Math Blog from the Other Coast

An excellent mathematics-based blog has come across my (virtual) desk:

What's new

The diarist, Terence Tao, a professor in the Mathematics Department at UCLA, explores mostly his research and related issues, so some of the posts may be stratospheric and out of the reach of many math enthusiasts. But there are also posts with good air-pressure, and some excellent advice on math-related careers and writing techniques, as well as mathematical constructs and such. For instance, his latest post on the nature of "proof-by-contradiction" makes for excellent reading. Give it a try!

Mathematics Study Tips - Banking practice problems

In the recent post Mathematics Study Tips - Scrimmaging, I talked about a method for studying for a mathematics exam by mimicking, as best as you can, the exam environment. Sounds obvious, right? One doesn't create a soldier by watching war movies. The idea of practicing for an exam by doing new problems in a timed environment works to gauge your understanding of the material while reducing the stress of an exam by acclimating yourself to the climate within the exam. Here is another idea, mentioned in that last post:

In an actual exam, there usually will NOT be a marker on each problem telling you, for example, that "this problem is from Section 3.2 on the Mean Value Theorem". Instead, all you will see is the statement of the problem (and an implicit promise by the Instructor that the problem falls within the scope of the course). Without the context of which section the problem came from, can you still manage to do the problem? One way to help you to be sure is to take your problems out of the context they are in. Try this:

After each section of a text has been discussed in lecture and you have completed the homework problems for submission, take some time to re-write some of the other section exercises (ones that are "like" the ones in your homework set) in a common place later in your notebook. Add other problems when other sections are completed. Rewrite these problems verbatim from the text, but do not write the section or problem number. Don't DO these problems, just bank them for later.

Now when the exam approaches, and you are looking for items to focus on, go to this section of your notebook, grab 5 or 6 of these "banked" problems, go to a quiet, distraction-less place, and time yourself doing these problems without notes, text, or any other aid. If you can do the problems with ease, you are ready for these types of problems on the exam. If you cannot, however, or if some of them prove difficult, then simply re-orient these problem problems with their original sections and note these sections as ones you still need to focus on. Out of context, these problems are much closer to what you will see on an exam.

Another way to do this is to work with someone else, who can grab a problem from the text without telling you which section it comes from. While this is easier and requires little prior planning, it does involve more than one person. But then again, talking mathematics with others is really how one learns, right?

Again, in bocca al lupo!

Math in the Media - The Professional Take

Got an interesting email blast on a service offered by one of the professional organizations for mathematicians, the American Mathematical Society (AMS). The heads-up comes from Dr. Ellen Maycock, the Associate Executive Director of the AMS. An excerpt:

The American Mathematical Society would like to remind you of a special service we offer, Headlines & Deadlines for Students, providing email notification of mathematics news and of upcoming deadlines that are of special interest to both graduate and undergraduate students. These email notifications will be issued about once a month, and when there's special news. Imminent deadlines will be included in these emails, which will link to a web page that's a centralized source for information relevant to students and faculty advisors, at

http://www.ams.org/news-for-students/.

We hope that you will share this email with the appropriate individuals in your department. It's not necessary to be a member of the AMS to sign up for this email service, at

http://www.ams.org/news-for-students/signup.

I will add that the news items highlighted on this website may also be of interest to pure enthusiasts of this discipline (and not just students of the field). It's good stuff, and I will be highlighting at times some articles mentioned here. But for now, take a look and sign up if you want the email service.

Beyond Calculus II: Honors Vector Calculus or not...

A common question I often get from our very ambitious undergraduates focuses on a choice of vector calculus classes we offer. Vector Calculus, Calculus III, and Multivariable Calculus are all names for the same basic study of the properties of functions of more than one independent variable. This material is required for most engineering disciplines, as well as mathematics, and most of the natural science majors here at Hopkins. Since the techniques and material makes a lot more sense once students have studied most of the properties of functions of one independent variable, this course naturally follows the Calculus I and II sequence.

We have two flavors of vector calculus here at Hopkins:

110.202 Calculus III and 110.211 Honors Multivariable Calculus

The basic question is; Which should I take?

The basic answer is: depends....

Both of these courses fulfill the same requirements for all majors and minors that require multivariable calculus. Both can serve as prerequisite courses for any higher level course that requires multivariable calculus. Both cover the same basic material over the length of one semester, and run from the basic notions of vectors, matrices and the real space R^n through notions of continuous and differentiable functions of more than one independent variable, ending the basic material with the final major theorems tying together major aspects of the course: Green's, Stokes' and Gauss' Theorems.

The major difference between these two courses is one of focus. 110.202 Calculus III is more of a standard Calculus course, developing a blend of theoretical background on the nature of functions of more than one independent variable and the actual calculations involved in solving problems pertaining to this material. 110.211 Honors Multivariable Calculus, on the other hand, spend much more time on the theoretical nature of the material, digging deeper into the "why" aspects of calculus instead of "how things work". Students in the latter will develop a better understanding of content like the Inverse and the Implicit Function Theorems, and learn better how to analyze functions and problems that are not so straightforward. Furthermore, the honors version goes a bit beyond Gauss' Theorem and 110.202, with an introduction to differential forms, and a basic development of a generalized unified theory of the latter three theorems entitled "generalized Stokes'". Both courses are a challenge, but the latter is more so.

