Sunday, April 29, 2007
There are many reasons to love Erdos. He had a wonderful sense of humor. He is credited with the notion that "a mathematician is a device for turning coffee into theorems." Coddled so much by his mother, who lost his two sisters to fever just before he was born, he remained delightfully helpless throughout his adult life. He is reported to have been incapable of buttering his own toast. Perhaps most famously, he was a man without home or country. He would show up on the doorstep of a friend with the announcement "My mind is open," which meant that he had a question that when answered would probably create an entirely new branch of mathematics. As I said, there are many reasons to love Paul, but to me his true legacy is not as a mathematician, but as a math teacher.
Erdos embodies all of the qualities a good teacher should. He was wonderful will children, entertaining them with original nursery rhymes and tricks of dexterity with his pill bottle. He was skilled at matching the right problem with the right person. If Paul posed a question, you could be sure that it was one which you were uniquely suited for and he was known to work on some 200 open questions simultaneously with friends around the globe. He communicated with the elites in the field and with high school students curious about the world. This indiscriminate mentoring led to his being described as a "bee spreading intellectual pollen." He would travel the globe with two half-full suitcases containing all his worldly possessions, and arrive just in time to ask the right question at the right time, thereby propelling his friends and students further in their understanding of the world. He had his politics, to be sure, but they did not stand in the way of the acquisition of knowledge. During the Cold War, when scientific discoveries were dashed against the impermeable Iron Curtain, Paul Erdos ushered them through, acting as the link between Uncle Sam and Uncle Joe.
I could go on and on about Erdos. He was kind, compassionate, never arrogant, and always generous with his time and with his mathematics. A math teacher couldn't find a better roll model.
Saturday, April 28, 2007
Quite a terrifying comparison, huh? Most bloggers have been giving detailed explanations of cultural differences that lead to this disparity, and while I agree whole-heartedly with them, I wanted to post about another contributing factor. In his excellent book, The Math Gene, Keith Devlin points out a simpler point. Math is an extension of language, and the Eastern languages are better for math. Chinese is a mono-syllabic language, which for many concepts can prove difficult, but is perfectly suited for counting. Here are the Chinese pronunciations of the Arabic numerals.
1 one yee
2 two uhr
3 three sahn
4 four suh
5 five woo
6 six lyo
7 seven chee
8 eight bah
9 nine jyo
10 ten shi
They are mono-syllabic and end in vowels, which make them easy for a child to pronounce. On top of that, the linguistic pattern for counting is identical to the base ten place value system. In English, we count to ten, use a couple of throwback contractions to the days of Beowulf, pass through the teens, and then carry on with 20,30,40,etc. Yet forty will be spelled without a "u" and there will be know eleventy unless you're a Hobbit. The situation in French is similar, where there exist remnants of a base-20 system. However, in Chinese, eleven is literally spoken as ten one. Twenty-two becomes two ten two. Four hundred fifty three breaks down as four hundred five ten three.
I suspect that somewhere in grade school this linguistic disparity ceases to be significant, as English speakers master the peccadilloes of their languages, but by then all of those powerful cultural differences take over. We in the West might just have to accept that we will continue to be worse at math on average than our Eastern counterparts. We should probably settle for being the best we can be and stop worrying about how we stack up to other countries.
Friday, April 27, 2007
I will begin with a personal anecdote. About a decade ago, I went to visit a close friend attending a prestigious liberal arts college on the outskirts of Atlanta. We quickly exhausted the recreational opportunities available on the small campus, and without much deliberation, agreed to escape to the nearest cineplex for a flick. My friend, the quintessential technophile, leaped to his computer in an effort to find movie schedules. Those being the early days of the Internet, the task was far more difficult than it is today. After watching him struggle in frustration, and despite his numerous appeals to the contrary, I picked up the phone and dialed the number for the theater. Within a minute, I had the showtimes.
Today that friend owns his own software company and has likely made more money than I will see in a lifetime. But the story illustrates my point. A computer is a tool. It is an extremely powerful tool and used correctly will allow humanity to achieve things few dreamed possible. Yet, as with all other tools, there is a time and a place for "high tech," and we must constantly know where that line is drawn. I have enjoyed the art of carpentry for as long as I can remember. As a child I helped my father build a deck on our house. He took me to the lumberyard and showed me how to pick out the wood. He taught me how to swing a hammer and how the principle of the lever could assist me in removing a nail. I learned with painful clarity the consequences of poor aim. This romance with construction continues today, and a trip to the hardware store an event to be savored. Yet despite gaining proficiency with all manner of pneumatic and power tools, and in flagrant disregard more the Toolman's cries of "More power!" there are times when I eschew those modern wonders for a simple hammer or hand-held screwdriver. Not for nostalgia alone, although there is a certain pleasure in it, but because sometimes they are the correct tool for the job.
As any skilled craftsman knows, sometimes "more power" is too much, and that mantra holds true for the computer age. Sometimes "low tech" gets the job done, and sometimes it is the best way to go. If you want a child to develop their creativity, want them to appreciate color and improve hand-eye coordination and take pride in their work, you may want to save the money spent on expensive Paint software and hand them a pack of Crayolas. I would much rather have a page torn from a child's coloring book depicting purple ducks hanging on my refrigerator than a cleanly rendered hodge-podge of Clip-Art any day.
It may not be readily apparent, but as Isaac Newton would attest, each action has an equal and opposite reaction. As our children shape the world with tools, those tools are simultaneously shaping them. Like fish growing to the size of their containers, the mind of a child grows to suit the environment to which it is accustomed. Few parents would approve of pre-schoolers using a power saw, but almost none would balk at extensive computer usage. In her book, Failure to Connect, Dr. Jane Healy explores quite throughly the potential benefits and possible dangers of computers in education. It is almost ten years old, but its message is still very relevant. When considering time and money spent on educational software in this country, we must learn to balance our unbridled enthusiasm for the digital age with constant review and skepticism. With great power comes great responsibility and sometimes more is just more. Our children are plugging-in earlier and earlier. We are seeing a generation of children with diminishing imagination and limited attention spans. Their ADHD brains hyperlink from one train of thought to the next. They are adept at search and navigation, but lack the ability to synthesize raw information into meaningful and original conclusions.
I still love computers. I use them daily. I believe they should be available to all students, but that they will never be a substitute for an able and enthusiastic teacher. When designing a system, sometimes the important question is not can you do something, but why. We must keep that in mind before we allow our young ones to give themselves over to the Matrix.
