A database of applied math problems

A friend of mine teaches at the British Columbia Institute of Technology. They are building a database of applied math problems, at the 11th or 12th-grade level. Their goal is to give students a better idea of "why do we need to learn this?", which is the bane of all math teachers.

They're not asking for calculus problems. But I've taught calculus and I often had the sense, while teaching the "applied" problems, that they were just straight-up asking the students to do derivatives or integrals, with some words added purely as a red herring to confuse the students. I mean, really, if a ladder leaning against a wall falls down, is there any situation in which one cares how quickly the area underneath the ladder is changing? My memory of pre-calculus classes is hazy, because I haven't taught at that level, but I do remember having a pervasive sense that the applications were contrived.

Counterexamples in X

Counterexamples in Probability And Statistics (Joseph P. Romano and A. F. Siegel) and Counterexamples in Probability and Real Analysis (Gary L. Wise and Eric B. Hall) both seem to be books in the tradition of Counterexamples in Analysis (Bernard Gelbaum and John Olmsted) and Counterexamples in Topology (Lynn Arthur Steen and J. Arthur Seebach. These are books that collect the examples just "outside" the boundaries of the various standard theorems, the point being to explain why one needs the seemingly strange collections of hypotheses that seem to begin every analytic theorem. (Hence the tags "education" and "teaching"; I've often seen these counterexample books described as "anti-textbooks", and as being complementary to standard textbooks which often spend most of their time telling you what's true.)

It seems that these books are concentrated on the analytic end of mathematics; I couldn't find, for example, books of counterexamples in algebra, combinatorics, or number theory. There is, however, Theorems and Counterexamples in Mathematics. My sense is that the nonexistence of these books is connected to the fact that those fields don't seem quite as rife with theorems where all the work is hidden in the definitions.

A surprise from differential equations

A problem which circulated among the grad students here at Penn today, which came up while somebody was teaching differential equations:

Consider the differential equation dy/dx = (y-1)². Separate variables and integrate both sides as usual; you get y(x) = 1 - 1/(x+C), where C is determined by the initial condition. Take the limit as x goes to infinity, and you get lim_{x -> ∞} y(x) = 1, regardless of C.

But now notice that dy/dx is positive for all y. (We're working over the reals here.) So if the initial condition is of the form y(x₀) = y₀ for some y₀ > 1, then we start at 1 and keep going upwards; how can the limit be 1?

Of cousre there's a mistake somewhere in here. But where is it? (I know the answer, but it took annoyingly long to figure out.)

edit: the differential equation was wrong.

Product rules for more complicated products

Michael Hardy wrote a paper entitled Combinatorics of Partial Derivatives. He gives Leibniz' rule for the differentiation of a product in the rather unfamiliar form

which I am calling unfamiliar not because it's a particularly deep result, but because it's just not something one usually writes out. It could be proven by induction from the more "standard" product rule (uv)' = u'v + uv'.

But why don't we teach this to freshmen? Sure, the notation might be a bit of a barrier; I get the sense that a lot of my students learn the Σ notation for the first time when we teach them about infinite sequences and series, at the end of the second-semester calculus course; of course they learn about derivatives, including multiple derivatives, sometime in the first semester. (If it is true that they are seeing the Σ notation for the first time then, it doesn't seem quite fair, because then we're asking them to assimilate this weird notation and some real understanding of the concept of infinity at the same time. Indeed, at Penn we often tell them not to worry so much about the notation.) But ignoring the notational difficulties, fix a value of k -- say, 4 -- and get

so basically we notice two things: there are four primes in each term, and the coefficients are the binomial coefficients, which are familiar to most students.

One doesn't take the fourth derivative of a product that often; but even knowing that might be preferable to

Also, one can expand a rule like this to products of more than two terms; we have

Again, this doesn't come up that often, and I don't want to try to write it for derivatives of products of an arbitrary number of factors. Still, the idea is fairly natural but how many freshmen would even know that

?

I really don't know the answer to this -- but products of three factors are not incredibly rare, and the rule here is quite simple -- just take the derivative of each factor in turn, and sum up the results. There's even a nice "physical" interpretation of it -- how much does the volume of a three-dimensional box change as we change its various dimensions?

