College Math Teaching

May 22, 2013

In the news….and THINK before you reply to an article. :-)

Ok, a mathematician who is known to be brilliant self-publishes (on the internet) a dense, 512 page proof of a famous conjecture. So what happens?

The Internet exploded. Within days, even the mainstream media had picked up on the story. “World’s Most Complex Mathematical Theory Cracked,” announced the Telegraph. “Possible Breakthrough in ABC Conjecture,” reported the New York Times, more demurely.

On MathOverflow, an online math forum, mathematicians around the world began to debate and discuss Mochizuki’s claim. The question which quickly bubbled to the top of the forum, encouraged by the community’s “upvotes,” was simple: “Can someone briefly explain the philosophy behind his work and comment on why it might be expected to shed light on questions like the ABC conjecture?” asked Andy Putman, assistant professor at Rice University. Or, in plainer words: I don’t get it. Does anyone?

The problem, as many mathematicians were discovering when they flocked to Mochizuki’s website, was that the proof was impossible to read. The first paper, entitled “Inter-universal Teichmuller Theory I: Construction of Hodge Theaters,” starts out by stating that the goal is “to establish an arithmetic version of Teichmuller theory for number fields equipped with an elliptic curve…by applying the theory of semi-graphs of anabelioids, Frobenioids, the etale theta function, and log-shells.”

This is not just gibberish to the average layman. It was gibberish to the math community as well.

[…]

Here is the deal: reading a mid level mathematics research paper is hard work. Refereeing it is even harder work (really checking the proofs) and it is hard work that is not really going to result in anything positive for the person doing the work.

Of course, if you referee for a journal, you do your best because you want YOUR papers to get good refereeing. You want them fairly evaluated and if there is a mistake in your work, it is much better for the referee to catch it than to look like an idiot in front of your community.

But this work was not submitted to a journal. Interesting, no?

Of course, were I to do this, it would be ok to dismiss me as a crank since I haven’t given the mathematical community any reason to grant me the benefit of the doubt.

And speaking of idiots; I made a rather foolish remark in the comments section of this article by Edward Frenkel in Scientific American. The article itself is fine: it is about the Abel prize and the work by Pierre Deligne which won this prize. The work deals with what one might call the geometry of number theory. The idea: if one wants to look for solutions to an equation, say, x^2 + y^2 = 1 one gets different associated geometric objects which depend on “what kind of numbers” we allow for x, y . For example, if x, y are integers, we get a 4 point set. If x, y are real numbers, we get a circle in the plane. Then Frenkel remarked:

such as x2 + y2 = 1, we can look for its solutions in different domains: in the familiar numerical systems, such as real or complex numbers, or in less familiar ones, like natural numbers modulo N. For example, solutions of the above equation in real numbers form a circle, but solutions in complex numbers form a sphere.

The comment that I bolded didn’t make sense to me; I did a quick look up and reviewed that |z_1|^2 + |z_2|^2 = 1 actually forms a 3-sphere which lives in R^4 . Note: I added in the “absolute value” signs which were not there in the article.

This is easy to see: if z_1 = x_1 + y_1 i, z_2 = x_2 + y_2i then |z_1|^2 + |z_2|^2 = 1 implies that x_1^2 + y_1^2 + x_2^2 + y_2^2 = 1 . But that isn’t what was in the article.

Frenkel made a patient, kind response …and as soon as I read “equate real and imaginary parts” I winced with self-embarrassment.

Of course, he admits that the complex version of this equation really yields a PUNCTURED sphere; basically a copy of R^2 in R^4 .

Just for fun, let’s look at this beast.

Real part of the equation: x_1^2 + x_2^2 - (y_1^2 + y_2^2) = 1
Imaginary part: x_1y_1 + x_2y_2 = 0 (for you experts: this is a real algebraic variety in 4-space).