Students getting a BC score of 5 (or a 110.109 grade of B+ or better) can be encouraged to take this version if they are so inclined. Students with less strong scores should stay in 110.202, or at least should inquire further with the Math Department before registering for the honors version. In either case, while 110.211 is indeed a great course in vector calculus, taught the way mathematicians really want to teach a math course, it should be understood that the course will be quite a serious challenge.

Course sizes typically run over 100 easily for each lecture of 110.202, with about 4 recitation sections of 25 each. In contrast, 110.211 runs with about 40 students in 2 recitation sections.

Though always self-selected, students are usually quite enthusiastic about the honors version. it is also great fun to teach!

Hope this helps....

Chicken or Egg or ...: Which comes first? CalcIII, LinAlg, or DiffEq?

If I can point to a set of questions that are asked most often of the Math Department, one definitely on the list is the following:

Is it better to take Calculus III or Linear Algebra first?

Throw Differential Equations into the mix, and you get a branching of one's math career into three distinct paths. All three of these courses, at least here at Hopkins, have a full year of single variable calculus as a prerequisite; necessary for technique as well as theory in the case of multivariable calculus (Calculus III) and Differential Equations, and necessary for a sufficient level of "mathematical maturity" in the case of Linear Algebra.

But for many majors, and interests, one must take courses in and well understand two, if not all three, of these topics. So what order makes the most sense?

Its a good question. It turns out, it is not really important....

I am starting a new series about these and other courses under the tag and title "Beyond Calculus II". In this series, I will explain better the idea and focus of these three (and other) courses taken after a full year of calculus is achieved. Here at Hopkins, a large population of our students start their tenure here at this level.

For now, though, let's stick with the topic above. To start:

There is neither multivariable calculus nor differential equations in linear algebra, yet there is a bit of linear algebra in both of the others. In contrast, linear algebra is a more mature course, sometimes requiring more in the way of expanding one's frame of reference mathematically than the other two.

That said, we actually took a look at performance among students who took the two courses 110.202 Calculus III and 110.201 Linear Algebra back to back over a two year period (there are quite a few of them). I will pass on the details of this study, but we found that there was no real preferred order to these two, at least as far as ultimate grades go.

Couple this with the fact that any linear algebra found in either calculus III or differential equations is essentially covered within the courses, and any of the three may be taken in any real order. Hence preference for time slots, professors, and/or friends in the course may be of higher priority in your choice than content.

And one last note, our course 110.302 Differential Equations, fairly standard in content with most sophomore-level courses at American universities, is a course in ordinary differential equations (involving functions of one independent variable, in contrast with partial differential equations). One can easily describe this course as Calculus II.5 (weird notation, hih?). It can be viewed as the proper successor to Calculus II, rather than Calculus III. Just sayin....

Math in the Media: Math and the City?!?

Drawing a crowd in close enough to hear what you have to say is what a headline is all about. A guest columnist for the New York Times' The Wild Side, Steven Strogatz, filling in for Olivia Judson, pulled me in easily with his headline in the column on May 19th:

Math and the City

Though not quite similar to Sarah Jessica Parker (et.al.)'s work, it is a good title.

Here, instead, Mr. Strogatz discusses a couple examples of the mathematics of life that mathematicians tend to see everywhere via their training; patterns, proportions and logical structure that show up again and again in disparate contexts. In this case, Zipf's Law on the frequency of word usage in a language, patterns in economies of scale, similarities in the energy needs of a city based on its size to the energy needs of mammals based on their size all share a remarkable one-ness in their structure. "Spooky" is Steven's word fo it.

Its a good read....

One personal note, though (said with tongue firmly planted in cheek): Steven leads the article with

"One of the pleasures of looking at the world through mathematical eyes is that you can see certain patterns that would otherwise be hidden."

Right, he is. Sometimes is seems kinda like what Neo sees at the end of The Matrix (although in our case there is no trace of any sort of messianic behavior, no doubt).

And it is quite a pleasure. That is, when it isn't a curse.

Sleeping; a good study tip?

Want to do better on your next math exam? Try sleeping well the night before. Easy right?

Apparently, cramming may not be the right approach to optimizing performance. This article in the Daily Telegraph today by Science Corrrespondent Richard Alleyne,

Good night's sleep adds up to better exam results - especially in maths

announces the results of a University of Pittsburgh study that says

a night of "high quality sleep" helps schoolchildren get better exam results - especially in maths.

Arguably, the study is small (56 students) and the article quotes another article from the Daily Telegraph who quotes the study. But the results make sense, at least from my perspective.

One caveat: The type of sleep most effective is not long in time, but restful in nature, with few if any awakenings or disturbances. However, knowing one has an exam the next day may be cause enough to make the sleep not so restful, no?

Still.... There is good advice in these results.