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Thursday, August 30, 2007

Tit for Tat

An old high school pal of mine has recently returned to the blogosphere. Reading Allen's blog was part of what got me interested in posting my own ramblings in the first place. He has been on a hiatus for a while, having become a first-time dad this past Fall, but he has triumphantly returned. Pencils Down was the subject of one of his first articles.

In a follow-up post, Allen referenced one of my posts regarding the famous Monty Hall Problem. He goes into great detail about his personal peccadilloes in the set-up of the problem, discussing the different strategies the game show host might take, and how they would affect the statistical outcome. As he says, the host's prior knowledge of the prize locations is a key element. This reminded me of some recent argument I've seen online as to whether Deal or No Deal comes down to a Monty Hall Problem. Any contestant lucky enough (or ballsy enough) to make it to the final two boxes is always offered the chance to switch by Howie Mandel. The general consensus seems to be that Howie has no prior knowledge of the game and so his offer can not affect the contestant's odds.

I'm not sure why Howie is believed to be any less duplicitous than Hall. Perhaps it is because he is Canadian.

Monday, August 27, 2007

Birds, Bees, and STEM

The National Science Foundation's (NSF) just released the results of their Research on Gender in Science and Engineering (GSE) program. It found that while things have markedly improved for women over the years, some myths still endure. But there is still hope for improvement, they say, and better education for women will translate into gains for both genders and all levels.

Thou Shalt Not Covet Thy Neighbor's Job

It's back to school time, and everywhere I turn, I find an edublogger lamenting some problem or concern they will have to face this year. While I certainly empathize with their worries/fears, I also would like to grab them by the shoulders and shake the hell out of them.

Good teaching comes with a sense of responsibility that eclipses many other professions. To stand before a classroom, means to tilt against an impossibly powerful opponent. It is an endless battle, and one which is predominantly beyond your control. You will be blamed for every failure, by critics at large and the one within. You will ask yourself, "Did I do enough? Did I ask the right questions? Could I have pushed harder? Did I push too hard?" You will beat yourself up over everything, agonizing over each lesson plan, focus in on excruciating details, in the hopes that the self-flagellation will make you a better educator. And when it's all said and done, it really isn't, because you get to do it all again in a few months.

What kind of self-loathing lunatic would sign on for this? Well, me for one. I know it's easy for me to be critical, safely on the outside looking in. Maybe I'll feel differently in a few years, but right now I am desperate to charge full speed into the fight.

I am a sucker for cheesy sports movies, especially underdog stories. I would say to my edublogger friends what those coaches say to their teams at half-time, when the deck is stacked against them, and winning seems impossible. The other team will always be bigger and stronger, more talented, better equipped, and have many more reserves. They will inevitably win 99 times out of a hundred. But that still leaves the one time. That one student on the verge of dropping out, the kid who doesn't think college is for kids like her, the child with the undiagnosed learning disability. A good teacher gets to win big every once in a while. They get to point to a child and say," There, that one right there. I helped that one." They may not earn a decent wage or get the thanks they deserve, but they know in their hearts that the world is a little better because they were willing to fight a battle when others said it couldn't be won.

That sounds like the job for me. Put me in Coach. I'm ready to play.

Sunday, August 26, 2007


Good thing she's pretty.

Nod to IB a Math Teacher.

Friday, August 24, 2007

I'm Talkin' Bout the Funk

Off pitch with some horrible fake guitar playing. But funny nonetheless.

Wednesday, August 22, 2007

The Best Laid Plans

School starts in just over a week. At that point, my plan was to cut back to a maximum of 30 hours a week at Pottery Barn. I figured that would allow me to keep my health insurance as well as help pay back the student loans I will be receiving. Most of my first year will be review, so I should be able to handle the load.

That was my plan, and like many plans it has gone awry.

