Tantric Topology

It what has to be my most bizarre and disturbing mathematical thought so far, I spent a good twenty minutes today in the Sexuality section of my local Borders puzzling over an illustrated copy of the Kama Sutra. After enduring many embarrassed glances from other shoppers and a few accusatory stares of employees, I satisfied myself that topologically speaking, there is really only one position, since all can be smoothly transformed into any other.

Summer School

Here's a interesting point-counterpoint on year-round schooling. I don't have any personal experience with this, having attended schools with more or less traditional schedules. The summer break, while enjoyable, has always seemed counterproductive to student learning to me, as well as more than a little archaic. We are no longer an agrarian society where kids need to quit their readin', writin', and 'rithmetic to help plow the field. I would certainly not be opposed to working in a school system with a year-round schedule, if it gets proven results.

Of course, many will argue that there is no proof of results. I certainly couldn't find any statistics on-line. There was plenty of anecdotal evidence on both sides, but no empirical data sources. I am beginning to suspect that summer breaks are just too woven into our country's culture for us to part with them. Does anybody out there have any ideas/experience with this issue?

With Limited Commercial Interruption

Occasionally an idea comes along that is so brilliant, so without flaws, that you wonder why exactly it hasn't already been put into place. This is one of those ideas. I'm not claiming it as my own, but I have decided to become its biggest campaigner. Let's begin.

I love to watch television. Whether its all-new or re-run, sitcom or drama, reality or scripted, educational or escapist, I love it. Like most people, I don't care for commercials. They spoil the flow of the show and they are largely a waste of my time. I also don't like paying for TV. I have tried surviving on broadcast only, with little success. I need the variety, and basic cable is a must. When I can afford it, I go for digital or better.

Cable companies want to make money. They have gone to a lot of effort to lay the lines and purchase rights to the programming. They are motivated by their desire to feed their families, drive fancy cars, and vacation in the Bahamas.

Businesses want to advertise their products. With the average pay-out per minute of commercial time climbing into the realm of scientific notation, their desire to get the word out is obvious. The more people who see the product, the more people buy it.

Now here's the plan.

What if you're TV worked just like your Google homepage? As you watch, channel surf, and mindlessly flip, it monitors your click-stream. Then it uses that viewing history to decide what products you might be interested in buying. It builds a list of those commercials most suited to you and puts them in a folder. Each month, you have the option of watching as many or as few of those spots as you choose. Every one you watch decreases your monthly cable bill by a certain amount. If you watch enough, your bill is zero. The businesses whose product you saw advertised make up the difference on your bill. Everybody wins. You get free TV with tons of variety, and you watch the commercials on your terms. The cable company gets to make tons of cash. Businesses get to advertise their products to their target audiences for a fraction of what they are paying now to blanket the market.

This is my dream. Please, share it with me.

Girl Power

It's pretty good to be a woman in America. I'm sure my female readers would beg to differ, and it's true that there is still rampant inequality and prejudice toward women. Women are overlooked for jobs, paid less than their male counterparts, and frequently deal with sexual harassment. They are criticized by one side for having a career and by the other for stooping to the stereotypical role of wife and mother. They are expected to do everything a man can do, only to do it backwards and wearing heels. But let's be realistic. Compared to most women around the world, American women are living the high life. I mean, there are still cultures where women are treated more as currency than as human beings.

Yes, it's true. Young girls in the USA can grow up to be electrical engineers, computer scientists, and applied mathematicians. The problem is, well, most of them don't. For whatever reason, little girls want to be Britney Spears or Paris Hilton. If that sad fact doesn't wake people up, I don't know what will. When we worry about how our nation can compete with the growing economies of China and India, most of us happily ignore the elephant in the room- that the most obvious and effective solution is to inspire more than 51% of our nation's children to pick up a calculus textbook and put down the Barbie doll.

Thankfully for the future of the world, there are organizations devoted to doing just that. Since 1950, the Society for Women Engineers has made it their mission to show young girls that careers in science and mathematics are not just for the guys. The host programs from kindergarten through college, and from coast to coast. Krystal Grube, an SWE representative says of an upcoming event in Minnesota,

On June 3rd, hundreds of girls in St. Paul, Minnesota will attend an event hosted by SWE called, “Wow! That’s Engineering!” Through hands-on activities, girls will learn how solar power works, the wonders of deep sea diving, and even develop their own lip-gloss. Most importantly, they’ll realize that engineering is not just about working behind a computer; it’s about making a difference in the world.

There has never been a better time for the next Marie Curie or Emmy Noether to show the world that women can not only compete with men in these subjects, but with the right touch of genius and self-confidence, can leave them in the dust.

Support our Troops

MarkCC over at Good Math, Bad Math pretty much sums up how I'm feeling today.

