Reforming the math curriculum

One of the problems with political weblogs, and with with general political commentary, is how so many people who don't know what they're talking about crawl out of the woodwork to spout the most bizarre, half-brained nonsense you could imagine. As exhibit A, I submit the comments to this article. (The article itself does not work wonders for my search for intelligent political commentary, but we'll pass over that for now.) Perusing the comments (an activity I strongly discourage) one finds a sad museum of exhibits of our culture's bizarre notion that permanent outrage is a virtue.

I don't want to talk about that, though; I want to copy below the comments I contributed because they illustrate my philosophy and practice of teaching mathematics, and any much-needed reform of the mathematics curriculum. I will preface them with the sad observation that I don't think I did a very good job teaching differential equations last month; after nearly two years of doing nothing but research, I was not in top form, and perhaps a little too overconfident for my own good.

I will edit the comments to protect the innocent and the not-so-innocent, to reflect a correction I made myself, to remove repetition of the main (and not-so-main) ideas, and to highlight what I think are the more important ideas.

One last remark before the comments: unsurprisingly, the one or two comments from mathematicians with (or working on) a graduate degree have made more or less the same observations that I made. If you're inclined to check that yourself, the link to their comments is above; feel free to expend the effort to sort the wheat from the chaff. :-)

On to the first comment. There had been some complaints about teaching long division in grade school (whichever grade it's taught), as well as what topics should be covered in a math class. I'm always wary of such discussions, because the frustrating result is that topics that ought to be removed — because they contribute nothing to understanding the material — are left in, while topics that ought to be left in — because they are essential for understanding the material — are taken out. There was, for example, some talk of long division; I'm not necessarily opposed to removing that from the curriculum, but I felt the discussion was progressing on completely the wrong lines. The point of my comment was to try and redirect it:

Sorry for the long comment...

I recently graduated with a PhD in math. I don't believe in the "appeal to authority" fallacy, but for what it's worth...

1. The long division algorithm, as I remember it being taught, did nothing to help me understand what division was. That doesn't mean it couldn't help me understand, only that it didn't.

On the other hand, if I hadn't learned the long division algorithm, I would have had a harder time learning long division of polynomials. That's not something most calculators will do, even most graphing calculators.

2. I recently taught differential equations to some students. In theory, the reformed mathematics means that they were supposed to learn about finding inverse matrices during their high school educaiton. In reality, they learned only how to plug it into a calculator and use the result.

This works fine at the high school level, but in differential equations, one has to find inverses of matrices that contain functions like sin(x) and cos(x). Again: calculators won't do that. Since students don't know how to find inverse matrices in the first place, you can imagine the fun.

3. Every time I teach calculus, I have to put the fear of the living God into my students by warning that since trigonometry is a prerequisite of this course, I assume that they know trigonometry. I don't teach it in Calculus; I review it. To students whose notion of trigonometry is to punch whatever they see into a calculator, it is a formidable task to learn the basics of trigonometry in the crash course that I subsequently give. I've had students drop the course because of that.

This summarizes the problems I have with most mathematics reform. I'm all for reform of the mathematics curriculum; I agree that too much is taught, at too shallow a level, with little or no understanding of the concepts involved.

The result, however, has been that even *more* material is taught (the addition of statistics, numerical programming [ed: I meant "linear" programming], modeling), at even shallower a level, with absolutely no understanding of the concepts involved.

IMHO, the lower mathematics curriculum should be designed with one overriding question: what mathematics would students need, in order to learn mathematics at a higher level? Designing the curriculum with the applications of mathematics to the hard and social sciences is a good idea, but it's not the most important question. Someone has pointed out already that all of the math that most people need, they learn in grade school. Let me raise another point: do scientists teach mathematics in their classes? not in my experience (unless the math teachers haven't done their jobs): rather, they show how to use mathematics that they assume the students have learned. Neither should mathematicians be expected to teach science in their classes.