Two weeks ago, my immediate supervisor put in his notice. If you've ever worked retail before, you know that the holiday season begins soon, and that put us without a stockroom manager for the busiest time of year. As it turns out there is already an intelligent, hard working fellow familiar with the stockroom. I think you see where I'm going with this. The promotion means that I will have to work a full 40 hour week, but I will be making half again as much as I am right now. The offer is just too good for me to pass up. My boss has known about my school schedule for some time, and she is perfectly willing to work around it. She has even given me the first week of classes off, so that I can acclimate myself to college without worrying about work. So come the second week of September, I will be working full-time and carrying a 16 hour course load.

Don't tell my adviser.

On the Same Page

Recently, I had the opportunity to work with a bright, creative student on improving his writing skills. We spent a great deal of time discussing how to assess the intended audience, and how diction, tone, and detail must be adjusted accordingly. In everyday conversation, a staggering amount of background information is assumed to be shared. When you write, you typically reach a larger audience, and you can assume far less.

Just for fun, I have selected a random reading comprehension sample from the net to see just how much core knowledge is required for real understanding. The test is aimed at students on the fourth grade level.

How many things can you see in the night sky? A lot! On a clear night you might see the Moon, some planets, and thousands of sparkling stars.

You can see even more with a telescope. You might see stars where before you only saw dark space. You might see that many stars look larger than others. You might see that some stars that look white are really red or blue. With bigger and bigger telescopes you can see more and more objects in the sky. And you can see those objects in more and more detail.

But scientists believe there are some things in the sky that we will never see. We won't see them with the biggest telescope in the world, on the clearest night of the year.

might find it hard to imagine that stars die. After all, our Sun is a star. Year after year we see it up in the sky, burning brightly, giving us heat and light. The Sun certainly doesn't seem to be getting old or weak. But stars do burn out and die after billions of years.

As a star's gases burn, they give off light and heat. But when the gas runs out, the star stops burning and begins to die.

As the star cools, the outer layers of the star pull in toward the center. The star squashes into a smaller and smaller ball. If the star was very small, the star ends up as a cold, dark ball called a black dwarf. If the star was very big, it keeps squashing inward until it's packed together tighter than anything in the universe.

Imagine if the Earth were crushed until it was the size of a tiny marble. That's how tightly this dead star, a black hole, is packed. What pulls the star in toward its center with such power? It's the same force that pulls you down when you jump — the force called gravity. A black hole is so tightly packed that its gravity sucks in everything — even light. The light from a black hole can never come back to your eyes. That's why you see nothing but blackness.

So the next time you stare up at the night sky, remember: there's more in the sky than meets the eye! Scattered in the silent darkness are black holes — the great mystery of space t
hat's because they're invisible. They're the mysterious dead stars called black holes.

  • I can certainly see how this would be an intriguing passage for a fourth grader, but there is a lot that the author has assumed. For example, the reader must know what the difference is between the Moon, planets, and stars. Although context clues such as "the" instead of "a" and the capitalization of "Moon" might be enough to suggest that Earth has only one natural satellite, I suspect that this fact needs to be understood ahead of time.

  • The fact that these things are visible each night requires an understanding of periodicity, if not necessarily rotation. The student must know that a telescope somehow magnifies images, and that the larger the telescope is, the more powerful its magnification.
  • They must have an understanding of scale as it pertains to the decimal system of measurement. "Billions" is a lot of years, more than most adults can conceive of, let alone a child.
  • There needs to be knowledge of the three common phases of matter, as well as chemical combustion. (Although it should be noted here that the flammability of certain gases in our atmosphere has nothing to do with the nuclear reactor of the sun. I don't know if the author is ignorant or finds it easier to massage certain key facts.)
  • Students also need to understand gravity. They need to know that it is often related to the size of objects, which for the purposes of this paragraph, serves as an indirect measure of mass.
All this needs to be firmly embedded in the child's brain before any of this passage will really make sense. We can teach all the tricks and parsing techniques we want, but unless students have a lot of background knowledge, they are still going to have trouble comprehending what they read. I think this is why we are having so much trouble raising our reading scores on standardized tests. There has been too much focus on "context clues" and not enough on the shear quantities of information that must be shared to even get young readers on the same page.