Now let’s look at the intersection of this surface in 4 space with some coordinate planes:
Clearly this surface misses the x_1=x_2 = 0 plane (look at the real part of the equation).
Intersection with the y_1 = y_2 = 0 plane yields x_1^2+ x_2^2 = 1 which is just the unit circle.
Intersection with the y_1 = x_2 = 0 plane yields the hyperbola x_1^2 - y_2^2 = 1
Intersection with the y_2 = x_1 = 0 plane yields the hyperbola x_2^2 - y_1^2 = 1
Intersection with the x_1 = y_1 = 0 plane yields two isolated points: x_2 = \pm 1
Intersection with the x_2 = y_2 = 0 plane yields two isolated points: x_1 = \pm 1
(so we know that this object is non-compact; this is one reason the “sphere” remark puzzled me)

Science and the media
This Guardian article points out that it is hard to do good science reporting that goes beyond information entertainment. Of course, one of the reasons is that many “groundbreaking” science findings turn out to be false, even if the scientists in question did their work carefully. If this sounds strange, consider the following “thought experiment”: suppose that there are, say, 1000 factors that one can study and only 1 of them is relevant to the issue at hand (say, one place on the genome might indicate a genuine risk factor for a given disease, and it makes sense to study 1000 different places). So you take one at random, run a statistical test at p = .05 and find statistical significance at p = .05 . So, if we get a “positive” result from an experiment, what is the chance that it is a true positive? (assume 95 percent accuracy)

So let P represent a positive outcome of a test, N a negative outcome, T means that this is a genuine factor, and F that it isn’t.
Note: P(T) = .001, P(F) = .999, P(P|T) = .95, P(N|T) = .05, P(P|F) = .05, P(N|F) = .95 . It follows P(P) = P(T)P(P \cap T)P(T) + P(F)P(P \cap F) = (.001)(.95) + (.999)(.05) = .0509

So we seek: the probability that a result is true given that a positive test occurred: we seek P(T|P) =\frac{P(P|T)P(T)}{P(P)} = \frac{(.95)(.001)}{.0509} = .018664. That is, given a test is 95 percent accurate, if one is testing for something very rare, there is only about a 2 percent chance that a positive test is from a true factor, even if the test is done correctly!

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March 18, 2013

Odds and the transitive property

Filed under: media, movies, popular mathematics, probability — Tags: — collegemathteaching @ 9:51 pm

I got this from Mano Singham’s blog: he is a physics professor who mostly writes about social issues. But on occasion he writes about physics and mathematics, as he does here. In this post, he talks about the transitive property.

Most students are familiar with this property; roughly speaking it says that if one has a partially ordered set and a \le b and b \le c then a \le c . Those who have studied the real numbers might be tempted to greet this concept with a shrug. However in more complicated cases, the transitive property simply doesn’t hold, even when it makes sense to order things. Here is an example: consider the following sets of dice:

dice

What we have going here: Red beats green 4 out of 6 times. Green beats blue 4 out of 6 times. Blue beats red 4 out of 6 times. All the colored dice tie the “normal” die. Yet, the means of the numbers are all the same.

Note: that this can happen is probably not a surprise to sports fans; for example, in boxing: Ken Norton beat Muhammed Ali (the first time), George Foreman destroyed Ken Norton and, Ali beat Foreman in a classic. Of course things like this happen in sports like basketball but when team doesn’t always play its best or its worst.

But this dice example works so beautifully because this “impossibility of the dice obeying a transitive ordering relation is theoretically impossible, by design.

Movies
Since the wife has been gone on a trip, I’ve watched some old movies at night. One of them was the Cincinnati Kid, which features this classic scene:

Basically, the Kid has a full house, but ends up losing to a straight flush. Yes, the odds of the ten cards (in stud poker) ending up in “one hand a full house, the other a straight flush” are extremely remote. I haven’t done the calculations but this assertion seems plausible:

Holden states that the chances of both such hands appearing in one deal are “a laughable” 332,220,508,619 to 1 (more than 332 billion to 1 against) and goes on: “If these two played 50 hands of stud an hour, eight hours a day, five days a week, the situation would arise about once every 443 years.”

