Friday, 18 April 2008

Riding The Thames

The old Vaudeville routine begins, “I’ve got good news and bad news, which do you want first?” This is a little different. I’ve got news, and we are not sure if it is good news, or bad news. Seahorses (Hippocampus hippocampus) have returned to the Thames River, in large numbers.

Most certainly there is one part that is good. The fact that they can now survive has got to be a sign of improvement for the river that was once so polluted that people jokingly dismissed the miracle of walking on the water. The little seahorse is sort of like the canary in a coal mine, when the environment becomes polluted they are one of the first to disappear.

The bad part??? Well, the little guys normally are found in the warmer waters farther south, so they may have become more common due to global warming and changes of the temperature of the water.

Now can someone down at Cambridge nano-engineering come up with a little sea-jockey, and we can do away with that silly annual boat race and have some real racing on the Thames. Saddle up and ride, boys!

Thursday, 17 April 2008

Prime Time Fun

Another from the same guy as last time, but with some questions. Is this funny to anyone who is not "mathematical"? Why is it funny to math-types? I worry that the later is wrapped up in some ego fed superiority concept.... I understand and "they" don't. For instance this is a tracer across the bottom of the cartoon site web page..

"Warning: this comic occasionally contains strong language (which may be unsuitable for children), unusual humor (which may be unsuitable for adults), and advanced mathematics (which may be unsuitable for liberal-arts majors)."

I hate that that might be true about me, but I still think it is funny.

Ok, as soon as I read it I started mentally checking for factors... eventually I decide that 1453 is prime. Now the math-teacher mind kicks in looking for variations of a theme. How common is it after 1 pm that the 12 hour clock is not prime and the 24 hour clock is, as in the example? The first example is at 1:19 (quick, factor that).

Is it more likely they are both prime? There are lots of examples of the latter right away, 101 and 1301 are both prime, as are 103 and 1303, 107 and 1307. Seems pretty common.

A reverse example occurs at 109/1309. The 12 hour clock is prime and the 24 hour clock is not, 1309=7 x 11 x 17; and again at 1:13.

Now admit it, you really can't get that much entertainment out of a typical comic.

Randall Munroe, who creates the XKCD comics says that he sometimes does this with mile markes on the highway. I guess I can admit that I find my self factoring the number on license plates and looking for interesting numbers. I still remember how pleased I was when I arrived on the base and received my PO box and the number was 1729; and only the math people know why.

Ok, while I'm admiting weird stuff... sometimes I just google a number to see where it shows up... 1729 was a good year for example, from my notes on Early American Math Textbooks, I find "In 1729 the first arithmetic published by a Native of Colonial America was published by Isaac Greenwood, a professor at Harvard. The title was Arithmetic, Vulgar (common) and decimal. Greenwood seemed to have a short and somewhat checkered life (1702-1745) He graduated from Harvard in 1722 and went on to become the Hollis professor of Mathematics, but was censured in 1737 for drunkedness and dismissed on August 30, 1738. He was replaced as Hollis professor, I believe, by John Winthrop of the Massachusetts Bay Winthrops." (by the way, 1722 is 42 x 41). 1729 is also the year that Danial Bernoulli invented a symbol for the inverse sine, he used AS. No one had ever made up an inverse sine notation before. Ok, just one more... "Benjamin Franklin advertised pencils for sale in his Pennsylvania Gazette in 1729"... and pencils brings us back to where we started, I used a yellow #2 to do all the factor doodles on the cartoon at the top... Life is a circle... or perhaps a Mobius Strip

Sunday, 13 April 2008

OOPS, The New York State Geometry Experiment,

Well, if you live in New York State, it may well happen that your kid will get a geometry teacher this year, who has NEVER taken a geometry class in his life… Honest…

A couple of decades ago, the schools in New York went to an “integrated” approach that covered a little algebra, a little geometry, a little something else each year, and broke the traditional cycle of Algebra I, Geometry, Algebra II, Pre-Calculus so common in much of the world. But in 2003, according to the NY Times, “State education officials created a committee to begin rethinking the math standards .., after two-thirds of the students who took the Math A exam failed, prompting a flood of complaints and criticism from parents and teachers.”