In unrelated news, I have tentatively formulated a loose course schedule for finishing my BA in Math in two years. I tossed in a minor in Physics just for fun. I'm starting off slowly with a lot of review. I may talk a big game here, but it's been over a decade since I even looked at a limit on paper. Even though I have credit through Calc III, I would just as soon retake it all. So without further ado, here it is:
Fall 2007: Calculus A
Structured Problem Solving (Java)
General Physics I
Spring 2008: Calculus B
General Physics II
Fall 2008: Calculus C
Intro to Probability
Foundations of Math
Intermediate Physics Lab I
Spring 2009: Statistical Inference
And there it is folks. Of course, when I get advised at orientation, they may tell me that I'm crazy. I know that last semester is going to be hell, but I don't want to have to prolong entry to grad school for a whole year because of a few missing credits. Wish me luck.
Wednesday, April 25, 2007
Tuesday, April 24, 2007
Regular readers will notice a curious amalgam of political ideologies spewing forth from my lips. I am generally a pragmatist, and I tend toward the center of the spectrum. I have friends who would happily play golf with Karl Rove and ones who will start showering again when Tibet is free. I am fiscally conservative and socially liberal, but I would just as soon get rid of states and take orders from Washington, so I can't be a libertarian. It all makes sense to me, even if it seems confusing or arbitrary to some. I mention of of this to establish some background for my opinions on genetic modification.
This past year, I thru-hiked the entire length of the Appalachian Trail. It was a wonderful experience and a life-long dream. I highly recommend it to anyone who enjoys being outdoors and wants to find out just how much they are capable of. While I was hiking, I spent some time with a guy known to me only as Cantaloupe. (We use nicknames only on the Trail.) He was trained in horticulture and it was common to see him reach down and scoop up a handful of trailside flora as a snack. He ate leeks, mushrooms, blueberries, bunchberries, snozzberries, and myriad other things. Each time he did it, part of me would cringe in fear. You see despite spending a good portion of my life in the outdoors and being an Eagle Scout, I had been taught from birth never to eat any wild berries, mushrooms, etc. After the novelty of it wore off, I started to think. I am an intelligent, capable grown-man. I have a college degree and am about to return to school for two more. But none of that makes up for this one atrociously humbling fact. I am incapable of distinguishing food for non-food. That makes me dumber than every animal on the planet.
Oh I can pick out a steak if it's wrapped in cellophane or chow down on a box of Cheez-Its, but if it isn't boxed, bagged, homogenized, pasteurized, pumped full of chemicals, and marketed to me during prime-time TV, it is not recognizable to my brain as being edible. Once I have a realization like that, I am not the kind of person to allow the situation to stand. I immediately embarked a quest to discover the origins of my food, and to decide if I really wanted to be eating it. I read several books, including Fast-Food Nation and The Omnivore's Dilemma. I watched several documentaries, including one called The Future of Food about genetic modification. After much deliberation, I decided to stop eating all the the processed crap that had constituted most of my diet. No more Coca-Cola, no more Doritos, no more corn-fed beef. I am trying to eat organic and local.
I do not consider myself a hippie. I bathe regularly and find the scent of petulie to be downright loathsome. In most other ways, I am a thriving Capitalist. I will happily save money at Wal-Mart. I am not making this change for health reasons, although I expect it to be a pleasant by-product. I am doing it because I feel that we have unnecessarily complicated to process of digestion and in doing so have developed an entrenched and thoroughly unsustainable system of agriculture that cannot be allowed to continue. Because I tend to look at things from a systemic perspective, I prefer to think of things as part of a whole. A cow is a machine for turning grass into meat. It is a highly efficient machine at that, and microbes in its gut allow it to get energy from cellulose that is useless to us. So energy originally from the sun is transfered to us through this process, and until we can teach ourselves to photosynthesize, we have to eat. Like all machines, what you get out of it depends on what you put in. Say you only a gigantic, gas-guzzling Hummer and your tank is empty. Assuming you have that kind of cash, the current oil prices probably don't concern you. Given the choice, do you put cheap low octane gas in your tank or spring for the high-test? That's what I thought. Well feeding a cow a diet of corn instead of grass is the biological equivalent of pouring some sugar into the tank along with that crappy gasoline. They weren't designed for it, and the meat suffers. They require more antibiotics to stay healthy and growth-hormone to achieve desired size. Feeding them this surplus corn allows you to raise more cows for less money, but if the food you are producing is lower quality and provides less energy, maybe there is a better solution.
There are many example of why I think we can raise food more efficiently and achieve better results from organic philosophies than from current commercial practices, but this post started with genetically modified crops at I'd better get back there. I don't have a problem with gene-splicing. I love technology and science, and I think that if our learned astronomers think they can build a better tomato, well then they should have at it. I do think that much more testing and evaluation needs to go into it, though. The current industry is propelled by profits and concealed by the same cultural compartmentalization that caused me to fear wild berries. Companies like Monsanto are allowed to grow their modified crops right next to other non-altered ones and then sue their neighbors when patented genes are found in adjacent fields. Those crops ought to be contained, and that's going to be tough, not to mention costly. If you have a dog who craps on your neighbor's lawn, is it your responsibility to fence your dog in or your neighbor's to fence him out? I believe the responsibility lies with the dog-owner just as it lies with the farmers raising Round-Up ready corn.
I understand that laws like this will throw up huge barriers to the agriculture industry and I am fully aware of all the commercial arguments against them. Still, I will quote a local organic farmer here in Maine.
"You grow what you want to grow, and let me grow what I want to grow."
Good fences make good neighbors.
Monday, April 23, 2007
Now granted, basketball is one of the few games that is pretty much guaranteed to have scores well into the double digits, but still, this is ridiculous. Even Crankshaft agrees.
Sunday, April 22, 2007
Recently, I've been thinking more about how I would like to design my tests. Instead of being the mean teacher who makes the hard tests, I was thinking about letting the students write it themselves. Each student, under my supervision, writes one multiple choice style question. Over the course of several drafts, I will help the student make any necessary clarifications or corrections. The student must explain why all answers were chosen. They must do their best to predict how a test-taker might make a mistake on their problem, thereby providing the "wrong" answers to go alongside the correct solution. This will not only help them better understand the techniques we are using, but will also help in taking standardized tests, where the show will be on the other foot. I expect they will design a test that is much harder than any I would give, and they will have only themselves to blame.
I will then compile the individual questions into a single test to be given at a predetermined time. Of course, there will be a certain amount of collusion on their parts. In theory, with total cooperation they could share all answers and each get a perfect score. To temper this, I will not only award points for correct answers, but will give each student one bonus point for each of his/her peers that was tripped up by the question he/she submitted. This should give an incentive toward competition. The opposing forces of selfishness and altruism ought to make for an interesting outcome, to say the least. In a class of 30 students, the top possible score would be 129%, a perfect 100% plus 29 bonus points, one for each classmate. The lowest score ought to be 3.3%, since we can assume the students will get their own questions correct.