Sights and Sounds

In an earlier post about the SETI project, I pointed out that at the level of electro-chemical signals, there really isn't much difference between sight and hearing. Both translate data from the world around us into electric impulses. There is no reason to believe that an animal that tracks primarily by sound, like a bat for instance, "sees" the world differently than we do. They just translate the echoes into a three-dimensional map, similarly to the way our brain does with our eyes. Scientists have recently discovered that our own brains can be trained to map the world with sound.

When you identify an object's shape, a particular part of your brain called the LOtv "lights up". At first this area was thought to be purely visual, but several years ago Amir Amedi, now at Harvard Medical School, showed that touch could also activate it. Now Amedi and his team have shown that even "hearing" a shape can activate the area.

The research group built a device that measures a subject's ability to mentally model space using sound, and found that they could in fact improve a persons results. This explains the conventional wisdom that blind people have a heightened sense of hearing. I don't think there are any would-be Daredevils out there, but it's still pretty cool.

Despite My Better Judgement

I have debated for several weeks about what I am about to write. I have tried to suppress my feelings about this for fear of permanently alienating my female readership. My most loyal blog-friends are women. The only blogrolls that link to me are authored by women. But blogger, after all essentially means someone who just can't keep their damn mouth shut. So after much deliberation, and despite knowing that this is a bad idea, I'm going to put it into words. It has nothing to do with math or education, but everything to do with morality, humanity, and justice.

There was a much talked about case in the Massachusetts courts this past month. It concerned the definition of rape and it has been attacked by women from all ends of the political spectrum. What allegedly happened was that a man masqueraded as his own brother in order to trick his brother's girlfriend into having intercourse. Here is the court's response, as told by the Boston Globe.

The Supreme Judicial Court unanimously ruled that a judge should have dismissed the rape charge against Alvin Suliveres, 44, of Westfield, Mass., because state law has for centuries defined rape as sexual intercourse by force and against one’s will, and that it is not rape when consent is obtained through fraud.

If the Legislature wants to make sex through fraud qualify as rape, it should follow the lead of several other states and change the law, the court said.

Now let me be abundantly clear. I have no doubt whatsoever that Suliveres did exactly what he is accused of having done. I think it is despicable and disgusting and thoroughly immoral. I still don't think it is rape.

The court's opinion that the state legislature ought to follow the lead of others is correct. The law, as it stands, was not intended to cover this kind of crime. And again, it is certainly a crime. But there has to be a legal difference between forced sex and that which occurs based on conditional consent. The first is rape; the second is fraud. Any effort to expand the definition is going to be an uphill battle, and will quickly lead to the slipperiest of slopes. To define sex as rape, simply because it was the result of duplicity is to suddenly add hordes of victims and predators to the ledger. The following is a brief list of common frauds used by both genders to solicit sex.

"I am over 18."
"I am a virgin."
"I am single."
"I love you."
"I am wealthy."
"I am on the pill."
"I am a powerful movie producer."
"I have no STDs."

Some of these lies are worse than others, but they are all designed to trick someone into doing something that they wouldn't otherwise do. Would couplings resulting from these subterfuges be rapes? If so, then we had better start building a whole lot of prisons, because the rapists are about to make the drug offender population look like small potatoes.

Again, I do not advocate the use of fraud to obtain sex. I feel that this man should be punished. But since we're talking about a serious capital offense, we have to define it very strictly with no room for error or interpretation. If we expand the definition of rape to include fraud, then all of the above lies could easily lead to life in prison. Is that a decision we are willing to make?

Pay No Attention to the Man Behind the Curtain

My regular readers know that as of September, I will be a college student again. I will be attending the University of Southern Maine in an effort to become a math teacher in our nation's public school system. In preparation for this, I have decided to brush up on my math a bit. I went to the used book store, and for under $20 I got textbooks covering Algebra, Plane Geometry, Trigonometry, and Calculus. I began with Algebra for no other reason than that's where I began the first time around. As I progressed through the disciplines, I began to wonder who has the job of deciding in which order students learn math. Of course, it makes sense to begin with arithmetic. Mathematics began when man started finding and manipulating patterns in the natural numbers. Formalized concepts and conventions like orders of operations and the product of negative numbers must be learned so that we can agree on what exactly we are talking about, and definitions must be drilled so that we have words with which to talk in the first place. All of this I get.

But then comes Algebra. Why is that exactly? Why don't we start with Geometry? Many concepts are necessarily presented in a linear fashion, but there are plenty that are fairly autonomous and are simply shoved in somewhere. Why is the chapter on Probability and Statistics at the very end of the College Algebra text I purchased? It doesn't have any prerequisites as far as I can tell, and I would argue that it is far more important than solving quadratic equations. The average person doesn't need to use very much of formal mathematics. Our brains have such well evolved pattern recognition algorithms that we can instinctively find practical solutions to problems without resorting to calculation. I doubt very seriously whether major league outfielders are solving quadratics to predict the path of a pop fly. Ironically, even though every decision we make is based in some part on probability, it is the area of math that most confounds our intuition. Shouldn't we learn that up front? Why wait until we've mastered more technical courses?