Monday, August 20, 2007

Fossilization and Memory

It occurred to me today that human memory is a lot like the fossil record. Those few vivid recollections we each possess are few and far between when compared to our lifespan, and there is almost no consistent recipe for making a memory. The right combination of sensory input, context, and repetition cause some snapshots of time to stick better than others. Even when we do remember an event, we are more likely remembering isolated fragments instead of a coherent whole. Then, like puzzle-loving scientists, we attempt to piece the tiny bones back together into a probable design. Often times we get it wrong, and our finished skeleton doesn't match those of other similar finds.

But like the fossil record, our memories, though incomplete, are all we have to go on. Even with the advent of audiovisual recording technology, the odds of having a truly important event caught on tape are slim. So we are forced to use recollections we know to be flawed to guide us in predicting future outcomes.

Oly Oly Oxen Free

It seems as though there is an endless game of tag running rampant through the blogosphere. At least once a month, one of my blog friends punches me in the arm as they zip past, shouting "tag, you're it." These memes still remind me of chain letters, which I have always hated. Therefore, I will respond to Lost Clown as I have with past rounds of the game, by posting my random list without tagging anyone else. The buck shall stop with me.

Four Jobs I Have Had
US Census Taker
Set Designer/Technical Director for Community Theatre
Tenor in a church choir (funny since I'm an atheist)
Salesman at baby/pregnancy supply store

Four Places I Have Lived
Fallston, MD
Knoxville, TN
the Appalachian Trail
Portland, ME

Four of My Favorite Foods
cheese ravioli
beef with broccoli stir fry
chicken Parmesan
Hot and Spicy Chex Mix (I make it a meal, believe me.)

Four Places I'd Rather Be Right Now
Grand Teton National Park
Times Square
the Outback
Anywhere in New Zealand

Four Movies I Can Watch Over and Over Again
Dead Poets' Society
Die Hard
Napoleon Dynamite

Four TV Shows I Like to Watch
Prison Break
So You Think You Can Dance

Four Websites I Visit Daily
Angry for a Reason

Four Early Musical Influences
the Beatles
Billy Joel

Four Computers I have Had
Commodore Vic 20
Tandy PC
Dell PC
iMac (soon)

And there you have it, folks. Yet another glimpse into the intricate and disturbing psyche of me.

Saturday, August 18, 2007

I Love Cats

This post in an experiment, designed to assess how friendly felines affect the popularity of blog posts. You see, I have grown somewhat obsessed with my Google Analytics profile of late. I visit it religiously each morning to determine how successful my blog is becoming. As the page loads each day, I am like an investor checking the financial ticker, praying for my stock to go up, preparing for it to go down. I celebrate every spike in ratings as though it were the ultimate tipping point, the threshold that stands between me and total internet domination.

The largest spike I have received to date was my comical take on Oscar, the death-sniffing cat. Though I was elated to have the readership, I was a bit saddened that it was thanks to a pet. Even my deliberately controversial post submitted to the Carnival of Education did not warrant as much attention. My girlfriend overheard me grumbling about what I felt was a peculiar disparity, and she matter-of-factly provided me with the hypothesis of this little experiment. "People love cats," she said.

So in an effort to test just how much the web-surfing community prefers felines to mathematics, I am writing this warm, fuzzy post about my own pets.

Zoe is an exceptionally small gray female with little white socks. She wandered into Sarah's house and despite the best efforts, could not be persuaded to leave. Sarah had recently lost two cats that had been with her for nearly 15 years, and little Zoe helped to fill the void in her heart. She and I became fast friends, but there has been a bump in our relationship which she has not as of yet gotten past. I left her to hike the Appalachian Trail for five months and she has never forgiven me for it. She will tolerate my attention now, but it isn't like it once was.