On the other hand, the university-level mathematics curriculum should be redesigned with the overriding question: what do engineers, scientists, etc. need from mathematicians? and what mathematics do our graduates need in the real world? Why do I emphasize these questions here, and not at the lower level? because college graduates are (presumably) studying this material in order to use it in the future. (Not counting the shameful practice of some departments to weed out weaker students by requiring irrelevant math classes, e.g. history majors in a trigonometry class.)

Again, I don't think math professors should be teaching science or engineering; rather the mathematics necessary for them. When I teach applied differential equations, I do a number of application problems, although I have no idea if a sane engineer would use the approach we use in class. I do demonstrate the physical necessity of the number i for the class (the so-called "imaginary" number; finally they learn! after first seeing it in high school) But I only work in as many "real-world" applications as I have time for. I don't have time to learn biology, physics, materials science, electrical engineering, environmental science, etc.... let alone teach it.

Once the university-level curriculum has been redesigned, the lower-level curriculum can be redesigned to prepare students for the university-level curriculum.

Unfortunately, most attempts to reform the university-level curriculum make the same mistakes that are made at the lower-levels: without thinking enough about it, they discarded things that are necessary added things that are unnecessary, and didn't teach anything very deeply. My former university experimented with one reform calculus textbook; it was such a disaster that they went to one of the "combined-approach" texts, which IMHO is one of the worst. But "sort of" satisfying everyone is easier to do than satisying a few and outraging the rest :-(

I agree with the person who said that what most high school students need is not more mathematics, but logic instead. I didn't really learn logic until I started studying advanced mathematics, and it was a revelation; I loved it. We should replace one year of the high school mathematics requirement with one year of logic.

Someone pointed out the Saxon books. My mother used the Saxon books on my youngest brother, and as a result he never had trouble with mathematics :-) Unfortunately, Saxon textbooks have long been eschewed by the educational establishment, because they don't fit fashionable theories. (Maybe things are changing, but I haven't seen that happen yet.) The fact that the students learn something, and learn it well, and enjoy learning it, is irrelevant.

Finally, I think that [the gentleman who wrote the article] has misunderstood something: when Dr. Neill writes [mathematics] teaches them perseverance, attention to detail, critical thinking skills and discipline, he's not lauding the sort of perseverance, attention to detail, etc. that can be taught by football. Physical perseverance, attention to detail, etc. are not much good at fostering intellectual perseverance, attention to detail, etc.

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Anyway, those are my thoughts, which certainly don't make for the last word :-)

Indeed. The first two comments were, in my opinion, intelligent and well-posed. This is my reply to them (other person's original comments in blue):

I suspect at the average public school, nothing is taught at a deep level, especially not math.

I agree. I taught high school math for two years, and I tried to make it lively and deep. Many students told me they liked math for the first time in their lives: advanced and remedial students alike. Alas, for the average students it didn't work so well, and I got tired of dealing with the whole public school issue (which is something else altogether), so I left to do something else with my life. It wasn't a salary thing: no amount of money would ever compensate for the nonsense I had to put up with from average students, their enabling parents, and administrators who felt that a certain level of misbehavior was tolerable and to be expected. I'm no Jaime Escalante :-(

Can you find out what percentage of your students have actually had a reform math course next time you teach?

[edit: notice that my reply shows that I hadn't read his last four words carefully]
Not really; the students would be no help at all, and I haven't the time to look up their backgrounds (assuming I even have access to such information, what with privacy laws and all) :-) It's a neat idea, though, thanks; I might try that at my next job as a research project.

I can say that my most certain experience was tutoring a high school student whose teacher was using a UCSMP text and supplementary materials. The UCSMP text did not give nearly enough exercises for the students to acquire mastery of a concept. Some students acquire mastery easily, most do not; this fellow was of the sort who absolutely needed as gradual a learning curve as possible. We both felt the UCSMP text was deficient in that regard. I could make up some problems, but that took wayyyyyy too long.

Moreover, there is, as has been noted before, a profound difference between the strength of the curriculum and what gets delivered in class.