But there is one remark from this Wikipedia article that seems interesting:

The unlikely nature of the final hand is discussed by Anthony Holden in his book Big Deal: A Year as a Professional Poker Player, “the odds against any full house losing to any straight flush, in a two-handed game, are 45,102,781 to 1,”

I haven’t done the calculation but that seems plausible. But, here is the real point to the final scene: the Kid knows that he has a full house but The Man is showing 8, 9, 10, Q of diamonds. He knows that the only “down” card that can beat him is the J of diamonds but he knows that he has 3 10’s, 2 A’s. So there are, to his knowledge, 52 - 9 = 43 cards out, and only 1 that can beat him. So the Kid’s probability of winning is \frac{42}{43} which are pretty strong odds, but they are not of the “million to one” variety.

March 5, 2013

Math in the News (or: here is a nice source of exercises)

I am writing a paper and am through with the mathematics part. Now I have to organize, put in figures and, in general, make it readable. Or, in other words, the “fun” part is over. 🙂

So, I’ll go ahead and post some media articles which demonstrate mathematical or statistical concepts:

Topology (knot theory)

As far as what is going on:

After a century of studying their tangled mathematics, physicists can tie almost anything into knots, including their own shoelaces and invisible underwater whirlpools. At least, they can now thanks to a little help from a 3D printer and some inspiration from the animal kingdom.

Physicists had long believed that a vortex could be twisted into a knot, even though they’d never seen one in nature or the even in the lab. Determined to finally create a knotted vortex loop of their very own, physicists at the University of Chicago designed a wing that resembles a delicately twisted ribbon and brought it to life using a 3D printer.

After submerging their masterpiece in water and using electricity to create tiny bubbles around it, the researchers yanked the wing forward, leaving a similarly shaped vortex in its wake. Centripetal force drew the bubbles into the center of the vortex, revealing its otherwise invisible, knotted structure and allowing the scientists to see how it moved through the fluid—an idea they hit on while watching YouTube videos of dolphins playing with bubble rings.

By sweeping a sheet of laser light across the bubble-illuminated vortex and snapping pictures with a high-speed camera, they were able to create the first 3D animations of how these elusive knots behave, they report today in Nature Physics. It turns out that most of them elegantly unravel within a few hundred milliseconds, like the trefoil-knotted vortex in the video above. […]

Note: the trefoil is the simplest of all of the non-trivial (really knotted) knots in that its projection has the fewest number of crossings, or in that it can be made with the fewest number of straight sticks.

I do have one quibble though: shoelaces are NOT knotted…unless the tips are glued together to make the lace a complete “circuit”. There ARE arcs in space that are knotted:

WildArc

This arc can never be “straightened out” into a nice simple arc because of its bad behavior near the end points. Note: some arcs which have an “infinite number of stitches” CAN be straightened out. For example if you take an arc and tie an infinite number of shrinking trefoil knots in it and let those trefoil knots shrink toward an endpoint, the resulting arc can be straightened out into a straight one. Seeing this is kind of fun; it involves the use of the “lamp cord trick”

Screen shot 2013-03-05 at 5.02.43 PM

(this is from R. H. Bing’s book The Geometric Topology of 3-Manifolds; the book is chock full of gems like this.)

Social Issues
It is my intent to stay a-political here. But there are such things as numbers and statistics and ways of interpreting such things. So, here are some examples:

Welfare
From here:

My testimony will amplify and support the following points:

A complete picture of time on welfare requires an understanding of two seemingly contradictory facts: the majority of families who ever use welfare do so for relatively short periods of time, but the majority of the current caseload will eventually receive welfare for relatively long periods of time.

It is a good mental exercise to see how this statement could be true (and it is); I invite you to try to figure this out BEFORE clicking on the link. It is a fun exercise though the “answer” will be obvious to some readers.

Speaking of Welfare: there is a debate on whether drug testing welfare recipients is a good idea or not. It turns out that, at least in terms of money saved/spent: it was a money losing proposition for the State of Florida, even when one factors in those who walked away prior to the drug tests. This data might make a good example. Also, there is the idea of a false positive: assuming that the statistic of, say, 3 percent of those on welfare use illegal drugs, how accurate (in terms of false positives) does a test have to be in order to have, say, a 90 percent predictive value? That is, how low does the probability of a false positive have to be for one to be 90 percent sure that someone has used drugs, given that they got a positive drug test?