Oz is our gentle giant. Sarah added him to the mix while I was hiking so that Zoe might have a friend. It was a risky venture, since we had already attempted to add a second cat the year before with disastrous results. Zoe had attempted to kill that cat. I'm not talking about your standard hissing, swatting, cat-fight. Usually they pin back their ears and box faces for a few seconds until the loser runs away. This was something else entirely, something I had not seen before nor hope to see again. This was a no holds barred cage match with blood and fur flying. Naturally, we expected a similar ordeal with Oz. Instead, she spit out only the slightest little hiss, and they quickly became pals.

Oz is an exercise in counter-intuitive psychology. When Sarah rescued him from the adoption agency, she told me he was the ugliest cat she had ever seen. He had been abused, and I mean seriously abused. He had been set on fire and still has a BB embedded in his left side. To add insult to injury, his size made him an ideal candidate for blood donor, so he was shaved in patches all over. Despite all this, he is the absolute sweetest cat I have ever known. I regularly wake in the night to find him bathing my head and he meets you at the door like a dog.

Now that I have completed the experiment, I thought I would toss in one more bio. This is Freckles. Freckles was meant to be a gift for Sarah, to keep her company while I was hiking. She is a Disney fanatic and had wanted a dalmatian since she was a girl. I had been apprehensive about owning a dog while living in an apartment, but one day while perusing the online edition of my local paper, a pop-up appeared for the county shelter. It was his face. I placed a call to the shelter and was told he was still available. I wanted to meet him first, in case it wasn't going to work out. He peed no fewer than twenty times between his run and the visiting area. I walked him through the cat room to test his demeanor and he seemed unusually calm for a dal. Finally, I gave Sarah the call. She hurried down to meet him, but after less than a minute with him, she declared, "I don't like him." I assured her it was just her nerves, that the idea of her not liking any dalmatian was ludicrous. We filled out the paperwork and took him home, and you know what, Sarah was right. Freckles immediately bonded to me and has not left my side since. He hiked the entire Appalachian Trail with me and is unquestionably the best friend I have ever had. Sarah has grown to love him, but he is definitely my dog.

So there you have it, the menagerie de Tony. I don't know what I would do without them.

Friday, August 17, 2007

The Jungles of the

The first of my textbooks arrived today: Physics for Engineers and Scientists and Building Java Programs. I've already started reading the first one, having decided that I would like to be one of those nerds that stays a chapter or two ahead of the syllabus. It is ridiculous how giddy I feel about all this. I know I ought to be embarrassed, but I am just too excited about school to care.

Nerds of the world unite!

Thursday, August 16, 2007

They Don't Know Numbers

"He doesn't know numbers; he only knows beads." -Matthew Broderick as Richard Feynman, Infinity

It seems the Arab world is attempting to take a step backward in order to go forward. They are adopting an adding technology that goes back millenia.

Mental arithmetic is a form of calculation that does not involve the use of any physical or external gadgets, such as calculators or computers. The skill is developed early through the use of the abacus. Eventually children are trained to calculate large numbers in seconds with accuracy and speed.

Though the popular image of an abacus is correctly attributed to the Chinese, the actual pebble replacement system was used first in Babylon. It still is the fastest pre-electronic way to do simple arithmetic, beating even today's standard algorithm. Though, the last time I checked, it is both a physical and external gadget. The above program, known as Universal Concept of Mental Arithmetic System (UCMAS,) allegedly uses the abacus to improve mental arithmetic. For the life of me, I don't see how the abacus could possible be useful for this. It only requires students to be able to count as high as the base of their numbering system (i.e. ten) and then to understand the concept of place value. Sure it's really fast by analog standards, but if it's speed you're looking for, go digital and electronic. If you want to teach mental math, teach some form of pencil/paper algorithm and provide memory drills.

Wednesday, August 15, 2007

Strong Men and Bearded Ladies

The 132nd Carnival of Education is up over at Education Matters, and it includes my recent post "M" for Mature. I am excited to be included, even though I know no submissions were rejected this month.