The coefficients seem kind of arbitrary, though; the point of Hardy's paper is that if things get recast in terms of partial derivatives they go away, both here and in Faa di Bruno's formula for the derivative of a composition of functions. One way to think of this is to imagine that, in the product rule, we have a "prime 1", a "prime 2", and so on up to a "prime k" if we're taking kth derivatives; we attach these primes to the variables in all possible ways, sum up the results, and then "collapse" them by reading all the primes to just mean ordinary differentiation.

Reference
Hardy, Michael, "Combinatorics of Partial Derivatives", Electronic
Journal of Combinatorics, 13 (2006), #R1.
http://www.combinatorics.org/Volume_13/PDF/v13i1r1.pdf

Teaching courses for non-technical students

I just learned that I have been assigned to TA a course called "Ideas in Mathematics" (which, if I understand correctly, meets some sort of distribution requirement for the college as a whole, so will mostly be non-technically-oriented people) in the spring. I'm looking forward to this -- it'll be a nice change of pace from my usual teaching assignments (calculus) and I have all sorts of half-formed ideas about how mathematics should be presented in a non-technical setting that I'd love to be able to test out.

But I've never taught a course which wasn't calculus. So I ask you, my readers -- what's different about teaching these two kinds of courses?

Two grading-related thoughts

In the class I TA, I drop the lowest homework grade (of eleven) for each student. But what does this mean? The meaning of "dropping the lowest homework" is clear enough if all homeworks count the same. But there was one homework assignment that was particularly long, and one that was particularly short, so I counted those for 30 and 10 points, respectively, while all the other homeworks were out of 20.

The first solution is to drop the homework for each student with the lowest numerical grade -- but that's just stupid, because for most students that actually did all the homeworks, that would be the single homework assignment that was graded out of 10. I rejected that without even bothering to do the code.

The next solution seemed to be to drop the homework with the lowest percentage grade -- so, for example, if a student had 23/30 on the single 30-point homework and all their other grades were 16/20 or 8/10 or above, then I dropped the 23/30. But isn't it possible that that algorithm might not maximize the score? Indeed, paradoxes like that are possible, as this Mathtrek article and this paper by Daniel Kane show. They remind me of Simpson's paradox, although I suspect there's no deep connection here.

The eventual solution I settled on was to drop the homework for each student that gave them the highest possible percentage; this wasn't hard to code, since I was dropping a single homework, but I wouldn't want to go to that trouble if I were dropping, say, the lowest two homeworks for each student.

Of course, I should have just made all the homeworks count the same amount. And I was even aware of this particular paradox! (The Mathtrek article, as you can see, is dated June 2006; I don't know if I read it then but I certainly had read it before the start of the current semester.

I also played around with the idea of dropping the lowest 20 points worth of homework, which would mean:

• if the homework with the lowest percentage was a 20-point one, dropping that one;


• if the homework with the lowest percentage was the 30-point one, rescaling it for that student to be out of 10 points;


• if the homework with the lowest percentage was the 10-point one, dropping that one, and reducing the weight of the homework with the second lowest percentage grade by ten points.

(This was inspired by a policy that at least one of my undergrad classes had, where the final counted twice as much as each exam, and the lowest exam was dropped, except if the lowest exam was the final, in which case the final was reduced in weight to have the same weight as a midterm.) But I wasn't sure what properties that has, and technically it is not what I said I would do. Of course the algorithm I choose isn't going to change any grades by much, and dropping the lowest homework is really something to take a bit of pressure off the students, but I wanted to stick to my word.

Furthermore, the method I chose has the nice property that no student can come to me and say "my grade would have been better if you'd chosen a different homework to drop".

Also, it's occurred to me that what we tell our students about grades is a lie. In the particular class I'm TAing, we tell the students that the homework counts 20%, the two midterm exams count 25% each, and the final counts 30%. But this ends up not quite being true, because homework grades are less narrowly spread than exam grades! (The mean overall homework grade was 84%, the standard deviation 12%; the mean grade on each exam was in the low sixties with a standard deviation of 16%.) This has actually generated complaints for me from students in the past, who assumed that we were curving each section of the course separately and were disappointed to learn that we curved the total raw score. (These were students who had relatively better homework grades than exam grades, and thus ended up being hurt by this policy.) I'm a probabilist, though, not a statistician, so I won't elaborate on this.