I'm just using Stat as an example here. I can't prove conclusively that is is the most important math skill and should therefore we taught first. All I want to know is who decides the order in which mathematics is presented? Somewhere, deep in the shadows of the mathematics priesthood, sits a little robed homunculus pulling all the strings. I just want to know who he or she is. If anyone out there has any clues to this, please let me know.

It's All Greek to Me

It's pretty common to hear mathematics referred to as a language. But how accurate is that statement? Hal Daume makes a valiant effort at answering the question, and is especially insightful in distinguishing between the math we speak and that which we right up in formal papers. Check it out.

To Infinity and Beyond

I just finished watching Infinity, a 1996 film directed by and starring Matthew Broderick about young Richard Feynman. The movie is mostly a love story, and depicts Richard's relationship with his wife Arline, her losing battle with tuberculosis, and his work at Los Alamos. It is a touching film and captures the greatness of the man. There is a particular scene early on that I think speaks toward the ongoing calculator controversy in our schools. Feynman challenges a Chinese shopkeeper to a duel of arithmetic, pitting his mental algorithms against the abacus. Feynman loses the addition round, ties on multiplication, and wins hands down on cube roots. Afterward, he explains to Arline that his win was inevitable, saying of the old man "he doesn't know numbers, he only knows beads." It is a wonderful scene, and a wonderful movie.

This Little Piggie Did Calculus, This Little Piggie Stayed Home

Scientists in Great Britain have just completed a study that shows a correlation between finger length and math ability. When I first came across this article, I was all set to be annoyed. I thought for sure that it was going to be another adventure in data mining, but I was wrong. The finger length is not the proximal cause, but rather a result of prenatal hormone surges. Testosterone, which is known to affect growth the right half of the brain where spatial reasoning is located, also causes the ring finger to be slightly longer than the index finger. A digit ratio of less than one correlates with left brain development, or language skill.

Now from the little bit I've read about this, I can find no obvious fault in the methodology. There was a predetermined hypothesis, evidence was gathered, and a conclusion was made. I do find major fault in the reporting. Here is how the article begins:

Parents may be able to predict how well kids will do in math and reading by measuring their fingers, British scientists claim.

Now I understand that journalists have to front load the hook. Their goal is to get people to read the story. Fine. But shouldn't they feel some obligation to present the material in a responsible manner as well? Most researchers don't get the opportunity to speak directly with the public and are forced to use the media as a go-between. For most people, the only science the read about is in their local newspaper. So placing a statement like this right up under the byline can be very dangerous. Sure the author used passive tense and employed words like may and claim to slightly erode confidence in the conclusion. But they should know that when it comes to predicting test scores, parents will latch on to anything remotely scientific, especially if it comes from the British, because we all know how smart they are. I mean, just listen to them talk.

Here is how those same British researchers present their conclusion in the final paragraphs;

Finger length is by no means a measure of intelligence or ability, Brosnan stressed.

"You may well look at a finger and say, all other things being equal, they may well have a leaning towards mathematics," Brosnan says.

"Or if they have a child who has a digit ratio that suggests very good verbal skills and a struggle with mathematics, this might suggest that, to explain mathematics to that child, you use a lot of verbal strategies and you cut down on the graphical, visual information."

By their own admission, the results of the study are less meaningful than this article would imply. The scientists appreciate the power that their work might have on desperate parents and rightfully suggest that not too much be read into it. These results are interesting at best. There is by no means a single definitive predictor of skill or intelligence.

I'm also very sensitive to boy vs. girl related math myths lately. After reading Rebecca's post earlier in the month, I have been thinking a lot about all things sexist in mathematics. Now to be fair, this was not a gender study, per se. It was a study of androgen surges, and because both males and females have both testosterone and estrogen, a continuum is possible which matches the evidence. Generally speaking, if the measured difference between groups is less than that within groups, then the phenomenon is fairly impotent. So if there is more variance between girls scores than there is between the average of girls and boys, than we can't really link the difference to gender. But that's not what this study is doing, so I can still only argue with the reporting.

Numerous studies have been done lately that conclusively demonstrate the harmful affects societal stereotypes have on test scores. Girls are known to do worse on a math test if you tell them before hand that boys do better. It's a horrible reality, and as a future male teacher (currently male, not yet a teacher,) I desperately want to avoid contributing to this problem. I wish that the authors of these articles felt the same way.

Joining the SF

If you still think mathematicians lack passion and emotion, read this. Few people are in a better position to appreciate the beauty of life than those who study its patterns.