With regards to that post, I have this to say. There is absolutely no evidence for or against the efficacy of my theory. I'm certainly not married to it, and no full well that it will probably never happen. It stemmed from my willingness to do whatever is necessary to eradicate math phobia, no matter how outlandish the idea. Take it with a grain of salt, as they say.

Monkey Math

Elsa Addessi, a researcher at the Institute of Cognitive Sciences and Technologies in Rome, Italy, has demonstrated a surprisingly advanced concept of numerosity in capuchin monkeys. The test subjects were given coins of differing "value," which could be traded for quantities of peanuts. While some of the animals showed preference for color or quantity, several were able to consistently maximize their payoff. While the monkeys are not technically adding, they are showing both the ability to understand the use of symbols and concepts like "more than" and "less than." This is still pretty impressive considering they diverged from us 65 million years ago.


This was staring me in the face at the grocery store check-out yesterday.

The jokes just write themselves.

Tuesday, August 14, 2007

Sleeping Around

There was an article in the Chicago Tribune the other day that called into question the results of nearly every sex survey from Kinsey to the present. It is popularly believed, and supported by surveys, that men have more sexual partners on average than women. Mathematicians know this is a logical fallacy.
He provided a proof, called the High School Prom Theorem:

"Each girl is asked to give the number of boys she danced with. These numbers are then added up, giving a number G. The same information is then obtained from the boys, giving a number B. Theorem: G=B. Proof: Both G and B are equal to C, the number of couples who danced together at the prom."

Researchers speculate that one of two things is happening, either the men are going outside of the interviewed population for their sexual escapades or both genders are simply lying. I'm sure it's a bit of both.

A couple of things bothered me about the article. First, the article interchanged the words median, mean, and average continuously, which I find quite irritating, since they are not necessarily the same. Second, although I can clearly see why the arithmetic mean number of copulations must be the same for both genders of a heterosexual population, I'm not sure that really tells us anything about relative promiscuity.

Let's say we define promiscuity as having more than one sexual partners in relative simultaneity. Then we imagine a graph of heterosexual pairings that reflects the common alpha-male situation. There are 10 men and 10 women. One studly/slutty dude hooks up with 4 of the women. Two other men and women engage in monogamous coitus. The rest go home alone. If we use a simple arithmetic mean, then both genders engage in 0.6 sexual encounters. However, that one guy accounts for 2/3 of the men's numbers. If you randomly selected a guy from the room, there is a 1/10 chance of him being promiscuous, by my definition. The same cannot be said for the girls, as none of them had more than one partner. So in this sense, men can be said to sleep around more.

You could translate this into a weighted average as well. If you assign weights of 0 to not promiscuous and 1 to promiscuous, then the men achieve a weighted average of 0.4 and the women of 0.0. This ranking puts the men ahead in the slut department.

It's important when applying mathematical computations to real life situations that the math is not only correct, but non-trivial. And the method you use can significantly affect the outcome, as is evidenced here.

Sunday, August 12, 2007

Look Before You Leak

Proving once again that mathematical thinking may be employed in the most surprising of places, a statistically minded blogger at Guns Guns Guns Bikes Bikes (seriously, that's the name,) has worked out exactly where each gender stands (or sits) in the age-old toilet seat debate. I wouldn't recommend citing this source during an argument with your significant other, but it sure is good for a laugh.

Saturday, August 11, 2007

Mathematics is Rated M for Mature

Pay close attention. I am about to suggest something that will make many of my math contemporaries put down their protractors and take up pitchforks against me. Before I get to the point, I'm going to attempt to outline the path that has led me to this heresy.

The following is a collection of ideas that have been espoused at one time or another on this blog.

  1. The math education requirements set for aspiring and current elementary school teachers are far too relaxed.
  2. Calculators are the latest in a series of tools, each adopted in turn for there superiority.
  3. "It takes a certain maturity level to comprehend certain types of math." (Comment from Andy)
  4. When engaged in the design process, sometimes weak links can simply be removed.
Now for some elaboration.

