Casting Out Nines

Blogging’s been light lately because of the push to finish spring semester courses. I did so today, turning in my last batch of semester grades. So maybe now I can get back into a regular swing of posting. 

My main role this summer will be that of the stay-at-home dad. Our two girls normally attend preschool and daycare during the week, our 4-year old full-time and our 2-year old part time. This summer, the 4-year old will be going just three days a week and the 2-year old just one day a week. I will have one day a week to myself (see below), but the other four weekdays will be spent either one-on-one with my 2-year old or two-on-one with both of them. It’s a role that I am greatly looking forward to playing. 

I will be “Mr. Mom” during the day, and then I will be teaching not one but two classes in the evenings. I signed up to teach our 8-week summer calculus course on Mondays, Tuesdays, and Thursdays from 5:30-7:30; and I am directing an independent study for one of our seniors, the meetings for which will be on Wednesdays 5:30-6:30. (We’ll be working through selections from Edward Bender’s Mathematical Methods in Artificial Intelligence.) 

The girls will remain in their current preschool/daycare schedule until the start of summer classes, so I do get a couple of weeks for my own “vacation” in the interim… which will be spent writing up a paper for the ICTCM proceedings and finishing up a book review for the MAA which I should have submitted back in March. I will also need to work on adapting my 14-week calculus course to an 8-week format, which is harder than it sounds. 

Between potty-training by day and derivative-training by night, I don’t expect to have tons of free time to spend on projects like I did last summer, when the girls went full-time five days a week to daycare while I whiled my days away as a gentleman of leisure read and studied for the Reconnect 2007 workshop. Actually, it was about three weeks into that summer break that I decided I would much rather have the girls at home with me than sit around surfing the internet and pretending to learn about machine learning. The bulk of the “projects” that I have for summer involve reading, and you’ll hear more about what I am working through as it happens. (Don’t want to be too ambitious at this point and lay out a reading list.)  

For the moment, though, it’s nice not to have the pressure of prepping for the next day’s classes and keeping up with the grading stream. 

Asking the following question may make me less of a mathematician in some people’s eyes, and I’m fine with that, but: How do you explain the meaning of the differential dx inside an integral? And more importantly, how do you treat the dx in an integral so that, when you get to u-substitutions, all the substituting with du and dx and so on means more than just a mindless crunching of symbols? 

Here’s how Stewart’s Calculus does it: 

• In the section introducing the definite integral and its notation, it says: “The symbol dx has no official meaning by itself; is all one symbol.” (What kind of statement is that? If dx has “no official meaning”, then why is it there at all?) 
• In the section on Indefinite Integrals and the Net Change Theorem, there is a note — almost an afterthought — on units at the very end, where there is an implied connection between in the Riemann sum and dt in the integral, in the context of determining the units of an integral. But no explicit connection, such as “dx is the limit of as n increases without bound” or something like that. 
• Then we get to the section on u-substitution, which opens with considering the calculation of (labelled as (1) in the book). We get this, er, explanation: 

Suppose that we let u be the quantity under the root sign in (1),  . Then the differential of u is du = 2x dx. Notice that if the dx in the notation for an integral were to be interpreted as a differential, then the differential 2x dx would occur in  (1), and, so, formally, without justifying our calculation, we could write …

So, according to Stewart, dx has “no official meaning”. But if we were to interpret dx as a differential — he makes it sound like we have a choice! — then using purely formal calculations which we will not stoop to justify, we could write the du in terms of dx. That is, integrals contain these meaningless symbols which, although they have no meaning, we must give them some meaning — and in one particular way — or else we can’t solve the integral using these purely formal and highly subjunctive symbolic manipulations that end up getting the right answer. 

Er, right. 

To be fair, my usual way of handling things isn’t much better. I start by reminding students of the Leibniz notation for differentiation, i.e. the derivative of y with respect to x is dy/dx. Then I say that, although that notation is not really a fraction, it comes from a fraction — and that much is true, since dy/dx is the limit of as the interval length goes to 0 — and so we can treat it like a fraction in the sense that, say, if then and so, “multiplying by dx”, we get . But that’s not much less hand-wavy than Stewart. 

Can somebody offer up an explanation of the manipulation of dx that makes sense to a freshman, works, and has the added benefit of actually being true? 

Here’s a problem I have with the way most calculus textbooks are written, and therefore by default the way most calculus courses end up being taught. Tell me if I am crazy or missing something.

We teach calculus from a depth-first viewpoint. That means that whenever we encounter a concept, we go as deeply as possible in that concept before moving on to the next one. There are some subjects where this makes sense, but in calculus we have a small number of main ideas that are made out of several concepts, and if we stop to attain maximal depth on every single thing, there’s a good chance that we never arrive at the main idea with any degree of understanding.

The big ideas of calculus — the rate of change (derivative) and accumulated change (integral) — are actually really simple if you consider them simply for what they are and what they were invented to do. Derivatives, for instance: You have a function, and it is changing in all kinds of ill-behaved ways. The object is to find out exactly how quickly it is changing at a given point. We quantify that rate of change by sticking a tangent line on the graph of the function at that point and measuring its slope. Really, that’s it. Slopes of lines. The rest are technical details on how to calculate this slope with some degree of accuracy, and those details range from graphical estimation to interpolation tricks to algebraic techniques.

But in Stewart’s Calculus book, the coin of the realm of calculus texts, here’s what students have to study before the derivative is defined: an entire chapter of precalculus review (a mind-numbing section 1.1 on functions and notation, mathematical models, families of functions, exponential functions, inverse functions and logarithms), then a chapter on limits in which students have to master finding limits from graphs, calculating limits using the Limit Laws, the epsilon-delta definition of a limit (mostly untaught these days), continuity, and limits at infinity.