Resistance is Futile

It seems the arrival of the new TI-89 has rekindled the debate over the use of calculators. I have to admit that I'm still undecided on this one, and like most things, I suspect a fairly straight forward compromise can be reached. On the one hand, the calculator is just the latest in a long series of math tools. It replaces the slide-rule, the abacus, and counting stones. There is no reason why we should suddenly object to making our lives easier. There are many concepts and tools integral to life that I have no idea how to use or that I use without total understanding. I have no idea how to grow corn, weave clothing, smelt metal, etc. I might have cursory knowledge of the processes and with plenty of time and money, I'm sure I could figure it out. But if someone held a gun to my head and asked me to make soap or milk a cow, I would be in just as much trouble as if they asked me to build a digital computer, even though I bathe, blog, and consume dairy daily.

On the other hand, I remember my immediate disgust when I was tutoring a girl in college algebra who had to reach for a calculator to find 1/3 of 3. How can students be made to understand math basics when they have a hand held computer that is one step away from Skynet?

To me, the problem results from teachers posing pseudo-mathematical questions. The problems look like math. They have numbers and operations and variables. But instead of requiring reasoning and abstract pattern manipulation, they are really only asking the student to operate an algorithm. When faced with this, why wouldn't an intelligent student reach for a shiny new TI-89? Mathematicians don't use calculators all that much, not because they are trying to stay true to their numerate roots, but because they simply aren't of that much use. If you're trying to prove a conjecture or sure up a hypothesis, a calculator is not going to help you. So instead of arguing the merits of technology, we should be pushing our young people to do real math and happily leave the algorithms behind.

I understand both sides of the argument, I really do. But I can't help but think that we are waxing just a bit too nostalgic over the whole thing.

The Price is Right

Everything I know about economics, I learned from Bob Barker. In what is probably the simplest of games, Bob will show two or three items and ask the contestant which is the Most Expensive. It's good clean fun for the whole family and teaches a little about supply and demand. So let's play.

One gallon of skim milk, one gallon of Starbuck's coffee, one gallon of regular unleaded gasoline.

Which is more expensive?

Not so subtle moral to the story: Stop complaining about paying at the pump. Prices are supposed to represent the relative cost of production.

To Be Continued...

Well I'm off to Trail Days. I'm so excited. It seems like forever ago that I summited Katahdin, and I really miss it. It will be great to see all my hiking friends. I am escpecially interested in talking to my friend Jebus (aka Matt) about his work in knot theory at UGA. Hopefully, I will be able to post some interesting tidbits from that conversation.

I'll start posting again next week.

Demystifying DNA

It's no secret that the misuse of statistics is a particular pet peeve of mine. I have often voiced my resentment that the general public's innumeracy is constantly exploited, and have expressed my desire to rearrange the high school math curriculum to focus more strongly on statistics, even at the loss of other disciplines. A particular issue where this constantly crops up is in DNA evidence.

Not long ago, the state of Maine sent a bill before its legislature that proposed mandatory DNA data be sent to the CODIS registry for all sex offenders. I was asked by the Maine Civil Liberties Union to help write testimony against this bill, which I happily did. I raised several concerns about the accuracy of the CODIS statistics and the way in which they would be used in court. I have read several articles suggesting that attorneys for both prosecution and defense tend to eliminate highly numerate jurors through the selection process. This allows them to manipulate statistics through several time-tested methods. Humans have a natural tendency to process stats in terms of frequencies. We intuitively understand what is meant by 1/100, but have to deliberately process the same stat if given as a percentage. Skilled attorneys are aware of this, which is why when they are presenting evidence as rock solid, they will employ natural frequencies, and when they are attempting to cast doubt they opt for percentages.

My second big concern is that random matches may be much more common than we have been led to believe. To support this, I must quickly sum up the CODIS process. When a sample is screened at the lab, the technicians aren't mapping the entire double helix. This would waste a lot of time, because even though there is a lot of variation between members of our species, there's still an awful lot of similarity. Natural selection sees to that quite nicely. However, as we have slowly cracked the genetic code, we have discovered elements of the genome that do not appear to do anything. We call it "junk DNA" and it appears to be then result of replication and transcription errors. Because these "genes" do not manifest themselves outwardly, they are not subject to selection pressures and are therefore highly variant. This makes them ideal for identification purposes.

Now I can get to my point.

When we are presented with the likelihood of a match randomly occurring in the population, we are often impressed by the enormity of the numbers. One in 15 quadrillion translates as "case closed and who's buying the beer?" There are two problems with this, the first of which I realized at the time I wrote the testimony and the second I just borrowed from Keith Devlin. Before you can judge guilt or innocence, you have to firmly establish which "population" you're talking about. The 1/15,000,000,000,000,000 stat refers to the probabilities of the entire human population. The stats change dramatically when you look at individual subsets. For example, while the 13 sites used in the CODIS registry vary dramatically as I said, they vary far less within particular ethnic groups. The closer the specimens are to one another on the family tree, the more likely a match becomes. A man with lineage to sub-Saharan Africa is much more likely to match someone of that descent than he would someone from Scandinavia. For close blood relatives the probability of a match sky-rockets.

