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Saturday, December 22, 2007

Let the Chips Fall Where They May

Well I've finished up all final exams and turned in all projects. Now I just have to sit back and wait for the grades to start rolling in. I'm not really worried about any of them. I worked really hard throughout the semester and I had enough of a cushion going into finals, that I would REALLY have had to bomb them to lose my A average. At any rate, I feel like I've learned a lot and that it has been a good start to my second collegiate endeavor.

My biggest regret is that I haven't been very good about keeping up this blog. I had made a lot of lofty goals and promises that seem to have fallen through. I think my schedule for next semester will allow me more time for personal reflection here. If I use the holiday break to get back into the habit of daily posting, maybe that will help.

Tuesday, November 27, 2007

Numbering from Zero

I've been noticing a trend lately, and I hope it's a fad. Several popular math books that I have read have begun with a Chapter Zero. I can rationalize several excuses for this, but I still think it boils down to nerds being nerdy.

The first excuse is that in modular arithmetic, a set of N numbers begins 0,1,2,3...N-1 so that it can include the identity. But a book isn't a clock. The chapters aren't going to start over at any point.

The second excuse is that the authors are also modern computer scientists. (Both of my computer science classes began with an Assignment Zero.) After the popular language C, all languages have begun their indexing sequences from zero rather than one. This is a consequence of the decision to have the memory address of an array element be the same as its index. But there's no reason why they couldn't have started indexing at 1 and defined the memory address using an imaginary number equal to one less than the index. So this isn't really any more rational.

The last excuse is that the numbers are meant to serve as markers instead of names, counting forward in a sort of greatest integer function scenario. Since the markers come at the beginning of the chapter, and the first page of the first chapter can be viewed as the origin of the book, it might make sense to label it Zero. However, if books were designed to be read from right to left (I know there are many languages which read right to left across the page, but I don't know if the books themselves read from right to left as well,) we wouldn't expect to start with Chapter Zero followed by negative chapters.

As far as I can tell, there is no good reason other than fashion why a math book should break the convention of using counting numbers to label chapters. I find it irritating and i wish it would stop.

Monday, November 26, 2007

A Letter to a Young Mathematician

Dear Gina,

I've been thinking a lot about our conversation the other day, particularly your question about proofs. Even though your tone suggested that you had already made up your mind that the entire process was useless to you, I thought I would take a minute to defend the mathematical community.

First, I understand how you could feel blindsided by proofs. After all, you've been getting along quite well in math for years without them, so why start now, right? You might even feel a bit betrayed. Up until recently, school was about getting the right answers, and no subject exemplified that more than math class. Even as English diverges from grammar, and into the realm of essays and theme papers, mathematics remains firmly shrouded in its safe cocoon of black and white, right and wrong. Unfortunately, that security blanket is in large part a lie, and you might as well learn that now. It isn't even your fault that you've gotten the wrong idea. Many of the teachers you've had so far, especially those in elementary school have the wrong idea as well. There are many definitions for the science of mathematics, but however you look at it, it is about a specific way of thinking. It is more about asking interesting questions than it is about finding the correct answers. It is as much a journey as it is a destination, and the concept of proof lies at the heart of it.

Proofs are curious things. They are perfectly ordered step by step accounts, a yet they often hinge on fairly large assumptions. There are proofs which show an answer exists, but give no clues how to find it. There are even proofs which show that there are some mathematical facts, which while true, can never be proven. And to top it all off, there's a proof to show that we have no way of knowing what those proof-less facts are. So it's entirely understandable why they may confuse you.

There are several reasons why your teachers feel it is necessary to torture you with proofs. The most obvious is that they are trying to prepare you and your peers for every possible future. Should you decide to go into on of the STEM fields (science, technology, engineering, or mathematics) you will be required to take upper division math classes and the ability to do proofs with be a prerequisite. By your comments, I think it safe to assume that your path lies along a different fork of the road, so I won't belabor this point.

