For some time now, I’ve been stuck on the most ridiculous of math questions:
in which I’m trying to determine what is the late-orders behaviour of this equation (as n tends to infinity).
On the surface, it seems ridiculous. The equation is linear, and third order. It must be easy! Computing things numerically tells me that
for any number b,c, and d. However, the second upper bound is close. So very, very close…
The correct form of solution almost certainly involves logarithms, and there is a general theorem by Birkhoff and Trjitzinksy from 1933 which says that the form of the solution is probably something like
where you have to determine all the associated constants.
But that form is just so unwieldy it makes it impossible to figure out what’s right-side-up or right-side-down.
This is one of those things that just makes you want to tear your hair out!
Edit:
About 30 minutes after I made that post, I realized that
is the asymptotic behaviour I wanted (multiplied by some appropriate error). Jeez Louise. I should have noticed earlier that
except that last greater-than sign was the ‘almost’ I was looking for above.
Moments like these don’t make me happy.
They only make me a little bit less self conscious of my surprising inadequacy.
Today, I finally wrapped up most of the missing details of the main result of my Ph.D. research—which, to describe it in simplistic terms, proposes the theoretical existence of new types of never-before-seen (or computed) water waves.
And to me, that’s a very special thing. Historically (and as far as I know), applied mathematics has typically not been used in the prediction of new physical phenomena. Generally, it goes the other way around.
Take, for example the discovery of the soliton, which is generally attributed to the Scottish engineer John Scott Russell. Russell had been traveling beside a canal in Edinburgh when he noticed that a “great wave of translation” was moving through the water—a wave which was, surprisingly, moving along with little change to form or speed.
People trying to re-create Russell’s great wave of translation
Fluid dynamicists and mathematicians then studied this phenomenon, which led to a great many new theories and in fact, new physical applications (solitons are now used in fiber optics, as they provide a very stable signal, not prone to degradation).
But this is generally how things proceed: First, there is a physically motivated observation which goes against conventional theory, then it’s handed off to the applied mathematicians, who model it and develop theories for its predictions, then perhaps it’s then even handed off to the pure mathematicians who continue to extend and rigorously prove these theories.
Rarely does it go the other way around. One historical example in which it did, and which I love to teach in my classes—is the discovery of the planet Neptune. The history, very briefly, goes like this.
Astronomers had been puzzled by the orbits of the planet Uranus, which led them to theorize that an unknown planet was perturbing the orbit through gravitational interaction. The French astronomer Urbain Le Verrier then produced theoretical predictions of the orbits of this unknown planet. He then sent an urgent letter to an astronomer in Berlin, telling him where to look and at what time.
And sure enough, within a percent of his prediction, Neptune was seen and documented for the first time.
But these events are rarities. In fact, you could even argue that the discovery of Neptune wasn’t truly all the work of mathematics—we knew something was perturbing Uranus’ orbit, and so it was then left to the theory in predicting where this perturbation was originating from.
In my case, it seems that as a result of my research (technically, I would use ‘our’ to encompass my supervisor and myself…but this is my blog, after all), I’ve predicted the existence of new kinds of water waves.
Sure, it’s exciting. But it’s also a bit worrisome.
What if I made a mistake? What if it was all a fluke of the pencil? It’s disconcerting to produce a physical theory without a physical observation to go along with it.
And more worrisome is that I’m quite pessimistic that they’ll be observed any time soon, either through experiments or through numerical computations. The difficulty is that they exist in a very difficult-to-reach regime. In essence, they would be very difficult to `see’ in real life, and very difficult to compute using simulations.
It’s sometimes said that a mathematician is a blind man in a dark room looking for a black cat which isn’t there.
In my case, that’s not quite true.
I’m pretty sure that damn cat is there. I can hear it meowing and everything.
I couldn’t help but laugh when I read this inscription written on the Facebook page for a group of 17,000 high school students:
We’ve got just 3 terms of school left and then we’re officially free. Get psyched for leavers and all that shiz. Whether you’re doing TEE or not, after year 12 we’ll have LIVES.
Of course, seventeen-thousand high school students (and more) will be in for a world of disappointment when they realize that life after high school is not all that different from life before and during. If anything, there’s such a nice orderly air to it.
But of course, when you’re at that age, all you can think about is getting out. All you can think about is how different your life will be without them stinkin’ rules and overbearing teachers and dreadful cliques.
Oh dear. What I wouldn’t give to have that sort of naive outlook on life, again.
There is an article here about how Ontario college students are barely making the grade in math.
A new study shows a third of first-year college students in Ontario are in danger of not graduating because they flunked or barely scraped through their math course.
Researchers at Seneca College who conducted the study say that equates to about 10,000 students.
About 67 per cent of students achieved good grades — As, Bs, and Cs — slightly better than last year.
The governing Liberals are focusing on post-secondary education as a way to pull Ontario out of a major economic recession.
Monday’s throne speech promised to increase the portion of the province’s population that has a university or colleges education to 70 per cent from 62 per cent.
The government also promised to create 20,000 new post-secondary spaces this fall and increase the number of foreign students to about 54,000 from 37,000.
The Seneca study focused on the math results of 30,000 college students, but also examined the records of almost 80,000 students who enrolled in college in the fall of 2008.
Is this really cause for worry? Or have things been blown out of proportion?