That's basically the concept, and I have no idea how it would turn out. I thought of trying to predict the outcome beginning with a typical bell curve and going from there, but there were way too many assumptions to make. I think I'm just going to have to try it and see how it goes. Comments would be very much appreciated on this post, especially from those of you who are have some experience teaching and are not simply talking out of your ass like me.
Saturday, April 21, 2007
In response to the decision, Kathy Miller, president of the Texas Freedom Network said,
"I think the committee got the message that families and churches don't want the government to tell our children what to believe about the Bible."
This quote embodies the true spirit of the separation of Church and State. It is not to protect Caesar from God, but to protect God from Caesar. Here's why. If you claim you are teaching the Bible as culture, you can't legitimately use the Bible as your textbook because there isn't just one Bible. There has been translation after translation, revision after revision, leading up to what today's Christians know as the one true book. Here is a brief time-line of the translation history brought to us by Greatsite.com:
Timeline of Bible Translation History
1,400 BC: The first written Word of God: The Ten Commandments delivered to Moses.
500 BC: Completion of All Original Hebrew Manuscripts which make up The 39 Books of the Old Testament.
200 BC: Completion of the Septuagint Greek Manuscripts which contain The 39 Old Testament Books AND 14 Apocrypha Books.
1st Century AD: Completion of All Original Greek Manuscripts which make up The 27 Books of the New Testament.
315 AD: Athenasius, the Bishop of Alexandria, identifies the 27 books of the New Testament which are today recognized as the canon of scripture.
382 AD: Jerome's Latin Vulgate Manuscripts Produced which contain All 80 Books (39 Old Test. + 14 Apocrypha + 27 New Test).
500 AD: Scriptures have been Translated into Over 500 Languages.
600 AD: LATIN was the Only Language Allowed for Scripture.
995 AD: Anglo-Saxon (Early Roots of English Language) Translations of The New Testament Produced.
1384 AD: Wycliffe is the First Person to Produce a (Hand-Written) manuscript Copy of the Complete Bible; All 80 Books.
1455 AD: Gutenberg Invents the Printing Press; Books May Now be mass-Produced Instead of Individually Hand-Written. The First Book Ever Printed is Gutenberg's Bible in Latin.
1516 AD: Erasmus Produces a Greek/Latin Parallel New Testament.
1522 AD: Martin Luther's German New Testament.
1526 AD: William Tyndale's New Testament; The First New Testament printed in the English Language.
1535 AD: Myles Coverdale's Bible; The First Complete Bible printed in the English Language (80 Books: O.T. & N.T. & Apocrypha).
1537 AD: Tyndale-Matthews Bible; The Second Complete Bible printed in English. Done by John "Thomas Matthew" Rogers (80 Books).
1539 AD: The "Great Bible" Printed; The First English Language Bible Authorized for Public Use (80 Books).
1560 AD: The Geneva Bible Printed; The First English Language Bible to add Numbered Verses to Each Chapter (80 Books).
1568 AD: The Bishops Bible Printed; The Bible of which the King James was a Revision (80 Books).
1609 AD: The Douay Old Testament is added to the Rheims New Testament (of 1582) Making the First Complete English Catholic Bible; Translated from the Latin Vulgate (80 Books).
1611 AD: The King James Bible Printed; Originally with All 80 Books. The Apocrypha was Officially Removed in 1885 Leaving Only 66 Books.
1782 AD: Robert Aitken's Bible; The First English Language Bible (KJV) Printed in America.
1791 AD: Isaac Collins and Isaiah Thomas Respectively Produce the First Family Bible and First Illustrated Bible Printed in America. Both were King James Versions, with All 80 Books.
1808 AD: Jane Aitken's Bible (Daughter of Robert Aitken); The First Bible to be Printed by a Woman.
1833 AD: Noah Webster's Bible; After Producing his Famous Dictionary, Webster Printed his Own Revision of the King James Bible.
1841 AD: English Hexapla New Testament; an Early Textual Comparison showing the Greek and 6 Famous English Translations in Parallel Columns.
1846 AD: The Illuminated Bible; The Most Lavishly Illustrated Bible printed in America. A King James Version, with All 80 Books.
1885 AD: The "English Revised Version" Bible; The First Major English Revision of the KJV.
1901 AD: The "American Standard Version"; The First Major American Revision of the KJV.
1971 AD: The "New American Standard Bible" (NASB) is Published as a "Modern and Accurate Word for Word English Translation" of the Bible.
1973 AD: The "New International Version" (NIV) is Published as a "Modern and Accurate Phrase for Phrase English Translation" of the Bible.
1982 AD: The "New King James Version" (NKJV) is Published as a "Modern English Version Maintaining the Original Style of the King James."
2002 AD: The English Standard Version (ESV) is Published as a translation to bridge the gap between the accuracy of the NASB and the readability of the NIV.
That's a lot of copying and recopying. Any child who has played Telephone (Chinese Whispers in the U.K.) or anyone who has tried to talk on a noisy dance floor knows how easily a message is obscured during reproduction. A famous and extremely pertinent example is the translation of the word almah from the original texts to mean "virgin." The word for "virgin" is bethulah. Almah is more properly translated as "maiden." That's seems like a big mistake to me. It turns a poor unwed girl who's gotten in over her head into a miracle. Then there are the deliberately altered portions, those changed by innocent monks "fixing" syntax and by Machiavellian monarchs seeking to sure up their "birth right."You also have to teach the Torah by design and the Koran for comparison, along with those excluded Gospels like that of Mary Magdelene and Thomas. Nor can you skip the religious history leading up to the Jews. There need be comparisons between Zeus and the Yahweh of Job. The stories of gods like Mythras and Baal, whose creation myths also have them being born of a virgin, must be told alongside that of Jesus of Nazareth. And certainly other works by secular historians would be imperative.
I am an atheist, and believe religion has absolutely no place in public schools. Yet I believe the Bible should be taught in schools, under the conditions I have mentioned. Despite the true believers' claims of divinely inspired scribes channeling the Word of the Lord, the Bible is and always has been a human construct. It is impossible to understand Western Civilization or the current political climate without a firm understanding of it.
Friday, April 20, 2007
He drew inspiration from his own childhood experience in rural Arkansas, where students must endure a 90 minute one way trip to school. Many kids in these small agrarian communities do not have the opportunity to take math and science classes in high school and may feel that college is out of their reach. Dr. Hudson wants to change all that. His new mobile classrooms will focus heavily on math and science and have been endorsed by two Nobel Laureates.
I suspect that some more cell-phone towers will need to be installed for this plan to be feasible, but I definitely like the idea behind it. It certainly makes the most of multi-tasking, although I do wonder why we can outfit a school bus with such high-tech digital technology, but find it so difficult to install seat belts.