My second and third reasons fall into what I'm going to call the Karate Kid category. Now because I acknowledge that our difference in age means that this brilliant film reference may be lost on you, I will now provide the key plot elements needed for understanding. In the movie, the new kid in town, Daniel, is being picked on by a band of bullies, who in addition to their snobby upbringing have been trained in karate at a local dojo. The scrawny hero befriends the lovably wise Okinawan janitor, and convinces him to pass on his family karate secrets. The following scenes show Daniel performing a series of menial tasks for Mr. Miyagi, including painting fences, sanding floors, and waxing cars. Daniel grows increasingly angry over his friend's abuse of their agreement, and finally confronts him. At that point, we learn that the repetitive motions of those chores mimic exactly the movements required to defend oneself against an attacker.

In retrospect, this beloved film from by youth is cheesy and not a little bit far-fetched. But it has always seemed like the perfect way to make the following point. In school, as in life, what you are really learning isn't always what you think you are learning. If you can think of your generation's equivalent of the Karate Kid, please let me know, and save my future students the agony of this comparison. Still, the fact of the matter remains that the skills you are learning through doing proofs are useful in more circumstances than you can possibly imagine. When a doctor makes a diagnosis, or a lawyer builds a case, or a football coach draws a play, they are using the kind of analytical thinking that you are practicing through proofs. You first begin with a small pool of facts or postulates, and then you use carefully constructed reasoning to arrive at a sound conclusion. As you continue through school, your skills in other subjects will be improved by your ability to do proofs. Your English papers will be clearer and better supported. Your debating skills will improve. Really, there's no telling how far proofs will reach into your life.

All of this talk of arguments and debating brings me to my last point about proofs. I've already touched on how proofs are built up from first principles with each layer relying on the strength of the one below. Because of this, it is important to be aware of keystone elements of each proof. For example, you have probably proved that every triangle has angles which sum to 180 degrees. This proof follows directly from what Euclid called the Parallel Postulate. Roughly, it states that given a line and a point not on that line, there is only one way to draw a second line through that point so that it is parallel to the first line. Without going into too many details, I want to make an example of this postulate. Like any postulate, it can not be proven. It is an assumption considered so obvious that it can stand alone without proof. The problem with these kind of assumptions, whether they are in math or English or History, is that if they turn out to be wrong, then any argument or proof based on them crumbles as well. In this particular case, there are several systems of geometry that have been shown to both exist and to be invaluable, in which the Parallel Postulate does not hold true. There are spaces and surfaces where it is impossible to draw lines which do not intersect. In these spaces, a triangle may have more than 180 degrees. There are other spaces where they can have less.

After all this, I doubt I have changed your mind much. You probably still hate doing proofs and even after my best effort, you still don't see the point of it all. One of the things you mentioned troubled me more than any other. You told me that your teacher required you to do proofs from memory, naming each Theorem and Corollary as you go. In this one regard, we are on the same page. To many teachers confuse memorization with learning. They think that as long as you have a head full of facts you are better off for it. I disagree. Wrote knowledge without the ability to synthesize and improvise does not in any real way demonstrate learning. To some extent, your teachers can be forgiven their slowness to realize this. You do not remember a time before internet search engines, but I do. Not so long ago, information was hard to find. It could take hours to find the specific piece of data you were looking for, so it was often easier to memorize it once and carry it around with you forever. It was a kind of "be prepared" attitude toward education. Those days are over. There is no longer a need to fill your head with facts on the off chance that they may one day be useful. You can sift out the necessary info in a Google search that takes a blink of the eye. You shouldn't have to remember the names of each Theorem. There are after all quite a few. As long as you can understand them and put them in the right order to build your argument, that is what is important.

I hope some of this has gotten through to you. I know you aren't going to share my love for math, but rest assured, you will never be beyond its sway, so gaining at least a passing familiarity with its methods will prove useful to you.

Your big brother

Saturday, November 24, 2007

Je ne parle pas francais.