1. The Chinese say that in order to give each of your students a cupful of knowledge, you must have a pitcherful. Clearly, the people who determine the educational standards for elementary school teachers disagree. I recently had an opportunity to peruse a Praxis II practice test for elementary ed, and I was astounded at the difficulty level. The hardest question on the test involved little more than correct application, not derivation, but application of the Pythagorean Theorem. I think it is important to point out that the ancient Babylonians had already mastered this much. I appreciate the fact that there is much more that goes into teaching this just content knowledge. There is all the pedagogy and psychology, especially with the little ones. But the knowledge of how to teach becomes useless without mastery of what you are teaching, and in many cases, what they are teaching is how to hate math. And their students are learning it well.

2. Every time a new technology edges out an old one, traditionalists cry foul. What of the information that will be lost? What if this new technology is suddenly unavailable? This is the argument that naysayers employ against the use of digital calculators today. It is a valid argument, which is merely to say that it is not an outright lie. If a student is taught to perform arithmetic primarily by calculator, than that student forfeits the ability to use the "standard" pencil and paper algorithm, should the need arise.

As I type this, I am within sight of three calculators. The first is the built in application on the computer itself, the second is on my cellular phone, and the third is an actual hand-held with a total of 24 buttons. This machine, which probably retails for two dollars, has the ability to perform 5 arithmetic operations, can store values between steps, and can perform any calculation that would be required of the average person. Calculators are so ubiquitous that to suddenly be without them would mean one of two things, either society has collapsed or you are stranded on a desert island. In the first situation, I suspect there would be more to worry about than the ability to do long division, and in the second, simple finger calculation should suffice for survival.

Progress requires that we give up knowledge that our parents and grandparents depended upon. For example, can you start a fire without a match, can you even start a fire with a match, can you identify edible or poisonous plants, can you drive a stick shift, and the list goes on. When we trade that knowledge, it is with the understanding that we get something more from the deal. Maybe that is a dangerous assumption, but it has brought us safely down from the trees and into the modern era.

3. Before Andy made this comment, it had never occurred to me that the ability to understand math might depend on the maturity of the student. I have read so many stories on prodigies like Gauss, that I had assumed even the most advanced math could be grasped by a child, would that they had the right teacher. Now I am starting to see this may not be true. I have said many times that mathematics is the science of patterns. In order to see pattern, you have to be able to make connections between often disparate things, and that requires a healthy base of facts and experience from which to draw. Maybe children struggle with math simply because they do not have the mental and emotional background necessary to bring meaning to the algorithms.

4. The design process is just that, a process. Ideas seldom spring fully formed from the minds of their creators. Instead, there is a tedious and painstaking struggle to turn the initial concept into the finished product, and there are often heart-wrenching decisions to make along the way. As you watch the deleted scenes on any DVD, imagine how the director felt as the cut was made. You will notice that sometimes a different variation of the scene appears in the final cut, but often times it has simply been deemed unworthy and removed in it's entirety. It just wasn't working, and the faulty part had to be removed for the good of the whole.

Now for the synthesis.

Brace yourself. I propose that math education be delayed until the secondary level. I know that sounds crazy, but the more I think about it the more I love the idea. The two reasons we teach arithmetic are practical application for its own sake and as a precursor to later concepts. As I mentioned earlier, the practicality issue can be solved with a rudimentary explanation of the various operations followed by a brief tutorial on the use of a calculator. The issue of laying a foundation is much trickier. I can't even begin to argue that concepts touched on in arithmetic will not carry over to algebra and beyond. The latter is just a generalized version of the former. What I am suggesting is that school children lack the emotional maturity that makes that transition work. They have no concept of delayed gratification. They do not see that they are working toward something which may not become clear for several years. All they understand is that they are being forced to agonize over multiplication tables and long division and fractions, when they could just punch in the numbers on a calculator and be done with it. To them, it must seem like torture, and who's to say it isn't.