Then there’s a section on “Tangents, Velocities, and Other Rates of Change” followed by two sections on the Derivative.*

This approach plays directly in to the greatest weakness of the average calculus student, which is algebra/precalculus content mastery and the ability to master technical details of calculations and theory. How likely is it, for the student who struggles to read mathematics or use algebra correctly, that this student will be in any shape to learn what a derivative is, and what one is for, by the time they get there?

You want students to master those technical calculations and theory, of course. But you also want those to be mastered in context, not just as mathematical tricks to be learned as parlor games. The few students who survive the onslaught of detail mastery and are still psychologically around to learn what a derivative is, often find it extremely hard to know what f’(3) = 2 actually means. All they know is that you bring the power down and subtract one, and maybe the Product Rule.

I’d prefer some kind of approach to calculus that is not depth-first but more like breadth-first, where students get a good grounding in the overall ideas of calculus and do some basic work before mining into the really deep details. Not all students really need those deep details, after all.

* OK, there is a section (2.1) where the ideas of tangent lines and velocities are briefly introduced. And then summarily ignored until the end of that chapter. The students typically ignore that material right along with the book.

I got a nice surprise in the mail this morning — a review copy of the fourth edition of Marvin Greenberg’s classic text Euclidean and Non-Euclidean Geometries. It seems like this book has been in the third edition since time immemorial. I used the third edition in my first year of teaching after graduate school, 10 years ago, and loved the depth and clarity of the writing. That much seems not to have changed. There are some significant rearrangements and updates to the material, and overall the book just looks a lot nicer (And the color scheme matches my blog, to boot!) There don’t seem to be a lot of good intro-level geometry texts out there — and there are a lot of bad ones — so a new Greenberg is a nice early Christmas present. It’s the kind of book that makes you want to sit down and work through it just so you can learn geometry from back to front.

Freeman textbooks are on a roll these days, what with this new edition of Greenberg and with Rogawski’s excellent new calculus text. (Disclosure: I was a reviewer for Rogawski.)  I don’t advocate for textbook use often, but if you have to use one, use a good one!

Editorial: This is the penultimate article in the retrospective series we’ve been doing all week here at CO9s. This one takes us back to 2006 one more time.

One of the things that fascinates me most about teaching math is seeing how people acquire and use problem-solving skills. And one of the things I like to think and write about the most is how people can approach problems in different ways — especially when those ways are not the standard ways of doing so — and why students make various conceptual mistakes when they try.

This article was written after a calculus homework set involving a pretty standard intro problem about the velocity of an arrow shot straight upward on the moon. (Where the **** do we math people get these problem ideas?) I was reading James Gleick’s biography of Richard Feynman at the time and was very keen on how important visualization is in problem solving. I had also been thinking (and posting) about how the ephemeral idea of “critical thinking” really just boils down to having a good intellectual B.S. detector and having the will to question whether your thinking can possibly be right or not. Put all that together, and this article is what you get.

Side note: One of the biggest sources of traffic for this blog comes from people who enter in “velocity of an arrow shot upward on the moon” into a search engine and end up with this posting. For those of you who have found yourselves here by doing so: DO YOUR OWN HOMEWORK.

Critical thinking, visualization, and physical intuition

Originally posted: October 11, 2006

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Yesterday I posted my belief that “critical thinking” has at least as much to do with intuition as it does with what we normally call “thinking”. Namely, critical thinking has to do with — is activated by — having a sense of when something can’t possibly be right. Question: Where does that sense come from? And importantly, can it be taught? I’ve been reading through Genius: The Life and Science of Richard Feynman by James Gleick and have been struck by Feynman’s reliance upon visualization to make his dramatic contributions to quantum theory and other areas, and it makes me believe there’s a strong connection between visualization and the ability to solve problems that I’ve not heard mentioned often. (more…)

It turns out that according to a recent discovery in an ancient manuscript, calculus might first have been discovered not by Newton or Leibniz in the 1700s but by Archimedes a millenium earlier:

For seventy years, a prayer book moldered in the closet of a family in France, passed down from one generation to the next. Its mildewed parchment pages were stiff and contorted, tarnished by burn marks and waxy smudges. Behind the text of the prayers, faint Greek letters marched in lines up the page, with an occasional diagram disappearing into the spine.

The owners wondered if the strange book might have some value, so they took it to Christie’s Auction House of London. And in 1998, Christie’s auctioned it off—for two million dollars.

For this was not just a prayer book. The faint Greek inscriptions and accompanying diagrams were, in fact, the only surviving copies of several works by the great Greek mathematician Archimedes.

The kind of mathematics that Archimedes was doing look a lot like standard problems on integration that Calculus II students mutter about today — finding the areas of curved figures, finding the volumes of solids via cross-sectional area sums, and so on.

The article goes into depth about what makes Archimede’s work such a breakthrough, namely the willingness to work with “actual infinity” instead of “potential infinity”. Contemporaries like Aristotle believed that actual infinity didn’t exist; instead, the world is full of potential infinities. For example, lines that are infinitely long do not exist, only lines which are finite but could be extended, hypothetically, to infinite lengths.  The article points out that today, we don’t use actual infinity but rather potential infinity, which is the basic underpinning of the concept of the limit which in turn is what all of calculus is based on.

This is a major discovery — the kind that makes you think about the what-if questions of how the world might have turned out differently if calculus had taken off 700 years prior to Newton. And it’s somehow fitting that high-tech microscopic methods, developed no doubt using the very calculus that Archimedes must have envisioned, were used to extract Archimedes’ handwriting from the copied-over manuscript.