Thursday, April 19, 2007
The bulk of the post is a letter from Jim Stone, a math and physics professor of 18 years, who suggest that the ongoing innumeracy problem in this country is not caused by the education system and therefore can not be fixed by the education system. Maybe the problem lies in our cultural acceptance of our math deficiency. Now, of course, I would point out that the education system goes a long way to creating that culture. Although the length of the school day and year are determined by individual states and private districts, the average American student spends an estimated 1206 hours actively receiving instruction each year. That amounts to 1.65 years by the time they graduate high school. That's just class time. If you add in time between classes along with activities before and after school, the average first-grader can expect to spend 4 years on campus before they graduate high school. So students spend 1/3 of their life in school. Maybe that doesn't sound like much, but since they spend another 1/3 sleeping, that time in school amounts to the time spent on all other waking activities combined. So school pretty much is the culture for most kids.
But I've gotten away from Jim's point, which is a good one. Why are people so quick to admit their innumeracy? Just out of college, I got a job with a direct marketing company. We sold products for AT&T, American Express, Coca-Cola, and other Fortune 500 companies. At one point, we were working one the Wal-Mart credit line and I found myself in rural Kentucky signing people up for credit cards. On our first couple of days the conversion rate was pretty low, but on the third day the store manager suggested we would do a lot better if we offered to fill out the applications for people. He was right and the number of applications tripled. He knew what we didn't- that many of his patrons in this impoverished area were illiterate.
These people could have admitted this fact to me and saved themselves a lot of trouble. After all, I was a total stranger that was leaving town at the end of the week. But their pride would not allow it. Imagine all the myriad tricks an active adult has to master to hide the fact that they can't read. I would argue that it probably takes a much smarter person to pretend to be literate than it does to actually read.
The pride that keeps their secret about reading doesn't seem to exist for math. If you were to walk into a room of strangers and ask who was bad at math, plenty of hands would shoot up. Why is this? I don't know the answer. In a recent post about quantum computing, I accidentally stated that 16 cubits represented 10^16 pieces of information rather than 2^16. I didn't realize it until I went to work and I was so ashamed that I rushed to a neighboring on-line gaming center and paid $5 for the 30 seconds of internet time it took me to fix the gaff. I did all of this so that you, my loyal readership, would not think less of me. Why, I wonder, do so many others not share my shame over mathematical errors?
This is another question without easy answers. It may be that Jim is right, that the schools are not necessarily to blame. Acceptance of innumeracy begins at home, and many parents will have their children believing that they are genetically "bad at math" before they ever set foot on the school bus. The worlds of pop music and Hollywood will toss their collective blond hairs in confusion as to why anyone would want to be good at math anyway. But it is the schools that are in the best position to do something about the problem.
It is, after all, their job.
Wednesday, April 18, 2007
I saw this posted by MarkCC over at Good Math, Bad Math and thought it warranted a rebroadcast. It's a video of the Kaye Effect, a property of fluid dynamics in which a thinly poured stream of sufficiently viscous fluid is seen to periodically bounce upon impact. It proves once again that math is everywhere, even in your bottle of Suave.
Tuesday, April 17, 2007
But I doubt it.
One need look no further than the state licensing requirements for the elementary level to see the truth. Now I have quite a few friends that are elementary school teachers, so let me clarify my point, lest I be burned at the stake. I certainly realize that what I am asking is more than a little unreasonable. No one can be an expert is as many subjects as they are expected to teach, especially while dealing with rambunctious children. I do not believe it is the teachers' fault. But I do have a problem with specific elements of the system.
Why do we treat primary and secondary schools so differently? Most secondary certificates mandate 24 credit hours of specified subject taken at the college level. Why do we expect less from primary? Why do we expect them to competently teach history, English, math, science, and anything else the school district might desire? Maybe the evidence shows that children that age learn better when they have one person they can trust. But then why do we give them to someone else for music or P.E? What if we had pairs of teachers? One to handle math and science and the other to tackle English and social studies? Or maybe they both teach everything, each yielding to the expertise of the other.
I certainly don't have the answers, but I sure as heck know there's a problem. Think of how many students claim to hate math or hate history. Isn't this patently absurd? I challenge you to find any human who doesn't use some form of mathematical reasoning on a daily basis, and history is everything that has ever happened! How on earth can that be boring? I'll tell you. Force someone to teach a subject that they are not passionate about and require them to be only a few steps ahead of their students, and you have all the ingredients for boredom. It is a feeling that will stick with those kids for life.
I am starting to see slow changes lately. In Massachusetts, the state Board of Education has passed stronger licensing requirements for elementary teachers, particularly in the area of math. I can't believe they will have anything but positive results and I hope more states follow suit. We will never stem the tide of math phobia in this country if we keep turning a blind eye to the obviously low-expectations placed on elementary school instruction. There's not much point in hiring a highly skilled roofer after you've settled for the low-bidding stone mason. Sure, the roof doesn't leak, but will that be much consolation when the foundation crumbles and your house falls down?
"Beauty in mathematics is seeing the truth without effort."
-- George Polya
If you think that mathematicians are emotionless automatons, humorlessly crunching numbers all day, well then you've obviously never met one. On average, they are some of the most poetic people on the planet. They use words like beauty and elegance with the same hallowed tones others save for prayer. Conversely, they revile the ugly.
A well-crafted proof is considered a paradigm of beauty in their world. Sometimes, a beautiful proof can overshadow what may be an ugly result. It's all very romantic and difficult to explain if you aren't one of them. I exist somewhere on the fringe, which will allow me to continue this post. You see, I fear that the constant progress of computer science threatens to crush the poetry of the proof. As the machines grow more powerful, the ability to attack problems with brute force will be much increased. Mathematicians aren't going to like this.
For example, the famous Four-Color Theorem is commonly known as the first major proof to be done almost entirely by computer. Imagine a map, say of the United States. How many colors does it take to color in all the states so that no adjacent ones are the same color? Now imagine an infinity of maps. How many colors will it take? The answer as it turns out is four. Essentially, the infinite number of maps can be reduced to a more manageable number, because a lot of them are just variations. Then those basic map types were fed into a computer with a coloring algorithm and it went to work. It reported that it could color all the maps with only four colors. Case closed.
Only it isn't for a lot of people. For one thing, humans do not live long enough to check all those maps by hand. We have to trust the computer executed its algorithm correctly and that it was given an accurate algorithm in the first place. But mostly it is disliked because it is ugly. Now I can get to the crux of the matter.
