Casting Out Nines

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? 

Sol Lederman, who runs the blog Wild About Math!, has started a new blog about something called Brain Integration, billed as “a revolutionary stress management process that permanently improves the flow of information in the brain in less than ten hours, with no drugs or surgery.” Sol describes the focus of his new blog:

It journals my personal experience with being CURED from a lifetime of ADD. Before Brain Integration I couldn’t sit still for very long, I’d shy away from detail-oriented tasks or tasks that required organization, my focus was poor, I would get easily distracted, and my self-esteemed suffered from all of that. Post-ADD, I’m calm, centered, can sit at the computer for hours at a time if I need to get something done, I don’t get distracted when I need to focus, I’m organized, and I’m willing to do detail-oriented tasks.

Sounds interesting, and somehow it seems counter-cultural to suggest that ADD is a  condition which ought to be cured and can be without expensive and pervasive drug therapy.

I remember working during graduate school as a tutor at one of those expensive tutoring service franchises in an affluent part of Nashville, and we’d get parents all the time coming in proclaiming — with an attitude that was approaching pride — that their child was ADD and/or Learning Disabled in Math. Whenever I’d suggest that, with proper tutoring and learning strategies being employed, the child might once and for all get over ADD or LD, the parents actually seemed appalled, like if that ever happened then their child would have to live up to normal expectations, God forbid.

(Disclaimer: Before anyone gets offended at that last statement — Two of the better students I have ever had in my career were clinically diagnosed with ADD/ADHD, and rather than accept that they simply were psychologically unable to perform well at college level math or insist that success in math be defined down to terms they could easily accomplish, they worked with me to learn how to study the subject in ways that worked for them, and then worked their tails off, and ended up outperforming most of the other non-ADD students in the class. I think that they might have discovered on their own something like what Sol is blogging about now. I’ve got no problem working with ADD students; what I do have a problem with is someone accepting an ADD diagnosis as though it were some divine revelation of their eternal destiny, forever capping their ability to work and think and totally out of their control.)

A key element in being a college-educated person, especially in mathematics, is what athletes call mental toughness. This term can be a pop-psychological artifact with no real meaning, but if you look here and here and other places on the web, the general idea is that mental toughness is a combination of resilience in the face of minor and major failures; the ability to cope with difficult and numerous demands; confidence; focus; and  determination. Or better yet, it looks like this:

I believe mental toughness is key because, in college, you are preparing yourself for the rest of your life out of school, where the edges are harder and the difficulties far greater than just doing well on the next exam or getting a decent grade in your calculus class. Real people in the real world have to handle adversity, especially the particular adversity that comes from having ideas, thoughts, and proposed solutions shot down in flames.

College is an excellent training ground to develop mental toughness, and in mathematics that development is particularly acute because of the clarity with which right things are right and wrong things are wrong. Math students, especially math majors, ought to have the toughest minds around, because they have been tested and pushed to their utmost, they have summoned the intellectual honesty to admit it when their work has flaws — sometimes major ones — and they have developed the habit of working on through the injuries to finally win the match, so to speak. They should not be the ones who, when confronted with flaws in their performances, simply take it as a personal offense and fold up, unable to summon the will to keep on working.

So, a question:

How can an academic course or program, accomplish this task, when the very thing that catalyzes mental toughness - adversity couple with reality — is seen as offensive and humiliating? I can understand it if the professor is visibly and intentionally acting to humiliate or intimidate students; but if the prof is impassively and objectively pointing out problems in a student’s work, and the student feels that the prof is intimidating and humiliating them, then what is to be done? Does the prof overcompensate and become a sort of Barney-like figure, exuding love and goodwill while at the same time pointing out that does not, in fact, equal ? At what point should the student just take his/her lumps and deal with it?