The other point, which I failed to realize at the time, is the implied precision of the stats themselves. As Devlin points out in his excellent article, numbers like 1 in 15 quadrillion come from extremely naive applications of the power rule. It is true that if the odds of rolling a six with a properly weighted die is 1/6, then the odds of doing it twice in a row is 1/6 x 1/6 = 1/36. The odds of doing it three times is 1/216, and so on. This holds true for any situation where the probabilities of multiple events are concerned. But this is a mathematical ideal. In practice, the actual outcome can be way off the prediction, although with enough attempts, the reality will move closer to the expected value. So even in this simplified case, empirical testing will yield far more "accurate" results than a simple application of the power rule. A number like 15 quadrillion implies a degree of precision that is so ridiculously beyond our abilities that, as Devlin says, it is laughable.

The most interesting figures in Devlin's article come out of empirical tests of the CODIS registry.

As far as I am aware, to date there has been only one attempt to do this, [an empirical test of the registry] and the results obtained were both startling and worrying. A study of theArizona CODIS database carried out in 2005 showed that approximately 1 in every 228 profiles in the database matched another profile in the database at nine or more loci, that approximately 1
in every 1,489 profiles matched at 10 loci, 1 in 16,374 profiles matched at 11 loci, and 1 in 32,747 matched at 12 loci.

How big a population does it take to produce so many matches that appear to contradict so dramatically the astronomical, theoretical figures given by the naive application of the product rule? The Arizona database contained at the time a mere 65,493 entries. Scary isn't it?

It is not much of a leap to estimate that the FBI's national CODIS database of 3,000,000 entries will contain not just one but several pairs that match on all 13 loci, contrary (and how!) to the prediction made by proponents of the currently much touted RMP that you can expect a single match only when you have on the order of 15 quadrillion profiles.

Some of you will have misread this entire post. You will accuse me of being a damn liberal, commie, lefty who wants child molesters to run candy stores. That could not be further from the truth. In fact, if someone lays one hand on a child I know, they had better hope that the authorities get to them before I do. But I simply can not stomach an innocent person going to prison, especially if it is the result of a misapplication of mathematics. Not even Hardy would
want to apologize for that.

Mathematician Moms

An excellent Q & A session on instilling a love of math in your children from someone who ought to know.

More resources for moms (and dads.)

Reincarnation of Clever Hans

It looks like our old friend Clever Hans has gotten himself reincarnated as a lovable Golden Retriever. Perhaps he thought that his mathematical fraud would be less conspicuous if performed by man's best friend.

Hans, the famous counting horse, fooled audiences and skeptics alike for years until someone tried a simple test. The equine savant could do just about any calculation as long as his handler knew both the problem and the correct solution. If the card was switched, Hans gave the answer that his handler expected instead. The handler didn't even know he was doing it and I'm sure was devastated to find out that Hans was a cheater.

What do you want to bet that something similar is happening here?

A 100% Sincere Letter to the MPAA

Dear Motion Picture Association of America,

I am writing to thank you for your recent decision to add cigarette smoking to your list of criteria leading to an "R" rating. I have been hoping that you would come to the rescue of America's lung health and only wish you could have done it sooner. Since the implementation of your benevolent ratings system, all but a handful of social ills have been completely eradicated from our culture. Premarital sex is a thing of the past. No longer are sickos and perverts allowed to freely engage in coital permutations outside of God's cherished missionary position. Violence and drug use have dropped so dramatically that some are advocating turning our nation's prisons into factories for Bibles and cotton candy.

None of these great achievements would be possible without your invisible hand. Proving that life imitates art, you have molded this world in your image and this movie goer thanks you from the bottom of the heart. I don't know where I would be without you.


Absolutely, completely, sincerely yours,

American film lover

This is Not a Drill

In case you haven't heard there was another incident involving a school shooting. Thankfully, no one was injured physically, though the possibility for psychological damage is still up in the air. The "gunman" was a teacher at the school, and the emergency was completely staged. Evidently, some teachers got together and decided to "punk" the kids into thinking their young lives were about to end. The educators are accused of using mad judgement in their attempt to create a "learning experience" for the Tennessee sixth graders. The whole thing has left me wondering exactly what the terrified students were supposed to be learning.

School shootings have become an unfortunate part of our culture. Every time I turn around, another frustrated student has decided to end his life, but not before taking a few fellow classmates along for the ride. Certainly anything that educators can do to prevent these tragedies should be done. I think that shooting drills would be a good idea. They would be at least as effective as the old air-raid drills, and might save a few lives. What these teachers did was entirely different and completely inexcusable. The only thing these students learned was the feeling of terror and helplessness. This lesson was roughly the equivalent of sending the school guidance counselor to tell a young student that their mother has just died in a fiery car crash just to teach them about the grieving process.