The other factor that conspires to defeat students from enjoying math is the poorly prepared elementary teachers. They often times don't understand themselves exactly how what they are teaching is laying the framework for what is to come, so all they can do is drill the lesson as it appears in their workbook. Reform math programs, which are well intentioned, often make the problem worse, because they require a greater mastery of subject matter from the instructor, not less.

Rather than spending those elementary years teaching students to despise math, we could devote that time to other ventures. Whether the extra space is filled up by music or reading or recess is a question for another day and another blogger. When the students reach the secondary level, then we can begin teaching real mathematics. It's true that they will lack the aforementioned foundation, but they will also lack the ingrained aversion to math. It should be a simple matter to teach long division algorithms along side polynomials or multiplying fractions with rational functions. Students will then be in a position to appreciate what the are learning and why they are learning it.

The math education system is broken. Certain links in the chain have rusted with time. Opinionated cognoscenti from all sides are locked in heated debate over how to repair it, but I think perhaps the solution may instead require total removal of faulty parts.

Or have I gone crazy?

Thursday, August 9, 2007

A Brief History of Luddism

Pre-History: "Boy lazy. He no wait sky fire. He rub sticks. Make flame."

Stone Age: "Man-child is lazy. He uses bow-drill to start fire. He is losing the old ways."

Medieval Era: "Methinks yonder child dost laze about. He summons flame with flint and steel."

Age of Enlightenment: "My son has grown complacent from the new technology of sulfur matches."

Victorian Era: "The kid can't even be bothered to strike a match. He just flicks his lighter."

Modern Era: "Kids today are dependent on calculators. The can't do math the old-fashioned way."

Wednesday, August 8, 2007

Simple Arithmetic

Most of you probably think arithmetic is easy. All you computer programmers and tenured professors out there can add,subtract, multiply, and divide in your sleep, and you scoff at people who can't make change quickly in their head. Certainly, our nation's public schools think arithmetic is easy. They require elementary school teachers to have hardly any math background at all. And yet our elementary school students are using math technology 50,000 years in the making.

How Mathematics Happened by Peter Rudman explores in depth why and how civilized man came to depend on math. He focuses on the early years, beginning well before writing was invented, and giving an intriguing account of the birth of the science. I am in the process of reading it and it is truly fascinating. This isn't the first early math history I have read, but it is by far the most involved. Check it out. You'll never underestimate the power of the third r again.

Tuesday, August 7, 2007

A Blessing and a Curse

Have you ever worked with someone that had been promoted to his/her level of incompetency? This idiot that you're thinking of was once a model employee. It was proven excellence at a lower level that got this person promoted in the first place. Unfortunately, for both of you, this person just tried to meet one too many challenges. When I took Calculus III in college, that incompetent was me.

I have always liked math, and I fancied myself pretty good at it once. I sailed through Algebra, Geometry, and Pre-Calc without having to do much homework. As I moved into Calculus, things started to break down. My intuitive understanding was slowly eroding and I found myself struggling to grasp what looked like simple concepts. I kept getting A's, but I was working harder for them than ever before. I remember the first week of calculus in infinite dimensions. It was the hardest I have ever slammed into an intellectual wall.

Since I am going back to school in less than a month, I have started thinking about what may have caused this roadblock for me, and I have come to two conclusions. First, students determine their feelings toward and perceptions of a subject very early on. I felt math was easy and should require next to no effort on my part, because that's the way it had always been. Once it started to get tough, I began to have low-grade self-esteem issues that affected my work. Perhaps I should have been challenged more in earlier grades, just enough so I knew that some work was necessary.

The second revelation is more intriguing to me, and it has come through several years of reading popular math books, the ones with no formulas or equations, just a lot of metaphors and lay-person explanations. They have helped me learn some things that I didn't know in school. Now I understand the difference between applied and pure mathematics, terms I didn't hear of in school. Pure mathematics doesn't have to have any practical applications, as Hardy was fond of pointing out. It is very common for tools of pure math to sit on a shelf for decades or centuries before someone finds a good use for them, and sometimes one is never found. In many ways, the correspondence between chalkboard and reality is accidental. For example, it is a fortunate coincidence that Euclid's geometry so strongly correlates to life in flat space. At the scale the Greeks were used to working, it was flawlessly accurate. For space-traveling moderns like us, Euclid will not suffice. We live in a world where space and time curve, and we have had to use other non-Euclidean geometries, geometries fortunately constructed long before Einstein took his mind-trip on a beam of light.