Computer technology is going quantum. Most everyone is familiar with the binary nature of digital computers. The word bit stands for binary digit. Everything a computer does, it does by simplifying things to ones and zeros, to yes or no. In binary logic, the gate is either open or closed. In the immortal words of Yoda, "Do or do not, there is no try." With quantum effects, things get a bit muddier. Without going into to much detail, the weirdness that is the quantum world breaks the law of the excluded middle. Superpositions are allowed, gates which are both open and closed. Quantum computers take advantage of this phenomenon to perform calculations at the same time, rather than in rapid succession like current supercomputers. In fact, a company has recently announced the release of the first practical quantum computer. It is a alleged to have 16 quantum bits or qubits. This would allow it to handle 2^16 pieces of information simultaneously. There is a lot of suspicion around this one though. It is at least fifty years ahead of the estimated state of the technology and the developers are being very secretive. Time will tell. But even if this one's a hoax, it will happen eventually.
How will that computing power affect things? Suddenly uncrackable theorems will fall left and right if we are willing to trust the computers. Public key encryptions that use the CSA method based on large prime numbers will be broken in seconds. The P vs. NP issue will be moot. The Traveling Salesman will have his answer as he turns the ignition. (For a detailed explanation of what I'm talking about, check out Rebecca's blog.)
Will mathematicians revolt? What will happen to the elegance? I guess we'll all just have to wait an see.
Monday, April 16, 2007
I usually get my news from the Web, but somehow during a disaster like this, I feel better getting my streaming commentary from the cable news juggernaut. As I flipped ever faster through the channels from CNN to FoxNews to MSNBC to CNBC and beyond, I tried to put myself in their shoes. I am planning on becoming a teacher after all. In a few years, I will be standing in front of a class, in the midst of my prepared lesson plan, and there will always be an outside chance that a discontent, disaffected, misguided student will burst into the room with semi-automatic weapons fire. I accept this. It is one of myriad ways that I may die. I am a human being and we are a notoriously fragile bunch.
But what will I do with this knowledge. Do I turn the school into a prison, hire armed guards, install metal detectors, conduct random locker searches, and profile "high-risk" teens? That's not a world I would want to live in, even if I thought it would make me safer, which I don't. Part of walking out the door every day is knowing that you may die. You can hide behind whatever protective facade you like, be it religion or the latest fad diet, and that won't change the fact that at some point you will cease to live. No amount of legislation or safety protocols is going to change that.
So I mourn the loss of those who were killed and sympathize with those left behind. But I choose to live my life on my terms. Ben Franklin said
There are some bargains I am willing to make. I will give up drinking soda for a healthier body. I will stop driving a car for a healthier world. But I will not give up my freedoms for anything.
"Those who would give up Essential Liberty to purchase a little Temporary Safety, deserve neither Liberty nor Safety"
"Every man dies. Not every man really lives." -Braveheart
"I ain't so afraid of losing something that I ain't gonna try and have it." -Firefly
They will begin offering the new single-sexed classes alongside traditional coed ones, giving students and parents the option. The focus seems to be on the girls. As one teacher put it
"I've got very, very intelligent young ladies who outperform (other students) in my classroom, but they are not as active in participating in class," said Ann Geier, a math teacher at Everest who will teach a class of up to 30 girls. "I'm always interested in seeing how my girls do, being a woman math teacher."
Amber O-Connell, a student who registered for one of the new classes, says
"Girls act differently. They are afraid to ask questions. They will think the guys will think they are dumb."
I'm really not sure how I feel about this and I don't want to make any predictions. I'm choosing to look at it as an experiment that may or may not have the intended results. I will say this. I think schools should try as much as possible to mirror conditions in the actual world. The worlds of science, business, and academia do not protect girls from boys. They will eventually have to learn how to deal with that fact. I can only imagine how hard it would be for a "smart" girl, especially in the field of mathematics which is so largely dominated by men. I suppose the Wisconsin educators hope that this small buffer will allow the girls to develop their skills and self-esteem for college.
I hope they're right.
Sunday, April 15, 2007
Look, I dislike W. I freely admit it. But I'm tired of hearing about how 9/11 was an inside job. It's demeaning to the memory of those who died and quite frankly my intelligence. Check out this brilliant satire of all those conspiracy theorists. It asks some tough questions about the Empire's role in the Death Star debacle.
(The video has nothing to do with it. But eetz-a funny, no?!
Saturday, April 14, 2007
Imagine a toddler out for a walk with Mommy. They encounter in their travels a slobbery Golden Retriever puppy. Mommy, in her most nurturing of voices, accommodates her child's inquiring mind by christening the bounding amalgamation of fur and drool with the moniker of doggy. After extricating themselves from this cute little scene, they proceed with their journey. A few blocks away, they are greeted by the franticly bellowing bark of a Great Dane, whose towering form threatens the structural integrity of its chain-link enclosure. The toddler points at the dog, and with sage-like wisdom proudly proclaims doggy. Mommy is filled with pride. "My child is a genius," she thinks, as she begins mapping out the path from pre-school to Nobel Prize. But the daydream is soon interrupted and all hopes dashed. The object of the Dane's ire turns out to be a mischievous orange tabby that has intruded into his domain. To Mommy's chagrin, her little prodigy raises her chubby little finger toward the fleeting feline and with as much confidence as the diaper-clad can muster, announces yet a third doggy. Mommy is crest-fallen. It looks like she can treat herself to that designer handbag she has been eying, because her little one is clearly not college bound.
The preceding anecdote replays itself in some form or another all over the world. The mind of a child is forming new connections at a rate that adults can only superficially comprehend. Sometimes those patterns are correct but situationally inappropriate. This does not make them wrong. The pattern denoted as "furry, four-footed creature with a tail" is correctly applied to doggy. Of course, it can be applied to many other animals. It will take time, practice, and myriad "wrong" assessments before the nuances of kitty, horsey, and other cuddly quadrupeds are established. Such is the nature of language. It is sometimes messy and confusing, despite our best efforts, and our keen pattern-sense and mathematical reasoning will be both a blessing and a curse.
Consider that in English, trough, bough, dough, and enough do not rhyme, while beau, go, and sew do. That's enough to send anyone screaming for the hills, yet few native English speakers will claim to be bad at it.
The inherent order of mathematics is much heralded by its practitioners. It is neat,clean, and makes sense- most of the time. Unfortunately, math is a human construct, just like language. Often times, our beloved mathematics is guilty of the same confusion and inconsistencies as we saw above. Consider the notation f(x). For many, it immediately brings back horrible memories of their first taste of algebra and the day they discovered they were bad at math. They were cruising along through arithmetic. They learned how to do long division and divide fractions. They even went with the flow as the symbols started changing. The symbol for multiplication went from x to a dot to an asterisk, then to parentheses as in 3(4). It was confusing, but they had hung in there. Now they were being taught to use letters sometimes to represent numbers. Now that was pushing it, but still they pushed on. Enter the dreaded f(x). It looks familiar and friendly, and when asked to find f(4), the students find a suitable pattern from their arsenal and proudly answer 4f. I'm sorry says the teacher, the answer is kitty.