To go back to the old way would almost certainly cause someone to lose face after telling the parents, teachers, and world in general that the “integrated” approach would be the salvation of math education. So the New York system has come up with an even better way, It begins with “Integrated Algebra I”, proceeds to “Integrated Geometry”, and then on to ….are you ready for this??? “Integrated Algebra II.” Now THAT is a creative solution. “Never look back”, Satchel Paige often said, “somebody may be gaining on you.”…

There is a hitch. Many of the math teachers, like math teachers in most places, have been teaching for less than four years , at least according to Alfred S. Posamentier, dean of the School of Education at the City College of New York. So they mostly can be assumed to have graduated from high school within the last ten years. Fact one, they never took geometry as a proof based course in high school. Fact two, most teacher training institutions do NOT offer traditional Euclidean geometry in their colleges. Most likely conclusion, there will be some kind of patch up training sessions combined with lots of geometry teachers trying to stay one day ahead of the kids in the teachers manual. Not exactly a recipe for success, so you gotta believe they thought things had REALLY gone downhill.

So if your kid comes home next year and needs help in Geometry, perhaps you shoud not have them follow the tradtional advice of "Ask the teacher."

Saturday, 12 April 2008

Stoop and Deprecate

I often show my calculus students "Stand and Deliver" during the weeks after the AP exam. Recently Konrad K, one of my ex-students, sent me a note to check out the new episode of South Park, in which they do a parody, in part, of Escalante in the show. I can't see the whole thing because the link won't work outside the US, but if you want to see it, tryhere.

If you can't get the whole thing (for whatever reason) you can see a short clip at the blog site here.I'm sure I don't like the pro-cheating theme, and I hope nobody interprets this to mean that Jaime or the kids at Garfield over the years ever cheated... but it is parody.. I guess these guys have to be a little extreme to overcome the reality of the world today. There was the supposed case of the kids from Singapore calling back to friends in California between the time the test finished in the orient and started in the USA. Supposedly it became common enough that now there are two versions of most tests, one for the students in the US, and one for overseas.

Thursday, 10 April 2008

Harmony, Record Floods, and Traffic Jams

I wrote recently about the study in Japan of traffic jams, and recently realized that the mathematics of traffic jams and record rainfalls (or any other kind of record) have some things in common. But to tie them together in a mathematical way, I'll give you a problem, but first a little information for those not rememering all the language you learned in Alg II.

The harmonic series is the sum of a sequence of unit fractions with the natural intergers as denominators, 1+ 1/2 + 1/3 + 1/4 + ..... and on and on "To Infinity and beyond" as my Buzz Lightyear toy calls out... Now it is well known that the function diverges. That means that there is no limit to the sum. If you go far enough you will pass any number you can name.

For some kids that seems perfectly natural. They kind of think that if you keep adding more and more onto it, you will just get bigger than any limit you could imagine, but in fact, high school math is full of series that DO have limits. Take the sum 1+ 1/2 + 1/4 + 1/8 + 1/16... Keep going as far as you want, you will never get past two. It is easy to see graphically that you can't. Draw a square and shade it (ok, that's one) and now draw another and shade 1/2 of it... now shade 1/4 of it (1/2 of what is unshaded) and continue forever. The second square will not get full in any finite time. Game over, you lose, do not pass 2, do not collect $200.

Ok, so now you know a little about the harmonic series. Each term gets smaller and smaller, but in the end it goes to infinity; so here is the problem. Can it get to infinity without ever landing on an integer (after one), or does the "getting smaller and smaller" part mean that it must land on one somewhere out there? If you take small enough steps you have to land on a crack in the sidewalk somewhere, or do you?

If you think you know, drop me a note and I will tell you if you have figured it out or not; but of course if you think it does, you should come up with a value, or at least an order of magnitude for what it will be. On the other hand, if you think it NEVER lands on an integer you ought to be able to come up with some explanation.

Ok, so what does that have to do with foods and traffic jams? Consider this problem for a moment; If we started keeping records today, how many years will have a record flood in the next ten years, or the next hundred years. Well, the first year MUST be a record, so there will be at least one. Now what is the probability that the second year beats the first? Sure, 1/2 is the obvious answer. Now what about the third year; how probable is it to beat both the previous years... 1/3... and you are beginning to see a pattern. So if we wait ten years, we would expect, on average, to have 1 + 1/2 + 1/3 +... + 1/10 records occur. My calculator picks 2.93 or not quite 3 record years. For 100 years, we would expect only about 5.18 record events.

Of course it doesn't have to be a flood. You could see how many heads you can juggle the family knife set (ok, maybe not a good ides). But how about a traffic jam. If we focus on a single lane of traffic there will be a bunch of cars all wanting to go their preferred speed. Some get stuck wanting to go faster than the one in front. How many bunches of cars will there be? Well, each front car in a bunch is a record breaker for slow speed. So with n cars, the expected number of bunches would be 1+ 1/2 + 1/3 + .... + 1/n....