If you've ever tried to explain non-Euclidean geometry to someone, you may have experienced how much a metaphor can help and hinder understanding. That's exactly what mathematical constructs are; they are metaphors or models for reality. If you are too wrapped up in the similarities between metaphor and reality, you may be blind to differences. That's what happened to me back in college. I had grown accustomed to thinking of all math spatially. Whenever I heard the word dimension, I was thinking of height, length, width, etc. Most of the examples in my textbooks applied the lesson to measurement of space, so when I got to Calc III, and the dimensions grew beyond the familiar three, I was lost.

I am not suggesting that we abandon spatial examples and metaphors. Student's inherent understanding of space is strong and math education is wise to piggy-back of of it. But maybe there ought to be more examples in the texts that have nothing to do with space. Comparisons of color to light/heat absorption, or age to bone density, or whatever. This way students will begin to understand that dimension can refer to any variable characteristic, not just space. This realization has certainly helped me, and I can't wait to get back into class and prove my competence.

Friday, August 3, 2007

Scene of the Climb

Tomorrow will mark the one year anniversary of the completion of my AT thru-hike. I've decided to celebrate by summitting Katahdin again. I'm going to head up tonight and do some trail magic for any of this year's hikers, although it's still pretty early in the season, so there might not be any. This time, I will have my girlfriend, Sarah, at the top with me. She had to sit at home for five months while I pursued my dream and she isn't going to sit this one out.

Wednesday, August 1, 2007

Tilting at Windmills

I wanted to respond to the following comment that appeared on a recent post.

The following authors have been banned in some public school classrooms:

1. Alvin Schwartz
2. Judy Blume
3. Robert Cormier
4. J.K. Rowling
5. Michael Willhoite
6. Katherine Paterson
7. Stephen King8. Maya Angelou
9. R.L. Stine
10. John Steinbeck

If you stick to private schools you will not only be able to teach the classics but be called by your first name!

The list of books is the only part of this comment that I do not take issue with. It comes from the American Library Association, and it is actually the list of challenged authors. According to the ALA, a challenge occurs when a book is recommended for removal, whereas a ban is a successful censoring. I searched their website for a breakdown of public vs. private schools, but they don't separate the data that way.

The distinction seems trivial to me, especially considering the fact that it in no way records books that were never made available in the first place. I suspect that if fewer books are banned in private schools, it is only because private schools give parents much more control in curriculum selection. I found many stories of parents switching to private schools because they had lost a challenge to a book in a public school, and especially considering that most private schools are still religiously affiliated, I am willing to bet that as a group, they offer far less intellectual freedom.

I will assume that the schools referenced in this comment are purely secular. Even so, I have no desire to teach at those institutions. Since I decided to devote myself to education, many people have recommended that I go the private school route. I don't see the point. The types of students that attend private schools are as varied as their reasons why, but they all have one thing in common- parents that care about their education. This is a better predictor of success than any IQ test. Private school students are probably going to succeed in life regardless of how competent the particular school is. To me, it would feel like a fireman rushing to save a baby from a building that isn't burning.

As far as the allegedly superior success rates of private schools, I think the evidence is dubious at best. These schools have the luxury of hand-picking students. Whether specializing in troubled students, gifted students, or religiously conservative ones, the schools get to choose the niche in which they will be most effective. Public schools teach everyone. They turn no one away, and their finished product suffers accordingly.

I am getting into this to make a difference, and the place where I can affect the most change is in the trenches of public education. Sure I could find a school that perfectly compliments my convictions, where my career would be trouble-free, but again, I really don't see the point.