For many, this was the point where math went from fun to completely mind-boggling. What these unfortunates fail to understand, largely because they have not been properly told, is that the problem is not that they are bad at math, but that they are so good. This is one of many confusing notations employed in such an allegedly orderly field. Others include
- sinn and tann
but sin–1x = arcsin(x) and tan–1x = arctan(x).
- The square root symbol Öb , which actually refers to "the non-negative square root of b" but is often lazily used to mean both the positive and negative roots by math teachers.
- Confusion abounds in order of operations: What is –32 ? Many think it is
(–3)2,and so they arrive at an answer of 9. But that is wrong. The convention among mathematicians is to perform the exponentiation before the minus sign, and so –32is correctly interpreted as –(32),which yields –9.
- Or 3/5x, which depending on whether you use the BODMAS (bracketed operations, division, multiplication, addition, subtraction) or My Dear Aunt Sally (multiplication, division, addition, subtraction) will yield (3/5)x or 3/(5x).
These are just a few examples. I'm sure there are many more. The important thing is to appreciate how difficult and often times contradictory math notation can be, lest young math enthusiasts become older math phobics, simply because their highly adapted pattern recognition instinct was confused by faulty patterns.
Friday, April 13, 2007
In honor of this misunderstood number, I thought I would give a few reasons why 13 isn't so bad. For starters, it's the first "teen." Our Germanic heritage saw to it that 10+1 and 10+2 had entirely different forms than the rest of the 10's. They are both throwbacks to Old English and completely isolate themselves linguistically from the others. But many an adolescent knows that with 13, there comes power- the power of the teens.
It's also a prime number, which makes it part of one of the most mysterious and captivating groups of mathematics. Great minds through the ages, from Gauss to Riemann, have gone in search of the secrets of the primes, only to return in failure.
Lastly, at least for my purposes, it appears in the Fibonacci sequence, and therefore is intimately connected to the Golden Ratio phi and through it the rest of nature.
So the next time you shudder in fear at the number 13, remember all that it does for you.
Thursday, April 12, 2007
From day-one, I was suspicious of the case against him. Too much emphasis was being placed on a somewhat discombobulated (you gotta love that word) account from the "victim." In the era of DNA testing, we have been forced to revisit the statistical nature of all forms of evidence. Make no mistake, all evidence is statistical, especially eye witness account. In 1996, 28 people were released from prison due to DNA test after having served a combined 197 years of their terms. Of those 28 cases, 24 hinged on eye-witnesses. Studies have demonstrated that the human memory, which is equally good at registering patterns and completing ones that aren't really there, is wrong up to 50% of the time. That is no better than random chance, and certainly not good enough to convict someone. Many people mistakenly believe that adding corroborating witnesses reduce the odds of inaccuracy, but that is true only if they are kept in complete isolation. Even the slightest bit of innocent discussion between witnesses is enough to sway opinion in one direction or another.
Good prosecutors realize all this, which is how I know that the DA in this case, Michael Nifong, isn't one of those. Police line-ups have to be very careful to keep the odds as even as possible. The idea is to put the suspect in a line-up with a group of very similar people. If the victim alleged that the assailant was black and all of the ringers are white, what odds does the accused have? Not very good. In the Duke case, the odds were even worse than that. Upon hearing the "victim's" story, Nifong arranged for the following line-up procedure. He acquired photos of the entire lacrosse team, not just those known to be at the party, but the whole team. Then he asked the accuser to pick the men who had raped her. Quick, if I ask you to number out of a hat, what are your odds of picking a number out of that hat? This investigation was unethical to the point of being criminal. I hope Nifong is brought up on charges himself.
Unfortunately, high-tech DNA evidence is just as subject to this kind of misconduct. Even assuming that all human error has been removed from the gathering and processing, there is still a chance of misinterpretation of the data. The current CODIS registry that the FBI uses looks at 13 sites on the human genome. These sites are what is known as "junk" DNA, meaning that they appear to be artifacts of the complex code replication system, and they do not code for any traits. This allows them to vary widely in the population and makes them excellent choices for identification. Only just like in a traditional line-up, one must be very careful. Those sites, which vary so widely, have less variation within certain subsets. You are much more likely to match someone who is descended from a similar world region as you(i.e. sub-Saharan Africa.) You are extremely likely to match someone in your own immediate family, so if you're paternal uncle were to rape someone, all the fancy DNA testing could easily confirm that you did it. Juries are specifically selected by both prosecution and defense to be largely ignorant of statistical thinking, so this is a fact which can't be taken lightly.
Statistics affect our lives in more ways than I can possibly describe or even imagine. Understanding this kind of math is imperative for everyone.
Wednesday, April 11, 2007
Well screw that! The Council for Industry and Higher Education in the UK has just released a report calling for the government to stop the ever-accelerating exodus from math and science at there A-level schools by providing cash rewards for good scores. I doubt very seriously that this is going to bring many new students into the fray, but it just may be the extra incentive that a good student needs to become a great student. I realize I am advocating something that many may find thoroughly atrocious. In many ways, I agree. But there's no arguing with the almighty dollar- or in this case the pound.
Seriously, throw a little of that cash my way.
Tuesday, April 10, 2007
I chose to analyze an episode of the UK version of the show, primarily because I found a fansite that had done most of the work for me, but also because there's only so much Howie Mandel I can stand. The British version is identical in theme, but differs in a few details. There are 22 boxes, instead of the 26 in the US. The currency values in pounds are as follows:
In this respect, the banker is really more of a psychologist than mathematician. Like a poker player upping the ante, his goal is to keep his opponent in the game for as long as possible. In the UK version, there are sometimes multiple contestants per show, ala Millionaire. Since the banker pays out the 25,712.12 average per contestant, not per episode, it is in his interest to limit the number of contestants, thereby increasing the length of each game. In the US, there is only one contestant per show, so the goal is simply to fit the time alloted.
I first began by tallying up the winnings from the first 20 games of the inaugural season to see how the banker did. His actual average payout was only 18,720 , which is roughly 75% of the expected average payout. So he's doing pretty good. I then chose a game in which the contestant was taken the distance and ended up passing on a deal that exceeded his/her ultimate winnings. Madie from November 15, 2005 satisfied these criteria. I'm not going to list each turn here, but you can find the breakdown in the link above. Instead I will verbally summarize.
Round 1: After eliminating 5 boxes, the banker offers 6,900.
Odds of beating the deal: 6/17 or 35%
Odds of beating expected value: 3/17 or 18%
Round 2: After eliminating 3 more boxes, the banker offers 1,600.