I just got home from a Thanksgiving trip to Quebec City. It was my first time visiting our neighbors to the north, and their hospitality was warm and welcoming. Neither Sarah nor I speak a word of French, at least not beyond the few phrases I printed out before we left. I had made up my mind to make an effort, but I rarely got out more than two words before they broke seamlessly into fluent and flawless English. It's funny that even in the province of Quebec, where French language and heritage is celebrated to the point of revolution, every school child is taught to speak English. Yet I live in a country which is nearly fifty percent Spanish speaking and xenophobic loonies treat Dora the Explorer like she's going to break up the Union.

After overhearing one restaurant owner speaking five different languages. I felt amazed and ashamed all at once. Something has to be done about this.

Friday, November 16, 2007

Lead, Follow,or Get Out of the Classroom

I don't have much to say lately. Even when I have the time to blog, I don't have anything interesting to report. Today happens to be an exception. First, let me quickly get the auto-biographical particulars out of the way. I am nearing the end of my first semester, still maintaining a 4.0 GPA, and registering for next Spring. There, now we can get on with the show.

Last Saturday I was invited to attend a conference on Teacher-Leadership held at Colby College in Waterville, ME. When I say that I was invited, what I mean is that the opportunity presented itself and I volunteered. (As with the last conference I attended, I was disappointed, but more on that later.) In addition to my desire to master the content I intend to teach, I am very much curious about systemic design of the current education system. From my lay perspective, it looks to me like many schools are still stuck in what is essentially a one-room schoolhouse paradigm. Although, there are more of those rooms under a single roof, the individual classrooms are run autonomously from one another, with little to no coordination between teachers. This self-isolation makes it very difficult for teachers to learn from one another.

Some schools are seeking to change that. They are transitioning into a new kind of structure, where teachers are encouraged to cooperate and "compete" in a less-superficial way. By "compete," I don't mean literally fighting for their very job, as the high-stakes testing/accountability advocates intend. I mean the kind of friendly challenge that helps both parties reach new and unexpected heights together.

That is what the conference was supposed to be about, and it started out well. The opening address was passionate and to the point. The speaker outlined what constitutes teacher-leadership. He pointed out the self-similarity between layers of a school. Just as students learn from one another and drive each other to greater standards of success, so should teachers. Currently, unlike students who are keenly aware of each others methods and their relative successes, teachers are in the dark when it comes to what is happening next door. Instead of the once a year visit from the administration spies, we ought to be volunteering to sit in on each other's classes and welcome peers into our own. Only in this way can we strengthen our own abilities and the cohesion of our departments and schools.

After that, it all went downhill fast. Immediately following the opening address was a panel discussion, only the panel was made up mostly of students with no real understanding of teacher leadership, so the audience questions went largely unanswered. Then we divided up into several break-out sessions which had been decided upon before hand. I had selected one entitled "What a New Teacher Needs," reasoning that I would soon be a new teacher and would like to know what I will need. Unfortunately, that question was never answered. The presenter based the session on her ongoing doctoral thesis, in which she interviewed first year teachers at several points during the year, and made note of their personal success and intent to continue. Now i know that sounds like a great opportunity to learn something, and I would agree, if the sample size had been greater than four.

That's right, four. Out of all the first-year teachers in the state of Maine, she interviewed exactly four. Two of them had never really intended to go into education in the first place and one of those experienced an unfortunate lawsuit during the first month of school. All told, only one of them continued into a second year. Now what can we possibly glean from all this? I was hoping to start with a large sample of eager, well-educated young teachers, find out what obstacles they might face, and some possible ways in which teacher-mentors can recognize and solve those issues. Instead, I got a lot of poetry and rhetorical questions. It was an utter waste of time, and I seriously doubt that her thesis will pass review.

After the session, and on the way to lunch, I fell in with a group of students and their teacher who were likewise disgruntled. It turned out that they were from the ETEP graduate program at my very own USM. I introduced myself and complained along with them. It was the highlight of my day.