Following up on his three posts on classical education yesterday, Gene Veith weighs in on mathematics instruction: 

I admit that classical education may be lagging in the math department. The new classical schools are doing little with the Quadrivium, the other four liberal arts (arithmetic, geometry, astronomy, and music). The Trivium, which is being implemented to great effect (grammar, logic, and rhetoric), has to do with mastering language and what you can do with it. The Quadrivium has to do with mathematics (yes, even in the way music was taught).

This, I think, is the new frontier for classical educators. Yes, there is Saxon math, but it seems traditional (which is better than the contemporary), rather than classical, as such.

Prof. Veith ends with a call for ideas about how mathematics instruction would look like in a classical education setting. I left this comment:

I think a “classical” approach to teaching math would, going along with the spirit of the other classical education posts yesterday, teach the hypostatic union of content and process — the facts and the methods, yes (and without cutesy gimmicks), but also the processes of logical deduction, analytic problem-solving heuristics, and argumentation. Geometry is a very good place to start and an essential to include in any such approach. But I’d also throw in more esoteric topics as number theory and discrete math (counting and graph theory) — in whatever dosage and level is age-appropriate.

At the university level, and maybe at the high school level for kids with a good basic arithmetic background, I’d love to be able to use the book “Essential College Mathematics” by Zwier and Nyhoff as a standard one-year course in mathematics (and in place of the usual year of calculus most such students take). It’s out of print, but the chapters are on sets; cardinal numbers; the integers; logic; axiomatic systems and the mathematical method; groups; rational numbers, real numbers, and fields; analytic geometry of the line and plane; and finally functions, derivatives, and applications. You have to see how the text is written to see why it does a good job with both content and process.

(I took out the mini-rant against the gosh-awful Saxon method.)

Any thoughts from the audience here?

One of my linear algebra students is an education major doing student teaching. Today he showed me this method of simplifying radicals which he learned from his supervising teacher. Apparently it’s called the “Illini method”. Googling this term returns nothing math-related, so I think that term was probably invented by his supervisor, who went to college in Illinois.

The procedure goes as follows. Start with a radical to simplify, say . Look under the radical and find a prime that divides it, say 5. Then form a two-column array with the original radical in the top-left, the divisor prime in the adjacent row in the right column, and the result you get from dividing the radicand by that prime number in the left column below the radical. In this case, it’s:

Now look for a prime that divides the lower-left term, say another 5. Again, put the dividing prime across from the dividend, and the quotient below the dividend. With our example, the array at this stage looks like:

In general, continue this process of dividing prime numbers into the lower-left entry in the array, writing the prime across from that entry, and writing the quotient beneath that entry, until you end up with a 1 in the lower-left entry. So the final state of our example would be:

Now, look at the left-hand column of the array. Group off any pairs of numbers you see. Multiply together all numbers which are representative of a pair. In our case, there is only one such pair, a pair of 5’s. Any numbers that occur singly are placed under a radical and multiplied. In our case, that’s the single 2. Then multiply the product of numbers which are in pairs times the radical which contains the singleton numbers. So we end up in our example with .

Here’s another example with a larger number, :

There are three groups of 2’s, so outside the final radical we’ll put . And the 3 and 11 are by themselves, so under the radical we put 33. Hence .

Pretty clearly, all this method is doing is presenting a different way to do the bookkeeping for doing the prime factorization of the number under the radical. The final step of grouping off the prime pairs and leaving the un-paired primes under the radical is analogous to finding all the squared primes in the prime factorization.

This method is nice and systematic, and I can see why students (and student-teachers) might like it. But it seems to be obscuring some important concepts that students ought to know. With the method of factoring, looking for squared primes, and then removing them from the square root, at least you are dealing directly with the inverse relationship between squares and square roots. The Illini method, on the other hand, uses an approach of “put this here and then put that over there” with minimal contact with actual math. It does work, and it does keep things in order. But do students really understand why it works?

Your thoughts?  What does this method make clearer, and what does it obscure? Should high school algebra teachers be teaching it?

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.