Lastly: Social Security You sometimes hear: life expectancy was 62 when Social Security started. Well, given that working people pay into it, what are the key data points we need in order to determine what changes should be made? Note: what caused a shorter life expectancy and how does that effect: the percent of workers paying into it and the time that a worker draws from it? Think about these questions and then read what the Social Security office says. There are some interesting “conditional expectation” problems to be generated here.

March 3, 2013

Mathematics, Statistics, Physics

Filed under: applications of calculus, media, news, physics, probability, science, statistics — collegemathteaching @ 11:00 pm

This is a fun little post about the interplay between physics, mathematics and statistics (Brownian Motion)

Here is a teaser video:

The article itself has a nice animation showing the effects of a Poisson process: one will get some statistical clumping in areas rather than uniform spreading.

Treat yourself to the whole article; it is entertaining.

February 10, 2013

Just for fun: no professors allowed!

Filed under: basic algebra, calculus, Fourier Series, media — collegemathteaching @ 2:32 pm

I came across this article which had this photo:

Screen shot 2013-02-10 at 8.19.07 AM

Ok, here is the quiz:

Aluminum medal:

What are these formulas?

Bronze medal:

Derive one of these formulas

Silver medal:

Derive two of these formulas (hint: one way to derive one of these involves a change of variables and polar coordinates.

Gold medal:

Assuming you have a piecewise continuous function (ok, make it piecewise smooth if you wish) and is periodic over [-l,l] derive the middle formula.

You win: all of the money I made writing this note. 😉

February 8, 2013

Issues in the News…

First of all, I’d like to make it clear that I am unqualified to talk about teaching mathematics at the junior high and high school level. I am qualified to make comments on what sorts of skills the students bring with them to college.

But I am interested in issues affecting mathematics education and so will mention a couple of them.

1. California is moving away from having all 8’th graders take “algebra 1”. Note: I was in 8’th grade from 1972-1973. Our school was undergoing an experiment to see if 8’th graders could learn algebra 1. Being new to the school, I was put into the regular math class, but was quickly switched into the lone section of algebra 1. The point: it wasn’t considered “standard for everyone.”

My “off the cuff” remarks: I know that students mature at different rates and wonder if most are ready for the challenge by the 8’th grade. I also wonder about “regression to the mean” effects of having everyone take algebra 1; does that force the teacher to water down the course?

2. What would college students want to tell their high school teachers?

By Drew Appleby

I read Epstein School head Stan Beiner’s guest column on what kids really need to know for college with great interest because one of the main goals of my 40-years as a college professor was to help my students make a successful transition from high school to college.

I taught thousands of freshmen in Introductory Psychology classes and Freshman Learning Communities, and I was constantly amazed by how many of them suffered from a severe case of “culture shock” when they moved from high school to college.

I used one of my assignments to identify these cultural differences by asking my students to create suggestions they would like to give their former high school teachers to help them better prepare their students for college. A content analysis of the results produced the following six suggestion summaries.

The underlying theme in all these suggestions is that my students firmly believed they would have been better prepared for college if their high school teachers had provided them with more opportunities to behave in the responsible ways that are required for success in higher education […]

You can surf to the article to read the suggestions. They are not surprising; they boil down to “be harder on us and hold us accountable.” (duh). But what is more interesting, to me, is some of the comments left by the high school teachers:

“I have tried to hold students accountable, give them an assignment with a due date and expect it turned in. When I gave them failing grades, I was told my teaching was flawed and needed professional development. The idea that the students were the problem is/was anathema to the administration.”

“hahahaha!! Hold the kids responsible and you will get into trouble! I worked at one school where we had to submit a written “game plan” of what WE were going to do to help failing students. Most teachers just passed them…it was easier. See what SGA teacher wrote earlier….that is the reality of most high school teachers.”

“Pressure on taechers from parents and administrators to “cut the kid a break” is intense! Go along to get along. That’s the philosophy of public education in Georgia.”

“It was the same when I was in college during the 80’s. Hindsight makes you wished you would have pushed yourself harder. Students and parents need to look at themselves for making excuses while in high school. One thing you forget. College is a choice, high school is not. the College mindset is do what is asked or find yourself another career path. High school, do it or not, there is a seat in the class for you tomorrow. It is harder to commit to anything, student or adult, if the rewards or consequences are superficial. Making you attend school has it advantages for society and it disadvantages.”