Lunch was adequate and featured round table discussions of teacher-leadership from those in the field. It turns out that many teacher-leaders are unsure as to exactly what their job entails. They are also reluctant to take the responsibility, fearing that the nail sticking out is the one to get hammered down.

To summarize, it was a well-intended conference which from my perspective fell short. I am still very much in support of the concept of teacher-leadership. At some point, an organization grows too large and bulky for top-down leadership to work. There must be a combination of bottom-up and top-down for the system to reach optimization.

Wednesday, October 31, 2007

Gravity Suspended

Physicists the world over have been scratching their heads this Fall, as our current understanding of gravity has been shattered. The most famous of the fundamental forces, gravity has long been a source of curiosity for the eager young scientist. Newton took the first successful crack at it with his infamous apple and Einstein came along to sure up the theory for later generations. All was going well until recently.

A month or so ago, an extremely focused disruption in the gravity field allowed blogger and student, Tony Lucchese, to literally fall off of the face of the Earth. It is unclear as to what caused the disturbance, but what is clear is that Lucchese has been MIA for several weeks. Scientists from CalTech to CERN are working around the clock to rectify the problem. Until then, we can only wait.

Wednesday, October 3, 2007

Random Samples: Making an Ass of You and Me

I first heard the colloquial definition of assume from my middle school band teacher. He relayed it during one of his notoriously crimson-faced tirades as a part of his tireless crusade to get us to practice our instruments. The mere mention of the word ass brought a cacophony of pubescent snickers, as you might well imagine.

Twenty years later, I find myself enrolled in school once again. This time around, I'm not taking any fine arts, and my musical stylings are reserved for my shower head and the lonely walls of the Pottery Barn stockroom. Still, the trite but true words of Mr. Danner stick with me. Daily I am confronted with a seemingly innocuous request to assume. Assume a friction-less surface. Assume no air resistance. Assume a random sample.

Of course, for the purposes of class, I play along. I understand that in order to proceed, a beginner like myself, must temporarily set aside the more confounding components of mathematical problems. Yet I do not take these assumptions lightly, and they hover somewhere in my psyche underlined, in both italics and bold-face. I fear that many people become far too accustomed to these kinds of assumptions, never truly revisiting their ramifications at that later date. Even seasoned professionals fall victim to this permanent credulousness.

My physics teacher relayed the results of a now famous statistical experiment whose findings declared that cats stand a greater chance of survival falling from higher windows than from lower ones. My hand shot up with such velocity that it nearly dislocated my shoulder. You see, that study has a huge hole in it. The sample space isn't really random at all. It originally appeared in a 1987 issue of the Journal of the American Veterinary Medical Association and was authored by two doctors at the Animal Medical Center of Manhattan. They published based on a small sample of cases they had seen at their practice, all feline victims of high-rise free-fall. On the surface, it all seems on the up and up, until you consider the one gaping hole in their methodology. In order to be a random sample, the study must include all instances of the event in question, which is a cat falling out of a window and hitting the ground. Ask yourself this, if your precious Fluffy were to fall victim to accidental defenestration, and upon reaching the street level, you found myriad biological parts where your furry friend was hoped to be, would you scoop up the pieces and take them to the vet? No, and neither do most of the pet-owners in Manhattan. The 1987 study is based on the assumption of random sampling that did not actually occur.

As I have said, these school room assumptions are necessary steps to greater understanding in any and all STEM related fields. But we must never forget that they exist. Our future failures hinge upon them.

Saturday, September 29, 2007

Choking Down Technology

I'm going to skip past the part where I apologize for my long absence. I've been busy, let's not over-analyze it.

The most blog-worthy event in recent happenings was my attendance at a T3 conference for pre-service math teachers. I'm not sure "conference" is the right word; in truth, it was a two day sales pitch. I was invited to participate by the faculty adviser for the math ed program. Mostly I was doing it for the free TI-84+ graphing calculator I would allegedly receive at the end of it. ( I still haven't received it, by the way.)