Maria at Teaching College Math Technology Blog has the scoop. The comparison between manipulatives and sex toys is particularly, er, interesting.

Check out the unofficial course description for Math 55 at Harvard’s math department web site (all emphases added):

This is probably the most difficult undergraduate math class in the country; a variety of advanced topics in mathematics are covered, and problem sets ask students to prove many fundamental theorems of analysis and linear algebra. Class meets three hours per week, plus one hour of section, and problem sets can take anywhere from 24 to 60 hours to complete. This class is usually small and taught by a well-established and prominent member of the faculty whose teaching ability can vary from year to year. A thorough knowledge of multivariable calculus and linear algebra is almost absolutely required, and any other prior knowledge can only help. Students who benefit the most from this class have taken substantial amounts of advanced mathematics and are fairly fluent in the writing of proofs. Due to the necessity of working in groups and the extensive amount of time spent working together, students usually meet some of their best friends in this class. The difficulty of this class varies with the professor, but the class often contains former members of the International Math Olympiad teams, and in any event, it is designed for people with some years of university level mathematical experience. In order to challenge all students in the class, the professor can opt to make the class very, very difficult.

The most difficult undergraduate math class in the country,  taught by faculty whose teaching ability is not necessarily guaranteed, designed for freshmen but requiring several years of university level math experience? I detect a distinct amount of satisfaction from whomever wrote this.

The funny thing is, according to the article I referenced earlier, dozens of Harvard freshmen sign up for the course each year — some with no intention whatsoever of staying on in the course but just wanting to say they’d enrolled in the course and watch the real math people at work.

Via Vlorbik, here’s a letter to the editor (PDF) of the AMS Notices by Seymour Lipschutz extolling the virtues of Schaum’s Outlines as course texts and giving some suggestions for those choosing textbooks.

I agree with Lipschutz’ feelings about Schaum’s Outlines, up to a point. I’m a big fan of Schaum’s Outlines; they cost less than $20 and are loaded with precise, succint summaries of course material and worked-out problems. I
survived college physics and advanced calculus largely because of my now-battered Schaum’s Outlines for those subjects. I ordered the latest edition of the differential equations Outlines as I was considering using it for my DE course next semester, and I liked what I saw very much; and the publisher sent me a gratis copy of the beginning calculus Outlines and it was very good as well. I will be suggesting these outlines strongly to the students in those courses.

But to use them as the textbook for a course? I’m a little skeptical.  They are, after all, outlines. I think that students in the lower-level courses like calculus, and to some extent mid-level courses like DE’s or linear algebra, would benefit from having a more fully-featured textbook.

On the other hand, a carefully-written set of course notes made up by the professor, augmented by Schaum’s Outlines and hand-picked resources from the web, make up a pretty good blueprint for a cheap, portable, and effective package of course materials that I think students would get a lot more out of than a single monolithic textbook that they can’t carry around easily and never read.

If you’ve been submitting mathematics articles to refereed journals only to have them sent back to you every time, there’s hope. You can try submitting them to the new journal Rejecta Mathematica, which will consist only of papers which have been rejected from peer-reviewed journals. From their web site:

At Rejecta Mathematica, we believe that many previously rejected papers can nonetheless have a very real value to the academic community. This value may take many forms:

• “mapping the blind alleys of science”: papers containing negative results can warn others against futile directions; 
• “reinventing the wheel”: papers accidentally rederiving a known result may contain new insight or ideas; 
• “squaring the circle”: papers discovered to contain a serious technical flaw may nevertheless contain information or ideas of interest;
• “applications of cold fusion”: papers based on a controversial premise may contain ideas applicable in more traditional settings;
  
• “misunderstood genius”: other papers may simply have no natural home among existing journals.

Rejecta articles also allow the authors to speak out in defense of their rejected articles and include an open letter from the authors describing any known flaws in the paper.

And yes, although there’s no formal peer review process to get a paper into Rejecta, you can still have a paper submission rejected.

[ht Math-Blog]