Here We Go Again

When I read stories like this, I actually want to call up the author and verbally berate them. Evidently, someone decided to look at a large group of school children and see if there was a correlation between math test scores and birth dates. Not a particular correlation, mind you. There was no hypothesis. They weren't looking for a particular pattern, just any relationship would do. Shockingly, they were able to find one. In this case, it's that summer birthdays lead to poor performance, and they actually attempt to pin the blame on pesticides used during this season.

I don't know how many times studies like this have to be shot down before people get it. If you look at a random group of people, some people will do better and some will do worse. That's what randomness requires. We would not expect those good and bad scores to be evenly distributed. They're going to clump up in some months. But they could easily clump up in different months with different samples. Unless you are looking for something in particular to back up a hypothesis, you aren't doing science.

I don't have to know when this journalist was born to know that they don't no a thing about math. They've proved my hypothesis for me.

What Not to Do

I asked my blog-friend Rebecca to do the next in her semi-regular series Ask an Applied Mathematician about sexism in the classroom. Her reply, while I'm sure completely off-the-cuff gave some good examples of what not to do if your intention is to promote a safe, nurturing, and fair learning environment.

On the Road

I just want to apologize in advance if my posting is somewhat sporadic over the next week. I am taking a little "vacation" and I don't know what my web connection situation will be. My trip will take me back down South for the first time since moving to Maine. I will spend the first half of the week visiting my mom in Knoxville, TN along with the few friends I have left there. Midweek, I will be catching a ride to Damascus, VA for Trail Days, the Appalachian Trail festival. It should be a reunion of last year's hikers. I can't wait to see everyone.

So I will post as often as I can, but if I miss a few days, please bear with me.

Capital Letters

I just finished reading Ian Stewart's Letters to a Young Mathematician. The book is a compilation of letters written to a fictional math student named Meg, but it is really aimed at the math student within us all. Though you never see an of Meg's letters, you can imagine her frustration and her failed efforts to understand exactly what mathematics is. It is a question that even the most mathematically inclined among us ask and one with no straightforward answer. Stewart weaves a thoroughly enjoyable yarn and leaves the reader feeling at least awed by mathematics even if they are still a bit confused. It should be required reading in any math class.

Meeting Me Halfway

Few subjects are as naturally apolitical as mathematics. It's hard to study history without framing things in terms of current issues and controversies. Even science classes are ripe for ethical debates that force students to choose sides. Not so with math. For example, several times in the last month, my daily random walk through the blogosphere has landed me here. Now politically, the author and I could not be more distinct. Displayed prominently at the top of his tags are posts concerning the ACLU. You don't have to read much to see that they're not his favorite organization. The ACLU is also displayed prominently in my blogroll, only I volunteer for them. In fact, the author and I disagree on almost every subject except math. If I am point A and he is point B, math is our midpoint M. In fact, I've had to add an entirely new page element just to correctly categorize his link.

Math-a uniter not a divider.

So Easy a Caveman Could Do It

On the way home, I heard an ad on the radio for All State, warning me that I may live in one of the most dangerous neighborhoods in the country. Of course, I feel quite safe in my neighborhood or I would move. But according to statistics, which never lie, I am more likely to have an accident the closer I am to home. I guess I'd better start looking for a safer address.

I can forgive this little piece of propaganda. I understand that it's all about marketing, and that when it comes to insurance ads, making you afraid is the number one goal. It did get me thinking about all of the other insurance related issues that bug me. The first of which is that lately people don't seem to understand the very nature of what insurance is and is not. It is a mathematical gamble, no different from a trip to the casino. It is not a guarantee of anything. The next time you hear a politician talk about universal health care and affordable insurance for all, ask yourself this. What possible incentive would anyone have for operating an insurance company other than to make money? Let's go back to my casino analogy and expound. The owner of a casino knows that people are greedy by nature and that if they can be cajoled into playing long enough, the house will always win. Gamblers play against the odds and hope to hit that big pay-off early and take the house for all they can. Insurance agencies operate under similar principles. They have complicated actuarial tables that predict the probabilities surrounding your policy. If it is a health insurance policy, they look at your age, gender, occupation, family history, and any mitigating factors such as smoking or obesity. They make money by charging you at rates that they know are likely to have you paying in far more than you will ever take out. You are keenly aware that an accident could be waiting around the next corner and are therefore insuring yourself against the chance that it happens sooner rather than later. If not for the possibility of impending danger, you might be inclined to simply forgo insurance altogether and pay your money directly into a savings account or other interest bearing fund. For most people, this would end up being the better deal, if and only if they were disciplined enough to regularly deposit and only withdrawal for medical emergencies.