Odds of beating the deal: 6/14 or 43%
Odds of beating expected value: 3/14 or 21%
Round 3: After eliminating 3 more boxes, the banker offers 4,800.
Odds of beating the deal: 4/11 or 36%
Odds of beating expected value: 1/11 or 9%
Round 4: After eliminating 3 more boxes, the banker offers 14,800.
Odds of beating the deal: 3/8 or 38%
Odds of beating expected value: 1/8 or 13%
Round 5: After eliminating 3 more boxes, the banker offers 28,000.
Odds of beating the deal: 1/5 or 20%
Odds of beating expected value: 1/5 or 20%
Round 6: After eliminating 3 more boxes, the banker offers 4,800.
Odds of beating the deal: 0
Odds of beating expected value: 0
Glancing at the other games, it appears as though the banker chooses his "deals" so that the contestant has a 30% chance or better of beating it, thereby continuing the round and building excitement. Once it is apparent that the contestant has "lost," the deals do indeed begin to drop off. There is clearly no hard and fast rule being used, and some huge surprises occur. I suspect an analysis of a poker shark would be more appropriate here. I'm afraid I don't play well enough for that.
Monday, April 9, 2007
Now obviously astrology is crap, and if you think otherwise, this isn't the blog for you. But I want to discuss the details of this study further, because it's only saving grace was its shear magnitude. (A more in depth discussion appears at the Skeptics Guide.) They got the data from a 2001 census, and out of the 10 million couples, they found no statistically meaningful connections. This is the merit of using such a large data set. Trivial coincidences are averaged out. Had they used a smaller set they would quite likely have found a more "meaningful" result.
You see, they weren't looking for a particular correlation. There was no hypothesis. They didn't say, "Astrology predicts that Cancers will be attracted to Capricorns," then look for that to bear out in the data. They were just looking for anything interesting. That's called data-mining. It's a common ploy of bad science. Let me give another example. Let's say you have a theory about family pets. You have notices lately at the dog park, that girls seem to prefer smaller dogs, while men prefer larger ones. You call up the local vets and convince them to give you information on the gender of the owners versus the weight of their dogs. Maybe your idea is supported by the data; maybe it isn't. But you've at least conducted a reasonable study.
Now let's go data-mining. You gather a large batch of data about people and their dogs, and after pouring through it you find something curious. Every dalmatian is owned by a man named "Steve." Most German Shepherds are owned by people with French sounding names. And more often than not, Chihuahuas like to paint their fingernails some shade of pink. Can any meaningful conclusion be drawn from this? Of course not. Yet this is exactly the kind of thing that certain researchers try to do each year.
If you go looking for an unspecific anything, you're bound to find something. It may sound obvious, but it's easy to fall victim to this kind of thinking. So be careful out there.
Sunday, April 8, 2007
When I watch this show, which is to say less than a handful of times, I feel physical pain. Real, horrible, blood pouring from my nose and ears pain. So much so that I find myself yelling at the screen, as though I were could some how will them into better judgment. (Don't go in the basement! Don't open that door! Stop having premarital sex while a camp counselor at this creepy summer camp don't you know there's a maniac with a predilection for machetes on the loose you moron!!!!!) In the few times I have watched until the end, I have never-I mean never- seen a contestant walk out on the top of the curve. The same pattern occurs over and over. At first, the odds are in favor of them at least walking out with some reasonable amount of money. After all, something is always better than nothing. Then the excitement builds, and this excitement it seems is indirectly proportional to deductive reasoning. The banker begins to make offers, which are invariable statistically better than the odds the contestant faces. Eventually, a tipping point occurs, and those deals become worse and worse. The action follows the outline of a wave, and no one ever seems to buy out on the crest. The bubble always bursts.
Much like my brain when I allow myself to watch.
And unfortunately, despite rapid advances in technology, that's how computers are being used in the classroom- as paperweights. Technology is treated as some great panacea, much like accountability and school choice, as though its mere presence will instantly raise grades and improve education. It would be like buying your child a brand new set of titanium golf clubs and expecting him to beat Tiger Woods. Tools, whether they are tangible like a computer or systemic like testing, are only useful to those trained to use them well. Most teachers see computers as word-processors or mailboxes or calculators. They scoff at the notion that a computer could teach a child.
A new study purports to demonstrate that they are right, but all it does is show lack of understanding of the issue. Classrooms were introduced to a variety of computerized lessons and tools, and then then their performances were compared to the previous year without technology. Surprise, no appreciable difference was found. A second study is planned to see if teacher familiarity with the programs makes a difference. Seriously, do we need a study to answer these questions? Don't things always get worse before they get better during a regime change? Isn't it fairly obvious that a teacher must be familiar with the ins and outs of a lesson plan before they can effectively teach?
Many of today's young people are building online social networks that rival those in the real world. They gather information with search engines and they have never heard of a card catalog. Computers are here to stay. The help doctors diagnose disease and help navigate the family vacation. They will never and should never replace the human instructor, but it is time to start letting them do what they can do. They allow students to move at their own pace. No more forcing kids to keep up with the rest of the class. They do not discriminate. They will never play favorites, singling out boys over girls or whites over blacks. They provide students with what schools otherwise can't do- a 1:1 student/teacher ratio.
Sure a human being must be there to manage the class, but in a time when many teachers face 40-50 kids at a time, I don't see any way to move forward that doesn't involve embracing technology. Studies be damned.
Friday, April 6, 2007
Every once in a while, I find myself sitting in front of the TV when CBS's crime drama NUMB3RS is on. Though I do not regularly tune in to the show, I have certainly enjoyed it when I did. According to the show's website,
NUMB3RS is a drama about an FBI agent who recruits his mathematical-genius brother to help the Bureau solve a wide range of challenging crimes in Los Angeles. The two brothers take on the most confounding criminal cases from a very distinctive perspective. Inspired by actual events, the series depicts how the confluence of police work and mathematics provides unexpected revelations and answers to the most perplexing criminal questions.
In any event, my respect for the show increased when I found out that Texas Instruments has been pairing with the shows producers to design a down-loadable lesson plan for math teachers, so that they can easily use the show during instruction. Although I would hope some enterprising teachers are already employing techniques like this, it is helpful of CBS to provide it for them voluntarily. Here is an example of a lesson based on the episode "Burn Rate."
NUMB3RS Activity: EnergyIf only Mythbusters would do this, math and physics teachers would be set for life.
In “Burn Rate,” Charlie helps the FBI investigate a series of mail bombings with seemingly
unrelated targets. Charlie explains “explosions are all about physics and math: burn rates,
brisance and pressure waves.” He compares the bomb to a hand hitting a table set for a meal.
“By the way objects are displaced, I can tell you the size of the hand, how tight the grip, or how
much energy was imparted on impact; just like I can analyze what kind of bomb we're looking at
here.” In this activity, students will use radical equations to solve problems about energy.