My two cents: it appears to me that too many of the high schools are adopting “the customer is always right” attitude with the student and their parents being “the customer”. I think that is the wrong approach. The “customer” is society, as a whole. After all, public schools are funded by everyone’s tax dollars, and not just the tax dollars of those who have kids attending the school. Sometimes, educating the student means telling them things that they don’t want to hear, making them do things that they don’t want to do, and standing up to the helicopter parents. But, who will stand up for the teachers when they do this?

then-and-now-comic-460x270

thennoweducation

Note: if you google “education then and now” (search for images) you’ll find the above cartoons translated into different languages. Evidently, the US isn’t alone.

Statistics Education
Attaining statistical literacy can be hard work. But this is work that has a large pay off.
Here is an editorial by David Brooks about how statistics can help you “unlearn” the stuff that “you know is true”, but isn’t.

This New England Journal of Medicine article takes a look at well known “factoids” about obesity, and how many of them don’t stand up to statistical scrutiny. (note: the article is behind a paywall, but if you are university faculty, you probably have access to the article via your library.

And of course, there was the 2012 general election. The pundits just “knew” that the election was going to be close; those who were statistically literate knew otherwise.

January 17, 2013

Math and Probability in Pinker’s book: The Better Angels of our Nature

Filed under: elementary mathematics, media, news, probability, statistics — Tags: , — collegemathteaching @ 1:01 am

I am reading The Better Angels of our Nature by Steven Pinker. Right now I am a little over 200 pages into this 700 page book; it is very interesting. The idea: Pinker is arguing that humans, over time, are becoming less violent. One interesting fact: right now, a random human is less likely to die violently than ever before. Yes, the last century saw astonishing genocides and two world wars. But: when one takes into account how many people there are in the world (2.5 billion in 1950, 6 billion right now) World War II, as horrific as it was, only ranks 9’th on the list of deaths due to deliberate human acts (genocides, wars, etc.) in terms of “percentage of the existing population killed in the event”. (here is Matthew White’s site)

But I have a ways to go in the book…but it is one I am eager to keep reading.

The purpose of this post is to talk about a bit of probability theory that occurs in the early part of the book. I’ll introduce it this way:

Suppose I select a 28 day period. On each day, say starting with Monday of the first week, I roll a fair die one time. I note when a “1” is rolled. Suppose my first “1” occurs Wednesday of the first week. Then answer this: “what is the most likely day that I obtain my NEXT “1”, or all days equally likely?”

Yes, it is true that on any given day, the probability of rolling a “1” is 1/6. But remember my question: “what day is most likely for the NEXT one?” If you have had some probability, the distribution you want to use is the geometric distribution, starting on Thursday of the next week.

So you can see, the mostly likely day for the next “1” is Thursday! Well, why not, say, Friday? Well, if Friday is the next 1, then this means that you got “any number but 1” on Thursday followed by a “1” on Friday, and the probability of that is \frac{5}{6} \frac{1}{6} = \frac{5}{36} . The probability of the next one being Saturday is \frac{25}{196} and so on.

The point: if one is studying the distribution of events that have a Poisson distribution (probability p ) on a given time period, the overall distribution of such events is likely to show up “clumped” rather than evenly spaced. For an example of this happening in sports, check this out.

Anyway, Pinker applies this principle to the outbreak of wars, mass killings and the like.

September 11, 2012

Two Media Articles: topology and vector fields, and political polls

Topology, vector fields and indexes

This first article appeared in the New York Times. It talks about vector fields and topology, and uses finger prints as an example of a foliation derived from the flow of a vector field on a smooth surface.

Here is a figure from the article in which Steven Strogatz discusses the index of a vector field singularity:

You might read the comments too.

Note: the author of the quoted article made a welcome correction:

small point that I finessed in the article, and maybe shouldn’t have: it’s about orientation fields (sometimes called line fields or director fields), not vector fields. Think of the elements as undirected vectors (ie., the ridges don’t have arrows on them). The singularities for orientation fields are different from those for vector fields. You can’t have a triradius in a continuous vector field, for example.