There was only one other pre-service teacher there. Everyone else came from local middle and high schools. The presenter/salesperson is a full-time professor at Drake University in Ohio, but I'd wager that she gets a healthy stipend from Texas Instruments, as well. She spent two days demonstrating all of the fancy things you can do with their various products. The central focus was on the Navigator hardware, which allow all calculators to be networked together. The teacher can view all activity on each student's machine at the press of a button. That image can be projected onto a screen or better yet, onto a Smartboard. It was all glitz and glamor, and the
audience ate it up. I was less than impressed.

The stuff she was showing us was neat, I'll admit it. But it wasn't as revolutionary as everyone seemed to think it was. There are any number of softwares available that could do the same thing with desktop computers. And those machines would still be able to perform a multitude of other tasks, rather than merely being graphing calculators. I asked if TI produced similar software, and was told that they did. Guess what. It costs exactly the same as a calculator.

The slogan of the event was "TI Cares." That's even their phone number: 800-TI-CARES. According to the presenter, Texas Instruments makes all of this technology available to schools because it wants to improve the world, and help train future employees. The fact that they are making money hand over fist doesn't factor into it at all.

If you know me, you know I love technology. I will use it in the classroom, and use it abundantly. I was just hoping to learn something new about how best to use it. Instead, what I got was a tutorial on several of the latest models. There was no discussion on where low-tech might still be superior. No statistics on the efficacy of technology heavy programs. The equipment she brought didn't even work consistently enough for us to do all of the activities that she had planned. All in all, I think I got out of it exactly what I had intended to- a new calculator. Except, I don't even have that yet, so I guess the joke's on me.

Thursday, September 20, 2007

Politics of the Professoriate

I have read a few articles from the right wing recently complaining about the politicking of liberal professors. Until yesterday, I didn't really understand what the were getting at.

It makes sense to me to ground your teaching in real-world examples. If you're teaching a statistics class, then analyzing welfare statistics seems reasonable to me. If you're teaching biology, then debating the ins and outs of stem cell research seems entirely appropriate. Though they are common battlegrounds of today's political realm, as long as they tie in with the lesson, I think any controversy can be excused.

In my physics class, we are covering motion in two dimensions. Basically, ballistics. In a simple example involving range-finding, my professor segued into a review of the documentary Why We Fight and the Iraq War. He repeated a quote from the film regarding the success rate of so-called Smart Bombs and without much transition at all, stated that "we shouldn't be killing people."

It was an awkward moment, I thought, even though I happen to completely agree with him. I didn't see how the side note appreciably increased our understanding in any way, especially since Smart Bombs have on-board guidance systems and are not simply launched projectiles.

I have emailed him about the incident, and have yet to receive a reply.

Tuesday, September 18, 2007

Choosing "Choice"

I know I've been promising to write some posts explaining some of the concepts I am covering in class. Unfortunately, all of the things I have thought about discussing involve me being much more familiar with the Latex formatting syntax than I currently am. For now, you'll just have to read another one of my extemporaneous ramblings.

My schedule has kept me off of the blogosphere lately. The only discussion I have really been able to participate in lately is one about school choice over at Mindless Math Mutterings. I thought I would take a few minutes to elaborate on that issue here.

There are a few misconceptions about school choice and what it can and can't do. The idea is that simply creating a competitive environment will force schools to fix all their problems and become successful. They will either sink or swim. Here is the problem with that reasoning. The choice model is patterned after good old-fashioned natural selection. Whether you prefer the metaphor of a biological system or the business world, the simple fact is this. The vast majority of species and businesses that have ever existed have failed. They are extinct. They caved under the competitive pressures.

Selection pressures do force the cream to rise, but it happens through a process that can best be titled "creative destruction." If you implement that kind of system, you have to accept the fact that most of the schools will fail and be forced to close. Most of them will not be able to magically adapt overnight. That will leave legions of children looking for a new educational institution.