I agree, but it is very difficult to overcome a curriculum, and/or a textbook, whose philosophy is based more on wishful thinking than on what students actually need to learn. This obviously applies to traditionalists as well as to reformers, since traditionalists are just as given to wishful thinking.

Indeed, it is math where rote is the rule, more than anywhere else, and thinking is secondary.... Sadly, most students just get that, and only that, out of math.

I agree to an extent with you (see below). In my experience as a high school teacher, there were two reasons for this. (1) The teachers themselves didn't know the concepts, nor were they particularly interested in them. A lot of them liked "math" because they found the skill-and-drill appealing. Some (like myself, the chair, and another) were people who fell in love with math as college students; the rest were people who enjoyed grade-school math drills and thought that's what counted as math. (2) The students aren't interested in the concepts, because understanding the concepts require effort. Skill-and-drill is fairly simple, once one has memorized it. It's a much clearer objective to master than "conceptual understanding". Correspondingly, discipline problems are often fewer when students are limited to skill-and-drill, than when they are doing conceptual activities. I have observed this at all levels: students are happy to listen while I explain "how to do it", but the moment I start to explain "why it works," they become bored and start acting up, because they perceive it as "unimportant".

The same things happens in real research, by the way. My research was on a topic called Gröbner bases, which have important applications in mathematics and science (or so I'm told ;-)). Do you think that I, or anyone else, understands the concepts behind Gröbner bases thoroughly? of course not; otherwise there would be no more reason to do research in the topic, aside from discovering ways to apply them to new areas. Yet people have been calculating with them for forty years, and people are still discovering ways to improve the computation of them, or even to avoid aspects of it altogether.

And when I explain my results to conferences and seminars and even my PhD committee (!), listeners are usually interested and attentive only while I describe the result itself; the moment I delve into the details, their eyes start to glaze over. What I have to do in presentations, therefore, is structure them properly (question-answer format), and make myself open to their questions.

Even the result itself came not from a profound insight into the concepts underlying Gröbner bases, but after a lot of hand computation of special Gröbner bases. I discovered my PhD result by, of all things, factoring some polynomials by hand.

What I'm trying to say is that in mathematics, conceptual understanding usually grows hand-in-hand with skill-and-drill, and the idea is preserved by good, solid practice. (I've forgotten a lot of vector calculus and abstract algebra, because I never use them in my work.) To exclude one at the expense of the other is wrong, and many well-meaning math reform advocates make the error of advocating the elimination from the high school mathematics curriculum of those computational algorithms that have no apparent application in their experience: addition of fractions, long division, ... That's all I'm addressing.

I do not mean to disparage math reform per se; I'm sorry if that wasn't clear. I agree that there are serious problems with the traditional approach; reducing mathematics to rote memorization does both the students and mathematics a terrible disservice.

At this point, we had the entrance of a frustrated teacher who favors one of the more extreme proposals of reform mathematics: removing factoring from the curriculum. My main point here is (a) to illustrate how ill-considered many proposals are; and (b) to build to the final comment.

To begin with, I sympathize with your unhappiness with both the CA curriculum (teaching too much, with not enough depth). On factoring, though...

Perhaps someone will enlighten me on why factoring is so important.

Factoring is useful for solving equations, but that's not its only purpose. It's also important for later mathematics, such as calculus: as a means of accelerating polynomial evaluation (from this, we obtain e.g. synthetic substitution); for making rational expressions easier to deal with (partial fractions); for analyzing and simplifying trigonometric expressions; for separating complicated polynomials into smaller polynomials that are more easily analyzed and dealt with (differential equations, algebraic geometry, etc.).

And of course, you can't understand how the quadratic formula works, unless you know how to complete the square, which requires a solid understanding of factoring.

Out of curiosity: How did you receive a math degree without learning any of these things? I don't mean to disparage you; I'm really curious. Didn't you study any of those topics along the way to your math degree?

Factoring is completely useless because you can use the quadratic formula for that stuff anyway.