Next time you have that pokey guy in front of you, instead of honking and screaming and having a brain seizure, just calmly think to yourself, "Well this record is not as bad as being in a 100 year flood."

Saturday, 5 April 2008

Math Problems Start with I

I’ve had a few thousand high school kids come and go in my classes, so I figure I’ve been around long enough to make one or two observations, and lately I’ve been wondering why it is that seemingly bright kids can sometimes have such difficulties in math, and other, seemingly less intelligent and certainly less hard working kids, seem to grasp math so easily. One of the things I notice is that the problems often relate to words that start with the letter I.

Invisible Lines….. One of the first places we seem to lose bright kids is in that point in geometry when we expect them to be able to look at a sketch and see a line that isn’t there, but should be added to make a proof or problem solution easy. I think one of the classics used to show up on SAT tests until it leaked out and became too well known. It gave you a circle with a diameter of 14 and a center at point O, and a rectangle drawn as shown. The question just asked for the length of the diagonal shown.

To the quick student of math, there is another line that makes the answer obvious, the other diagonal is a radius of the circle, and since the diagonals are equal, the requested length must be 14/2 = 7 units. Bright kids who know all the above geometric relations miss problems like this because they seem only to see the lines drawn. It seems that geometry is where this shows up first, but the similar idea shows up repeatedly in math. Mathematical thinkers look for what might be there, or what we would like to know.

The second irritation “I" for the non-mathematical thinker is impossible. Some folks have a difficult time accepting that anything can be proven to be impossible. Their mind can’t get beyond the “maybe you didn’t try the one special way that works yet.” The early Greeks wrestled with the problem of “squaring the circle”, for instance. It is a simple enough idea, take the classic instruments of geometry, a compass and straightedge, and construct, in a finite number of steps, a square whose area is the same as a given circle. We have learned a lot from pursuing the challenge; Hippocrates of Chios (not the medical one from Cos) discovered lots of nice properties about the Lune, but he could never square the circle. Why?...because it is impossible. In 1882 Ferdinand von Linderman proved that pi was transcendental. All that means is that you can’t write a polynomial with rational coefficients (you remember those, things like x2 + 2x + ¾ = 0) which has p as a solution. But every construction with a compass and straightedge can be shown to be expressible as a term of a polynomial with rational coefficients, so no solution, no pi construction, and no pi construction, no pi r-squared. OK, that was over 100 years ago, but college professors are still receiving “proofs” from mathematical armatures who think they have solved the problem, and there is a thread going on one of the geometry news groups where someone is trying to convince those still patient enough to respond that he has conquered the problem. His problem seems to have been an overlooking of the finite part. The problem is more general, because in early geometry, one of the proof methods is called reducto ad absurdum. It is a method by which we assume something we want to prove true, is in fact false. We then have to show that that leads to an impossible consequence, and so we can assume the veracity of our original conjecture. The student who rejects the idea of finitely proven impossibility has forfeited one of the strongest of mathematical tools.

I’ve talked before about the problems students have with the “imaginary” numbers. Ok, everyone in the business thinks it was an unfortunate choice of terms for pedagogical usage, but it is just a name. Still the idea that anything derived from the seemingly impossible (sure… NOW they believe in impossible) square root of a negative number, could prove useful. I suppose that the concept would have to be expected to be difficult for students if only 130 years or so ago, the famous mathematician de Morgan described them as “self contradictory and absurd”

Friday, 4 April 2008

Moments... Nice songs

Came across a couple of Great Country songs... ...

Just wanted to share.

And everybody wishes they could send a message back to their younger, dumber self... which we probably wouldn't listen to,... but this is still a nice song... and there is a truth in there...."These are no where near the best years of your life".

Thursday, 3 April 2008

A little Irish Humor

Ok, BEFORE you send the hate mail... I married into an Irish-Catholic family, so I'm allowed to "take the micky out" once in a while as the Brits will say. And besides, I thought this one was just too funny to keep to myself. Thanks to Steve Kantor for sharing:

A drunk staggers into a Catholic Church,
enters a confessional booth, sits down, but says nothing.
The Priest coughs a few times to get his
attention but the drunk continues to sit there.
Finally, the Priest pounds three times on the wall.
The drunk mumbles, "ain't no use knockin,
there's no paper on this side either!"