That population overflow will cause the second major problem. Will the few remaining schools be able to scale up their success? Most of the schools that are doing really well are small schools. Whether they are private, public, or charter, they have small teacher to student ratios and they are able to cater to specific demographics. They do creative things like using students to clean up the physical plant instead of hiring full-time janitorial staff. But will they be able to do the same things with a student body that's been increased tenfold?

If you have any experience in the kitchen, you know that some recipes can be doubled or tripled, while others can't. What was exquisitely delicious when prepared for two is revolting when prepared for two hundred. Or a political example, if you prefer. Communism works extremely well in small groups. Contained familial styled villages can live quite happily using that political system. But I think we all remember what happened when that compound was scaled up to the size of a country.

Choosing "choice" would provide our nation with better schools over the long run, of that I have little doubt. But in the meantime, we will be closing a lot of schools and failing a lot of kids. I just don't think it's the answer.

Friday, September 14, 2007

I Pity the Fool

If you have a problem,

if no one else can help,

and if you can find them,

maybe you can hire....

A few weeks ago, my co-workers asked me if I had organized my Trapper Keeper for school, yet. I told them that I was all set except for my A-Team lunch box.

Today they gave me this belated birthday present.

Please Stand By

Sorry posts have been light this week, folks. I'm still trying to get used to the new schedule. I plan to post the first "lesson" tomorrow. I think it will be a short one on probability. It should be simple enough that a total beginner could understand it, so even if you don't "get" math, try to follow along. That means you, Mother.

Wednesday, September 12, 2007

Happy Birthday to Me

I received this lovely image from a friend today to commemorate the start of my fourth decade upon this Earth. Many people have asked me how I am handling this milestone birthday, and my typically nerdly response has been that it only appears significant due to our use of the base-10 counting system. Seriously, though, I see it as just another day in a life that is being lived well. I have no real regrets and I have been steadily crossing things off of life's "to do" list. My plan is to devote the next thirty years to helping kids understand how math can help them cross things of of theirs.

So here's to the big 3-0.

Monday, September 10, 2007

Politics on the Brain

Occasionally, Pencils Down strays away from its math education mission statement and ventures into the world of politics. I don't see any way to avoid this, since part of being sentient is having one's own opinions and part of being a blogger is voicing them. My own position on the political continuum is roughly at the midpoint. I tend to be fiscally conservative and socially liberal. I am a registered Independent and have happily voted for members of both major parties and several third-party candidates. I have friends who fall to the far right and far left of the field, and I argue with all of them.

Curiously, one of the most common arguments is of what exactly defines a Liberal and a Conservative. Many people choose to define it by what positions they take on certain issues, but that doesn't speak to controversies yet to be unveiled. Others use those political labels and the names of their party interchangeably, despite the fact that the party platforms have slid up and down the spectrum, even flip-flopping over the years. I read definitions in a political science textbook once that have stuck with me for both simplicity and accuracy. In short, a to be conservative means to look for the solution to a problem in the past. When faced with a new dilemma, they will attempt to apply the solution that has always worked with similar issues before. A liberal, when confronted with a new problem, will tend to dream up an entirely new solution. They prefer the untested to the status quo.

There is nothing to stop people for employing both methods in differing areas of life. I have already admitted to doing so. But what exactly causes a person to be one way or the other. It's the age-old nature vs. nurture argument, and nature has recently released some new results. Some psychologists at New York University have employed a simple test to investigate political persuasion. Test subjects were first asked to rate their political persuasion, 1 being conservative and 5 being liberal. Then the were each shown a different series of two letters, M and W. Regardless of the pattern, one letter was always more prevalent, showing up 80% of the time. The researchers found that when asked to match letters with the computer, subjects identifying with conservativism were more likely to "incorrectly" choose the dominant letter even when shown the other one. Liberals had a slightly "better" results, showing a greater ability to choose the "correct" letter even though it appeared far less frequently.