The quadratic formula works only with quadratic polynomials. It doesn't work with cubic polynomials, or quartic polynomials. The formulas for those polynomials are far too complicated to teach in high school (they were taught once upon a time) so it's best to handle them via factoring by grouping (or something of that sort).

The concept of factoring polynomials, and applying the zero product rule to find the solutions, is essential for understanding many tools of higher mathematics. You can't really understand how to solve linear differential equations with constant coefficients unless you understand how to solve equations by factoring.

Finally, I repeat that factoring polynomials helped me discover the main result of my PhD thesis. :-)

I hope you can see now that factoring isn't useless, unless your idea of mathematics is that something is useful only if it gives you the solution to an equation.

Oddly, after conceding that s/he was wrong to say that factoring was useless, s/he then dismissed that error and tried to say that it was still more or less useless in a high school curriculum. I tried to be polite with my subsequent reply:

I thought the goal of math reform was to make students understand mathematics better. Your suggestion for understanding the quadratic formula is that they use it without ever seeing how we find it? This is not understanding; this is indoctrination. It is not a pedagogy for forming thinking human beings; it is an approach to programming machines.

Again, if geometry students have a hard time solving a linear equation with one unknown, I think we need to spend time on other stuff.

Geometry students who have a hard time solving a linear equation with one unknown shouldn't be in a geometry class; they should have failed algebra I.

What on earth is going on at your school?!? An algebra I teacher who gives a passing grade to a student who can't solve a linear equation in one variable is as irresponsible as a high school that awards a diploma to an illiterate senior.

Do you honestly believe that the only things taught in algebra I should be those things necessary to a student for whom algebra I is a terminal class?
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That last question of mine is a challenge to all would-be reformers of mathematics. I alluded to this in earlier comments, but in this one I bring it out to the fore: frequently, reformers talk as if the only material taught in any class should be the content of that course necessary for using it as if it were a terminal math course. Hence the emphasis on shallow understanding but lots and lots of applications. Many reform texts give the impression that the science is more important than the mathematics! (and in a science class, it would be! — assuming the two can be separated, which I'm not so sure that they can.)

In my opinion, this is an error. Factoring should be taught in algebra I — not so much because it's necessary for non-mathematical applications of algebra I (it's not, usually) but because it's fundamental to so much of the math that follows algebra I.

How important is factoring, by the way? besides its contribution to my PhD thesis, I have officemates and professors whose entire PhD research has been based on faster factorization algorithms. I can't think of a single class, after algebra I, where factorization was not assumed. (Exception: the "contemporary mathematics" course that many universities teach, which I jokingly call "math for people who hate math." It's a great course — I've taught it two or three times — and a good example of how state boards of education should rethink the curriculum for students who are simply not disposed towards learning algebra I. I will note though that it's not a very easy course, and factoring may be required for some topics, just not the ones that I remember teaching.)

Let me also point out an important distinction. It may appear that I am at first arguing that concepts are less important than skill-and-drill (I say that much research can be classified as "skill-and-drill", because the concepts aren't completely understood) while in the later comments I am arguing that concepts are more important (factoring is necessary for understanding, or doing, later topics/skills). This is a false paradox; in each case, I am addressing two different issues. In one, I am insisting on the necessity of general skill-and-drill; in the other, I am insisting on the necessity of a particular concept. There are a lot of concepts I would like to dump altogether, or restructure in the overall curriculum, and in general I think both approaches are essential to a proper mathematics education. I am convinced that one of the weaknesses of my graduate education (and a reason it took 6 years to graduate instead of 4 or 5) was the lack of skill-and-drill in certain classes where professors valued "conceptual understanding" to the very exclusion of homework.

I mention this apparent paradox because I almost confused myself while assembling these comments :-)

In conclusion (for now), I would also like to highlight the fact that our math teacher friend talks about geometry students who can't solve linear equations in one variable, and thinks the problem there is the curriculum. Au contraire; the problem there was an irresponsible teacher.

I may update this with further comments, depending on how useful the further conversation is.