Nested radicals, smoothness, and simplification

I saw an expression involving a nested radical, namely

.

(Write φ = (1 + φ)^1/2 and solve for φ.) The Wikipedia article on nested radicals led me to Simplifying Square Roots of Square Roots by Denesting. The authors tell us that:

The term surd is used by TeX as the name for the symbol √ Maple has a function called surd that is similar to the nth root defined here; like all good mathematical terms, the precise definition depends upon the context. In general, a mathematical term that does not have several conflicting definitions is not important enough to be worth learning.

This reminds me of a couple things that happened in my class yesterday. First, I was defining what it means for a curve to be smooth; our definition was that the curve given by the vector function r(t) is smooth if r'(t) is continuous and never zero, except perhaps at the endpoints of the interval over which it's defined. (This makes smoothness a property of a parametrization, which is a bit counterintuitive. I suppose that one could define a curve -- as an abstract set of points -- to be smooth if it has a smooth parametrization. Although I haven't worked it out, I assume that if a curve has a smooth parametrization, the arc-length parametrization is smooth.) One of the students said "but the professor said 'smooth' means something else!" I'm not sure if the professor actually said "smooth means X" or if he said "some people think smooth means X", but it's a good point. (In particular, "smooth" often seems to mean that a function has infinitely many continuous derivatives.)

Second, the article is about using computer algebra systems to simplify expressions like

where the left-hand side is "simpler"; sometimes my students worry that they are not presenting their answer in the simplest form. While I'll accept any reasonably simple answer (unless the problem statement specifies a particular form), it is remarkably difficult to define what "simple" means.

One rule I have figured out, though, is that 4x - 4z - 8 = 0 should be simplified to x - z - 2 = 0 by dividing out the common factor. In general, given a polynomial with rational coefficients, one probably wants to multiply to clear out the denominators and then divide by any common integer factor of the new coefficients, so the resulting coefficients are relatively prime integers. The article addresses this sort of "canonicalization" in the context of nested radicals. I keep telling my students that they should keep that sort of thing in mind, especially since our tests will be mostly multiple-choice.

(Sometimes I'm tempted to define "simplest" as "requires the fewest symbols"... but how does one prove that some 100-character expression one has written can't be written in 99 characters? And how do you count something like "f(x, y)= (x+y)^1/2 - (x-y)^1/2, where x = foo and y = bar?" ("foo" and "bar" are supposed to be very complicated expressions.) Do you plug foo and bar into the original equation and then count the characters, or do you count the actual characters that are between the quotation marks?)

Some thoughts on coordinate systems

I'm teaching multivariate calculus this term. The course, as it's taught here, begins with a unit on geometry in three dimensions; here we introduce cylindrical and spherical coordinates. (I'm not entirely convinced that they should be introduced at the point they are, because basically all we can ask the students to do is to convert between the various coordinate systems, but my hands are tied. It almost seems to make more sense to wait to teach the coordinates until we get to the point where we're doing integrals over regions that are best expressed in cylindrical or spherical coordinates.)

Anyway, a student e-mailed me a question today. One problem on the homework was to find equations in cylindrical and spherical coordinates for x² + y² + z² + 2z = 0. In cylindrical coordinates, (where x² + y² = r² this becomes r² + z² + 2z = 0. (My aesthetic sense is that this is perhaps better written as r² + (z+1)² = 1, because then it's immediately obvious that there are in fact solutions to the equation.) She asked if it was acceptable to leave it in one of these forms, or if it needed to be converted to, say, something with z in terms of r and θ (which would involve square roots).

I responded as follows:

In rectangular coordinates it might be preferable to solve for z in terms of x and y if it's possible to do so without making things too ugly, since we have a tendency to think of surfaces given in Cartesian coordinates as a "graph" of a function of x and y. The same thing is sort of true in spherical coordinates, in that ρ is often seen as a function of θ and φ (This is like polar coordinates, where r is usually a function of θ.) In cylindrical coordinates, though, none of the three coordinates really seem to be "dependent" or "independent".

Of course, all these rules can be violated; for example, one would never write the equation of a unit sphere centered at the origin as

z = ± (1-x²-y²)^1/2

unless it were to graph it on a system that can't handle implicitly defined surfaces. The guiding principle should be that you want the simplest equation possible, i. e. the one which takes the least writing.