So when you start talking about insuring everyone, you are missing the point. Humans are fragile. We will get sick and we will die. The question is not if, but when. That is the question that the entire insurance industry hinges upon. If you try to force the insurance companies to enter into a gamble that they have a poor chance of winning, you have removed all incentive for them to offer their service in the first place. In the future, medical science may advance to the point when we can consider this. There are many who feel that conditional immortality is around the bend. If we no longer die of old age, then we only have to insure ourselves against accidental death and foul play. Until that time, we need insurance companies, and I can assure you they are not motivated out of the goodness of their capitalist hearts. Case in point- the act of God clause.

I just saw a great movie called the Man Who Sued God, which stars Billy Connelly as a lawyer turned fisherman. In the opening sequence, his fishing boat is destroyed by a bolt of lightening, and as you may have inferred the insurance company refuses to pay out, deeming it an act of God. Connelly decides to sue God and hauls the local heads of the major religions into court to act on the Almighty's behalf. Very amusing film; I highly recommend it. All insurance policies have this clause in their contract, and though they never specifically define what it means, essentially it is their way if insuring themselves. There are some events whose very nature depends on chaotic systems that can not be predicted using current technology. Insurance companies don't want to be held accountable for what they can't predict, so they put in this sweeping escape clause.

Now this pisses me off for several reasons. One, and these are i no particular order, they provide no evidence that God in fact exists. Two, if he does exist, aren't all events acts of God? How do they distinguish the ones they're going to pay out from those they won't? Third, isn't they entire point of a contract to strictly define the letter of the agreement so that both parties know exactly what they are getting? This practice seems more than a little archaic to me, and I really can't believe it is allowed to continue.

I feel I've reached the end of my rant about insurance. The conclusion has snuck up on me, and I find I don't have anything profound with which to close. I will just say this. Insurance is a necessary evil, just like death and taxes, and intimately related to both. Unfortunately, it is an institution which has not yet outlived its usefulness. In trying to revamp the system, we need to keep in mind the realities that it has been based upon. Some changes we have the power to make, but others will bring the whole building crumbling down around us.

The Univeral Language

From the dawn of time, mankind has looked to the heavens and wondered Are we alone? Our desire for galactic companionship is so strong that we are universally given toward imaginary friends. These interstellar neighbors come in various forms. Some reside in magnificent halls high atop mountain peaks. Others float upon fluffy nimbus clouds in the ultimate gated community. Still others sail about the universe in search of new an exotic orifices to probe. Thus far, these neighbors of ours have proved to be reclusive to say the least. They appear only to the few of us deemed worthy. Worthiness, though, does seem to be subject to a certain degree of relativity. The chosen are selected for their great deeds, their royal lineage, or their wisdom. Other selectees are marked primarily by ownership of a pick-up truck, distinct absence of teeth, and a fundamental lack of language ability. Regardless of the criteria, one thing is constant, we are always on the receiving end of the introduction.

In the age of science, we are trying to change all that. The SETI project, made famous by Carl Sagan's Contact, is keeping an eye out the window for visitors. They are our neighborhood watch and welcome wagon rolled into one. No the universe, of course, is a big place, and we can't look everywhere at once. In an effort to streamline the process, these scientists have to make a lot of guesses. They use their best estimates to guide them, which means excluding some possibilities in favor of others. Despite constant setbacks to our self-esteem, humans tend to be a fairly arrogant species. Even now, in an age when cosmologists are reluctant to place much significance on us at all, the anthropic principle heralds us as unique. We use evolutionary metaphors like ladders and trees, firmly placing ourselves on the highest branches and rungs. In our minds, we are the paradigm by which sentient life ought to be measured.

So we look for class M planets, those that are the most Earth-like. They have a similar temperature range and relative position to a star. The have liquid water and atmosphere. These are the parameters of life. Or are they? Now that science has discovered terrestrial organisms thriving in temperatures well beyond those thought conclusively deadly, and others living deep within the earth eating rock or digesting chemicals thought universally poisonous, how will that affect our search of the skies? Will be recognize a silicon based life-form if we happen upon it? What safeguard can be taken so that we don't overlook our target due to faulty assumptions?

Fortunately, the folks at SETI have a pretty powerful redundancy built into their system. It is the language of mathematics. Despite what science fiction would have us believe, it is unlikely that their are races of humanoids skipping about out there. There is only one hope of a Universal Translator, and that Rosetta Stone is going to be the science of patterns. Any form of life with the technology to send a signal powerful enough to reach us must possess a system of advanced mathematics. That system will be one that we can recognize. It may have more power than ours does, but it will be just as true for us as it is for them. They will not add 2 plus 2 and get 5.

There will be superficial differences, of course. There is no reason to believe that they will use a base-10 system. Even here, we have used base-12, base-60, and with the advent of computers, base-2. This is a minor discrepancy, and is easily overcome. Beyond that we can expect the math to be the same. How can I be so sure? Isn't our mathematics largely a function of how we perceive the world? Yes, and no. Our senses are the windows through which we see the world, and there is no reason to think that alien life would see the world exactly as we do, or for that matter to "see" the world at all. The real question is, could a life form achieve sentience without developing an accurate depiction of the world in which it was evolving. The answer is no. As long as we are talking about life that inhabits a three-dimensional world, we are talking about life that can see, hear, touch, taste, and smell.