It is important to note that even if this experiment can be repeated, it doesn't imply that one position is evolutionarily more adaptive. There have been plenty of times in history when the combined selection pressures have favored either a liberal or a conservative approach, so one can not be said to be more adaptive than the other. I suspect that applying a healthy dose of both is the most pragmatic way to go about it.

Sunday, September 9, 2007

Great Expectations

If you recently felt an inexplicable tremor in the tectonic plates of the blogosphere, and are curious as to exactly what might have occurred, wonder no longer. The shifting was caused by the much anticipated arrival of newcomer and reformed lurker, Jackie. Her comments have brought insight to numerous blogs over the last year, and she's finally ready to unleash the full measure of her intellectual fury. Continuities is sure to make its mark and I recommend that edubloggers go ahead and get in on the ground floor. I'm putting her in my blogroll and RSS feed immediately.

Friday, September 7, 2007

Back to School Week: Day 4

This will be the last post in this series. I'm getting sick of writing a diary, and frankly, I don't know how people do it. I've grown bored with myself after only four posts.

After I posted yesterday, I downloaded the first two Calculus lectures off of iTunes. It was basic algebra and I fast-forwarded through most of it. This class will be a review the biggest challenge for me is going to be scheduling times to take the tests.

This morning, I experienced my first extended use of the clicker. After we answered the questions, our results appeared on the screen. Then we argued with each other for a few minutes, and voted again. We continued until one answer received 100% of the vote. It was really quite fun, and definitely a useful teaching tool.

Now, I am off to work. For the first time since school began, I will be working a full shift.

Thursday, September 6, 2007

Back to School Week: Day 3

In the traditional Tuesday/Thursday repetition, this morning found me once again in general Physics. Today was our first foray into the use of the infamous "clicker." I was still flying high from my fortuitous clicker purchase yesterday. I had planned on making a quick stop at the bookstore before my Java class, expecting it to be a fairly simple transaction. The powers that be had decided to trump my hand by packaging the clicker with the textbook rather than stand-alone. Just as I was cursing the gods, a classmate who already had the clicker from last semester arrived hoping to buy just the book. It was one of those happy little accidents that rarely occur outside of Bob Ross paintings.

Anyway, as I said, we used the clickers to answer some simple questions. I kept expected Regis Philbin to ask me if it was my final answer. We quickly moved on to some basic definitions and set the stage for kinematics in one dimension.

Statistics is moving along at a slow and steady pace. I expect we'll be into simple probability by by next week. Today we were defining events and sample spaces and variance, etc. Still no end in site to the stupid questions from the geriatric brigade. I suppose I'm just going to have to learn to deal with it.

On an unrelated note, I found a new apartment today. It is everything I was looking for-3rd floor, utilities included, off street parking, pet friendly, and less than I'm paying now. There's even free wireless and the landlady will let me paint. I'll have a two week overlap, so I can decorate and move at my leisure.

The future's so bright, I gotta wear shades.

Wednesday, September 5, 2007

Back to School Week: Day 2

Only one class today. Java programming. Seems like it's going to be pretty easy. The teacher is aiming the class at people with zero programming experience. Personally, the last time I wrote a program it was in BASIC on my Commodore VIC 20, so I don't mind taking it slow. Things got off to a rocky start when our classroom door was locked. We decided to commandeer the adjacent room, and began the lecture. It soon became apparent that the lesson plan was dependent on the LCD projector locked next door, so we waited on security to come with a set of keys.

In addition to the aforementioned snafus, there were many stupid questions asked by my fellow students. I need someone to tell me how I can get over my immediate and transparent physical response to these inane queries? As a teacher, I can't afford to react this way.

Tuesday, September 4, 2007

Back to School Week: Day One

For the next week, my posts will be purely autobiographical. Feel free to tune back in next week if you begin to grow bored with me.