Comment by Steven Strogatz

Our local paper had a nice piece by Brian Gaines on political polls. Of interest to statistics students is the following:

1. Pay little attention to “point estimates.”

Suppose a poll finds that Candidate X leads Y, 52 percent to 48 percent. Those estimates come with a margin of error, usually reported as plus or minus three or four percentage points. It is tempting to ignore this complication, and read 52 to 48 as a small lead, but the appropriate conclusion is “too close to call.”

2. Even taking the margins of error into account does not guarantee accurate estimates.

For example, 52 percent +/- 4 percent represents an interval of 48 to 56 percent. Are we positive that the true percentage planning to vote for X is in that range? No. When we measure the attitudes of millions by contacting only hundreds, there is no escaping uncertainty. Usually, we compute intervals that will be wrong five times out of 100, simply by chance.

Note: a consistent lead of 4 points is significant, but doesn’t mean much for an isolated poll.

January 17, 2012

Applications of calculus in the New York Times: Comparative Statics (economics)

Paul Krugman has an article that talks about the economics concept of comparative statics which involves a bit of calculus. The rough idea is this: suppose we have something that is a function of two economics variables f(x,y) and we are on some level curve: f(x,y) = C_1 at some point (x_0, y_0, f(x_0, y_0) = C) . Now if we, say, hold y constant and vary x by \Delta x what happens to the level curve C_1 ? The answer is, of course, C = C_1 + (\Delta x) \frac{\partial f}{\partial x} (x_0,y_0) + \epsilon where \epsilon is a small error that vanishes as \Delta x goes to zero; this is just multi-variable calculus and the idea of differentials, tangent planes and partial derivatives. The upshot is that the change in C , denoted by \Delta C is approximately (\Delta x) \frac{\partial f}{\partial x} (x_0,y_0)  .

It isn’t every day that someone in the mainstream media brings up calculus.

September 1, 2011

Classic Overfitting

Filed under: media, news, popular mathematics, probability, statistics — oldgote @ 1:33 am

One common mistake that people sometimes make when they model things is the mistake of overfitting known results to past data.
Life is complicated, and if one wants to find a correlation of outcomes with past conditions, it really isn’t that hard to do.

Here Nate Silver calls out a case of overfitting; in this case someone has a model that is supposed to be able to predict the outcome of a presidential election. It has been “proven” right in the past.

Mr. Silver writes:

If there are, say, 25 keys that could defensibly be included in the model, and you can pick any set of 13 of them, that is a total of 5,200,300 possible combinations. It’s not hard to get a perfect score when you have that large a menu to pick from! Some of those combinations are going to do better than others just by chance alone.

In addition, as I mentioned, at least a couple of variables can credibly be scored in either direction for each election. That gives Mr. Lichtman even more flexibility. It’s less that he has discovered the right set of keys than that he’s a locksmith and can keep minting new keys until he happens to open all 38 doors.

By the way — many of these concerns also apply to models that use solely objective data, like economic variables. These models tell you something, but they are not nearly as accurate as claimed when held up to scrutiny. While you can’t manipulate economic variables — you can’t say that G.D.P. growth was 5 percent when the government said it was 2 percent, at least if anyone is paying attention — you can choose from among dozens of economic variables until you happen to find the ones that pick the lock.

These types of problems, which are technically known as overfitting and data dredging, are among the most important things you ought to learn about in a well-taught econometrics class — but many published economists and political scientists seem to ignore them when it comes to elections forecasting.

In short, be suspicious of results that seem too good to be true. I’m probably in the minority here, but if two interns applied to FiveThirtyEight, and one of them claimed to have a formula that predicted 33 of the last 38 elections correctly, and the other one said they had gotten all 38 right, I’d hire the first one without giving it a second thought — it’s far more likely that she understood the limitations of empirical and statistical analysis.

I’d recommend reading the rest of the article. The point isn’t that the model won’t be right this time; in fact if one goes by the current betting market, there is about a 50 percent chance (slightly higher) that it will be right. But that doesn’t mean that it is useful.

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