At this point, you may feel my arrogance is beginning to show. I will explain. When I say "see," I mean detect electro-magnetic radiation. It may not be in the same range of the spectrum that we think of as being visible. They may have an analog to "bee purple." They may even be able to see the fields themselves, like electric eels or platypuses. When I say "hear" and "touch," I mean sense pressure waves through fluids and solids respectively. When I say "taste" and "smell," I mean assess the chemical components of matter. They may perceive these things differently than we do. They might "see" sound or "smell" time. But regardless of how they paint their picture of the world, it will have to be accurate enough for survival, and that means pattern recognition, and that means math. For life to evolve, each generation has to get better at passing on its genes. That amounts to finding food, avoiding predators, and sometimes locating a mate. All of these things require an internal map of three-dimensional space, and as long as their world occupies the same unbounded three-spherical space that ours does, we can rest assured that their math will be the same as ours.

My Hero

Check out Dan's graphing lesson over at Dy/Dan. So far, of all the teacher blogs I've been reading, he most strikes me as the kind of educational leader I want to be. I plan to steal extensively from him, so I might as well start kissing his ass to avoid the copyright infringement lawsuit.

Counterinsurgence

I found this photo and simply could not resist posting it. Try to guess what you are looking at? If you have a sufficiently dirty mind, you might guess that what you are looking at is somewhat phallic in nature. You're half right. The one on the right is the penis of a particular species of waterfowl, whose scientific name escapes me. Henceforth, I will refer to it as . The one on the left, strangely enough, is the corresponding vagina of the species. We'll call it, oh I don't know, .

I have never in my life seen a more perfect depiction of evolution in action. Seriously, what kind of depraved deity would do this on purpose? Most birds do not have protruding genitalia, but waterfowl are evidently among the 3% that retain that link with their reptilian ancestors. This peculiar happenstance has allowed them to engage in a sexual arms race resulting in the insanity above and give the word kinky a first rate double-entendre. Like all males, only wants one thing, and he's not terribly polite about asking for it. As a defense against forced copulation, and knowing that no self-respecting male of any species will ever ask for directions, developed a labyrinthine target, full of twists and turns, jug-handles and dead-ends. Without her cooperation their is no way is getting his groove thing on. In retaliation, developed his own advanced weaponry, which cockscrews around in a similar fashion.* This brand of sex game may explain why waterfowl tend to mate for life. Once you find the lock that fits your key, there's not much point in jamming it in to other doors.



*This is not a Freudian slip. I'm just feeling terribly juvenile.

My Greatest Fear

I'm scared...

• that I won't be able to get through to my students.
• that I'll give everything I have and more, and it still won't matter.
• that as hard as I fight, the system will fight back harder.
• that I will fail myself and fail them.
  
• that no matter how hard I try, I will never be able to persuade my students to call me Oh Captain, My Captain.
• that despite my best efforts, my students will hate math.
• that the only thing my students will hate more than math is me.
• that I will get so wrapped up in changing the world that I will lose her in the process.
• that I will sacrifice having a family of my own for the chance at making a difference in the lives of children I may never see again.
• that being the change I wish to see in the world won't be enough.
• that the only reason I have been successful so far is that I haven't tried to do anything really hard yet.
• of more things than I can think of.

But I will not let my fears stand in my way.

Safety and Shallow Water

The other day I was having a polite, yet throughly innocuous conversation with an acquaintance of mine concerning phobias. This person admitted several paralyzing fears to me, and not wanting them to feel self-conscious, I mentioned my "apprehension" of deep water. Now strictly speaking, it is far from a phobia. I have been in boats on the open sea several times and did much splashing and swimming. I simply push thoughts of my total and utter vulnerability to denizens of the deep to the back of my mind. Upon hearing this, my acquaintance sought to reassure me by stating that I shouldn't worry because most shark attacks happen within a few feet of shore.

I will pause momentarily for you to digest the implied stupidity.

Of course most attacks occur within a few feet of shore; that's where all the people are! It's not as though some meandering bull shark in search of a meal will stop pursuing you as soon as you get fifty feet out. Sharks eat wherever they are when the happen to be hungry. This misunderstanding of statistics falls into a larger category of math confusion which includes the "facts" that most drownings occur in shallow water and most car accidents happen less than a mile from home. These relative danger doesn't change, just the amount of time you put yourself at risk. If this weren't the case, you could drastically cut down on your chance of auto collision by moving to a new house, yours being so horribly disaster prone and all.

Needless to say, I doubt my associate has a career waiting as an actuary. Equally obvious is the fact that I am going to try much harder to avoid stupid conversations.

This Just In

Research shows math anxiety ends in poor test performance.


Duh.