"Takes the least writing" is admittedly a bit sloppy here; how does one define it? The amount of "writing" any mathematical expression takes depends on one's notation. For example, subscripts or superscripts take up a lot of keystrokes in HTML, not so many in TeX (which is of course optimized for mathematical writing), and if one is writing by hand writing a subscript is just one character. (Indeed, one could even argue that subscripts and superscripts require less writing than full-size characters in handwriting -- they take less ink! But I shudder at the thought of taking this to its logical conclusion.)

Another principle I might invoke is "symmetry" -- another reason the equation about for a sphere looks wrong is because it fails to capture the fact that x, y, and z all behave "the same way". The form x² + y² + z², on the other hand, doesn't have this problem. But I think "symmetry" might not be so useful a concept when trying to teach students; the ones who will understand this rather vague idea are the same ones who would probably just give the equation in a "more symmetric" form even if I didn't tell them to.

And why do cylindrical coordinates not seem to have a "dependent" and an "independent" variable, whereas three-dimensional Cartesian and spherical coordinates do? (I admit that this might just be my intuition, and I don't know if this agrees with anyone else's intuition.)

links for 10 August

Why Stuff Is Hard, at the Everything Seminar. I asked this in physics class in high school and never got a satisfactory answer. I suspect I'm not alone here. I was told it was some sort of electromagnetic repulsion between the outermost electrons, but apparently the Pauli exclusion principle is really doing most of the heavy lifting.

Mark Chu-Carroll at Good Math, Bad Math comments on the way math is taught at certain religious schools. If you don't want to bother reading the post, check out the course descriptions at one such school. They all start out "Students will examine the nature of God as they progress in their understanding of mathematics". The descriptions of the non-mathematics courses begin similarly. Today I was reading parts of Laplace's A Philosophical Essay on Probabilities (available in Hawking's anthology God Created the Integers: The Mathematical Breakthroughs That Changed History); among other things, Laplace mocks the idea of Pascal's wager. I couldn't help but thinking of what Laplace is said to have said to Napoleon when asked why he didn't mention God in his work on celestial mechanics: "I had no need of that hypothesis."

Compound interest isn't intuitive, from Adventures of BruteForce; if you invest a little money now that's like investing a lot of money later. People just aren't set up to understand exponential growth, which isn't surprising; unrestrained exponential growth isn't common in the situations for which we evolved. A population can't keep doubling every ten years without pretty quickly running out of space; a sum of money can. There's a persistent rumor that Ashkenazi Jews are actually better equipped for understanding this particular sort of abstraction than other classes of people, because of certain unique historical circumstances -- for quite some time they lived among Christians, who were forbidden to lend money for religious reasons, but these same Christians wanted to borrow money, and therefore turned to the Jews, who were not subject to those same religious laws. The Jews who were better at understanding this fact ended up with more money themselves, their kids didn't starve, and supposedly this explains why about a quarter of Nobel laureates are Jewish. I can't find exact numbers overall. One thing I can find is in this Wikipedia article which says that "Of American Nobel Prize winners, 37% have been Jewish Americans (19 times the percentage of Jews in the population) [...]" But I'm not sure who they count as "American". (Wikipedia used to have a list of Jewish Nobel laureates, but it's been deleted.)

It would be interesting if this were true, because it seems to imply that evolution can work ono the scale of a few hundred years. As humanity heads more and more towards working with its brains instead of its hands, will we get smarter? (On the other hand, at least at the present time in the United States, intelligence and number of children seem to be inversely correlated; it seems difficult for Darwinian evolution to work in a population when almost everyone survives long enough to have children.)

a thought on teaching

From today's Philadelphia Inquirer: Masters of "spring" theory, about how physics is now being taught in some high schools by a "modeling" approach, in which students basically try to figure out the principles of physics for themselves, by teacher-facilitated experimentation. This requires the teachers to go through some intensive training themselves in order to teach this way.

To what extent is it possible to teach mathematics this way? I've heard of the Moore method, which seems like an analogue of this, but from what I've read that sounds like it's been most successful at the post-secondary levels. I also don't know if this particular page is the best explanation of the Moore method, because I've never experienced it myself.

(Also, apparently "string theory" is well-enough known that a mainstream news paper feels like they can make puns on it in their headlines. I'm not sure what to make of this.)