia and overcomplication of certain topics. Manipulation of percentages is one example. Reducing fractions “to simplest form” is another (more of an elementary school thing). The SOHCAHTOA version of trig is vastly more confusing than presentations based on the unit circle in my opinion. In fact, you could probably substitute several weeks of trig with a couple of lectures on the complex exponential function and de Moivre’s theorem from which pretty much all of it follows.

I’m not an educator but FWIW I studied mathematics at university. I don’t totally agree with you on this. Drill is necessary to some extent, but I’ve heard that improvement really comes from gradually pushing the boundary of one’s abilities, not repeating the same thing hundreds of times. So solving 10 algebra problems of increasing difficulty, each one slightly beyond one’s current ability, might improve you more than solving 50 routine and trivial problems. In any case, the amount of drill should vary by student. A student who can ace an algebra exam after working two problems should not be required to solve 50 routine exercises. But the US high school system (at least when I went through it) seems to see homework as an end in itself.

I’m not a math major. I am a student who is and will be required to take what I think is fairly high level math at least for a non math major. What I’ve noticed when it comes to classes I’ve failed in math, and classes I’ve done quite well in is my own attitude yes, but also having a professor who knows not just the material, but also how to teach.

I guess part of the problem is what I’ve always heard about students learning in different ways. What I learn best is math that seems to come from something. Solving derivatives using the difference quotient? That was easy because it made sense. The difference quotient was easy to remember because I could visualize where it came from thanks to quality lectures.

I had another math professor at a lower level, and all that was done in that class was homework. Here are the homework problems, let’s do them on the board. Oops you failed a test. Is it common for math professors simply to go over homework in class with no lecture? I ask because I only saw this with one professor.

There will be another problem. At least in my opinion. People who understand material too well may be the least likely to be able to teach it because they do not see where it is their students struggle. There are some things that I can do second nature, but do not ask me to show anybody how to do them.

All of that said I’ve actually found much of my calculus instruction (and this is by no means universal) has been better than any other math instruction I’ve received.

Also, sorry if I seemed a bit harsh in the other comment on another post.