Last year, I remember attending an math-education talk at the British Applied Maths Colloquium. The talk, like many of the educational talks these days, was a long complaint about how today’s educational system is abysmal and about how mathematics as a field is suffering. The speaker’s examples were not taken out of courses taken by students majoring in mathematics, but rather math courses taken by a variety of students.
At the end of the talk, I raised my hand.
“Can you tell me,” I asked, “How much of an effect these trends have had on the number of research mathematicians produced by the school?”
The speaker didn’t answer the question particularly very well. He mumbled something about the fact that there probably wasn’t much of an effect (!) on high level research and moved on.
My point was simple: In today’s day and age, people may have lost appreciation for mathematics as a population, but the impact on actual research—that is, mathematics of the highest level—was minimal.
The first thing we have to address is the sample group in the above study.
In Canada, “college” doesn’t refer to undergraduate universities, but rather to post-secondary community or technical colleges.
With this in mind, it’s not a stretch to remark that the students who attend these colleges are usually not as academically strong as their high school peers who chose to pursue a university education—at least, in the sciences and mathematics.
How many of these college students will go on to pursue careers in Engineering, Physics, Mathematics, or one of the other sciences? Very few down to nil would be my guess.
Later down in the article, I see the problem:
Monday’s throne speech promised to increase the portion of the province’s population that has a university or colleges education to 70 per cent from 62 per cent.
Today, everybody wants numbers. It’s not enough to provide equal opportunity to education, we need the numbers to rise. And so we’re widening our net.
Now don’t get me wrong. This is not a bad thing. It’s good to push students to learn mathematics. But the problem is that we expect everybody to be good at it. The statistics are misleading because we’re opening our doors to the rift-raft and allowing pretty much anybody to get into colleges and universities. So of course when it comes to a subject like mathematics, the number of people who flunk the course is going to increase!
Some people might get offended by what I’ve just said.
The funny thing is that intellectual ability is not seen in the same light as physical ability.
That is to say that people freely accept the fact that a sports superstar like Usain Bolt is genetically designed to do things none of us are capable of—and yet, people are surprisingly resistant to the fact that certain intellectual activities may be beyond them.
The key [to solving a math problem] is often brilliant but difficult. There may be a psychological unwillingness to accept that there is a world of brilliance and of understanding which may exceed [the student's]. There may be a sudden revelation that some higher mathematics is beyond them completely and this comes as a shock and a blow to the ego.
[...]
It is commonly thought that there are “math types” and “non-math types”. No one knows why people take to math easily and others with enormous difficulty. For non-math types, resistance may be the honest reaction to innate limitations. Not everyone becomes a piano payer or an ice skater. Why should it be otherwise for mathematics?
Davis and Hersh in the mathematical experience
I wanted to talk about the very nice article “Recruitment and retention of mathematics students in Canadian universities” by Fenwick-Sehl et al. (available here for educational purposes). In the article, they make the following points:
1. The number of mathematics degrees at the undergraduate and graduate levels remained relatively constant between 1992 and 2005
and,
2. The total number of mathematics degrees as a percentage of all degrees awarded has slightly decreased over the same time period.
The explain this trend through various theories, among some of them:
A. Recruitment and retention are not at the top of the priority list for most mathematics departments.
B. There is a migration to other math-related disciplines, such as biology or medicine.
C. Self-selection out of mathematics: The very best students in mathematics may believe they have a wider range of career options outside of mathematics (such as in engineering, medicine, and so on).
And so just a quick glance at this article ‘confirms’ my suspicions: All this hubbub about the abysmal performance of students in mathematics is taken out of context. Surely in 2010, we’re getting significantly more people into higher education, and outputting significantly more math-related researchers.
The government needs to understand that these statistics are taken out of context. The fact that most community college students are failing math does not imply that the research output of mathematics-related disciplines has faltered.
If anything, it’s increased significantly in the last few decades.
And finally, my criticism about natural abilities stands firm. I’m not saying that we shouldn’t aim to improve mathematical literacy. I’m saying that simply increasing the number of students who are required to take a first-year Calculus course is not going to suddenly produce an explosion of mathematicians.
If you were to suddenly put every single member of a school’s population into a rigorous track and field program, you would not suddenly produce a dozen superstars, nor could you realistically expect every student to have the natural athleticism to make it through the program. It’s a good thing, of course. People are exercising more and you’re going to expose the sport to people who may not have normally had the chance to participate.
But you’re also going to have to deal with a lot more wheezing, fainting runners than usual.
It’s okay for people to admit that they can’t sprint a hundred meters in under 12 or 13 seconds. So why is it not okay for some people to admit that there are some intellectual sprints that are beyond them?
The most recent episode of Community, NBC’s comedy about a rag-tag (Breakfast Club-like) group of community college goers, has cemented its reputation as one of the hot new TV shows.
In the episode, which is called Physical Education, the main storyline involves one of the character’s discomfort when his gym teacher tells the class they must wear shorts in order to participate in the college’s “Billards” class.
And like a lot of the brilliant episodes, it involves a terrific musical montage at the end, culminating in…well, you just have to watch.
Spoilers!
The show is centred upon a group of seven students: Jeff, the lawyer and arguably the main character; Brita, the aimless twenty-something activist; Pierce, the old buffoon; Troy, the former high-school star athlete; Abed, the pop-culture guru; Shirley, the recently divorced mother; and Annie, the repressed Jewish high-school dropout.
The show has a lot of heart and even more pop culture references. It’s quite zany and is just a laugh riot.