The equation below gives a relationship between the energy E of a certain bomb, the mass m of
a particle in the bomb, and the distance d the particle travels. Energy is measured in joules
(kg m2/s2), mass is measured in kilograms and distance is measured in meters. The constant of
25,000 arises from physics formulae. The physics principles and a unit analysis are presented in
more detail in the extensions section.E= 25,000md
1. A piece of shrapnel weighing 50 grams is found 10 meters from the site of the bomb.
a. Determine the value of m, remembering mass is measured in kilograms.
b. Calculate E, the energy of that portion of the bomb.
2. To get a sense of the size of a joule, match each item below with its energy level.
Energy level, in joules Item
a. 80 lightning bolt
b. 1,400 kinetic energy of a car at highway speed
c. 4,000 energy consumed by average automobile in US
in one year
d. 350,000 annual power use of one clothes dryer
e. 4(106) average person swinging baseball bat
f. 1.5(109) 1 gram of TNT
g. 3.2(109) 1 kg of TNT
h. 7.2(1010) eruption of Krakatoa
i. 1.5(1017) bullet traveling at 900 m/s
3. Using the energy level you determined in Question 1b, the equation for the situation
presented earlier is 4,000 = 25,000m d . Recall that a 50 gram piece of shrapnel traveled
a. Should an 80 gram piece of shrapnel travel a longer or a shorter distance than the
50 gram piece?
b. Calculate the expected distance from the bomb of the 80 gram piece.
4. a. Solve the equation E= 25,000md for d. Check your rewritten equation by verifying
that when m = 0.05 kg, d =10 m.
b. Calculate the distance a 25 gram piece will travel and the distance a 100 gram piece will
c. Use the results of Question 4b to determine a rule about the distance a piece will travel
when its mass is doubled.
The goal of this activity is to give your students a short and simple snapshot into a very extensive math
topic. TI and NCTM encourage you and your students to learn more about this topic using the extensions
provided below and through your own independent research.
• The original equation for energy is given below. Using the estimated values given, verify the
equation E= 25000md used in the activity, where a = 10°, g = 9.8 m/s2, l = 0.1 m, and
t = 0.0000107 s.
E = t a a
• The units of a joule are (kg)(m2)/s2. Verify that the equation below results in the correct units
for energy, measured in joules, where E is energy, C is a constant, m is mass in kg, l is
length in m, d is distance in m, g is acceleration of gravity in m/s2, and t is time in s. This
equation is an expansion of the one used in the activity.
E Cml dg
• An online unit conversion tool can be found at http://www.onlineunitconversion.com.
• An introduction to solving radical equations is available at
Wednesday, April 4, 2007
Tuesday, April 3, 2007
The line between church and state must be firm; the line between religion and politics and education doesn't exist. Faith is part of our culture, and as much as I wish it gone, it is a profoundly important part of the lives of most Americans. It is so important that the want to shout it from the rooftops, or in absence of a rooftop, distribute some snazzy fliers. This is their constitutional right, and it doesn't end at the schoolhouse door. And the great thing about those rights is that we all have them. So if another student, say, wanted to print up fliers saying that Jesus probably never existed and that the entire history of Christianity is a sham which continues to pollute our society today and into the future, well then that's fine, too. (But they probably ought to pick a really slick font and some fancy-ass paper, because that's going to be a tough message to sell to most.)
Fast forward to the present, when the American College of Physicians has released new recommendations regarding preventative mammograms. They argue that women under 50 with no medical history of cancer or other high risk factors need not be subjected to yearly mammograms. The current recommended age is 40, supported by the American Cancer Society. This is a case where mathematics must influence our decision making process. More and more, the medical profession seeks to back up its dogma with statistics. This is a good thing if it is done correctly. Unfortunately, most doctors do not understand the math themselves and certainly can't explain it to their patients. Let's look at the numbers for mammography.
The rate of false positives for this test is estimated to be between 7-15%. This is higher than in most other first-world countries. Why? What is a false positive rate? A false positive, or Type I error, is a positive an erroneous positive result. The patient does not have cancer, but the test shows that she does. No test is perfect. All medical tests give a certain amount of false positives. The rate is calculated by dividing the number of false positives by the number of total negative instances.
Then there is the problem of over-diagnosis.
The widespread and virtually unchallenged acceptance of screening has resulted in a dramatic increase in the diagnosis of ductal carcinoma-in-situ (DCIS), a pre-invasive cancer, with a current estimated incidence of about 40,000 annually. DCIS is usually recognized as micro-calcifications and generally treated by lumpectomy plus radiation or even mastectomy and chemotherapy. However, some 80 percent of all DCIS never become invasive even if left untreated. Furthermore, the breast cancer mortality from DCIS is the same— about 1 percent— both for women diagnosed and treated early and for those diagnosed later following the development of invasive cancer. That early detection of DCIS does not reduce mortality is further confirmed by the 13-year follow-up results of the Canadian National Breast Cancer Screening Study.These are the facts. So before you decide to get a test that is embarrassing, uncomfortable, or even painful for some, do the math. Decide for yourself if it is necessary for you. Ask your doctor for his/her false positive and false negative rates. If your doctor stares at you with a blank expression, you may want to look for another physician.
Monday, April 2, 2007
In a fitting tribute to science fiction's most beloved engineer, the ashes of James "Scotty" Doohan have been sent into space. The Star Trek icon will join the series creator Gene Roddenberry, who blasted off in 1997. For fans worldwide, this is the only imaginable send off.Doohan's portrayal of a starship engineer set the bar so high that none have since come close. Though there is no evidence that the phrase "Beam me up, Scotty" was ever actually uttered during the 5-year mission or subsequent movies, it has nevertheless become part of our culture. Somehow, amidst modifying the dish to emit tachyon pulses and keeping those darned dilithium crystals from failing, Scotty won our hearts. He will be missed.
Sunday, April 1, 2007
Unlike some government agencies, NASA is in danger of losing it's funding. Americans have lost much of their romanticism for space travel, and with ex-astronauts popping up in mugshots instead of on Wheaties boxes, who can blame them. So the struggling space agency loves to spark controversy in the press. I suspect that's exactly what they intend from the latest Cassini probe videos. The images clearly show a giant hexagon rotating above the north pole of Saturn. My main reason for this suspicion is that the same images were recorded by Voyagers I and II with little fanfare.
I'm sure it's just a matter of time before this one gets linked to little green men. Unlike the alleged face on Mars, at least this feature actually exists. While it is bizarre, embedded within a chaotic system like weather, I feel pretty confident in saying that it is a natural phenomenon. Look at how common the hexagon is in nature. Both diamond crystals and honeybees use its architecture.