Today began with Physics (Calculus based.) There are about 80 people in the class and there's a good chance that I'm the oldest. Most of them are either Physics majors or engineers of some kind. The teacher seems pleasant enough, insisting that we call him by his first name, Paul. His teaching style is somewhat discombobulated. He pulls examples out of the air, rather than preparing them ahead of time. Because of this, he often confuses himself. Much of the class consisted of us struggling to follow his running monologue. I did a lot of erasing, and in the future, I probably will let him get a bit ahead of my note-taking in order to preserve rubber.

I was not surprised that technology is going to factor heavily in the course. For starters, people don't answer questions with raised hands anymore. I have to purchase a radio-frequency clicker that will allow me to answer multiple choice questions. It feels a bit like Who Wants to Be a Millionaire. Also, all homework is done online. Since I purchased my textbook via Amazon, I did not receive the passcode for the homework database. I will have to buy it stand alone from the website.

After class, I watched as he answered a handful of questions. He was patient and willing to restate himself. This is something that I'm going to have to work on as a teacher.

Between classes, I got my student loan refund and deposited it in the bank. I also timed out the trip from school to work. It took me about 20 minutes, and even with a new apartment, I expect to be able to do it in 30.

I had learned in Physics that lab classes will not meet until next week, so Statistics would be my last class of the day. The demographics are very different from the first class. Out of roughly 20 people, 6 of them are obviously much older than me. I can tell those older people are going to annoy me. Several of them talked nonstop, in that lonely, awkward sort of way that people advancing in years tend to do. I'm sure one day I will fall victim to this verbal diarrhea, but for now it is irritating. There are also many more women in this class, close to 50%. This tells me that the class is probably required of many majors, including biology and history. I know that sounds chauvinistic, and I hope my regular readers know that I wish that were not the reality. Like it or not, women are not currently flocking to the STEM fields. I hope to change that, but for now, I think it's a fair assessment that the math in this class will be geared to a wider audience.

So that was Day 1. I'm still on cloud nine, and part of me really can't believe that I'm finally back in school. Tomorrow, I have Java programming and that's it. Hopefully, my deposit will have cleared so I can buy my clicker and do my physics homework. If not, I could be off to a bad start.

Monday, September 3, 2007

Twas the Night Before College

Visions of derivatives dance through my head.

Saturday, September 1, 2007

Math for the Cosmo Girl

I wrote so many posts (here, here, and here) in anticipation of Danika McKellar's Math Doesn't Suck that many of you may be wondering why I haven't reviewed it yet. I have no excuse other than that I've been busy. But I finally got around to reading it tonight and it's about what I expected.

It looks and reads exactly like an issue of Cosmo. The front cover sports a sexy photo of McKellar with superimposed titles in a variety of fonts and sizes, exactly like the glossy covers of fashion mags. Inside, there are explanations and examples of math concepts, broken up by testimonials and horoscopes. I read it pretty much the way I read Cosmo*. I scanned through the testimonials, skipped over the quizzes, scoffed at the horoscope, and put it back on the shelf.

It is definitely geared toward the girly-girl and I think that's just fine. Boys have plenty of other books directed at them. Although, the quotes from school aged girls got a bit redundant, and Danika is not nearly as funny as she seems to think she is, the explanations were presented in a clear, concise manner with lots of great tips and short-cuts. The book will definitely be a powerful weapon against math anxiety for young girls.

*My girlfriend has a subscription and I like to know what manner of rubbish is going to somehow get me into trouble each month. And as an aside, she always manages to stack the issues out of order, which bugs me because it jumbles up the Guy Without His Shirt on the spines.


After several years of plotting and planning, only one holiday weekend stands between me and my second collegiate experience. That being the case, I thought now was as good a time as any to discuss some changes to Pencils Down.
Up until this point, I have been sharing my own assorted layman's ramblings concerning math education. All that I know has come from either first hand experience as a student or from various books I have found at the local library. Many of my posts have seemed somewhat outlandish, even to me, but they have all stemmed from my sincere belief that math education in this country can be improved. There are still way too many people who cower in fear when presented with even the most vaguely math related conundrum. We can do better.

On Tuesday, I will take one step closer to my goal, and I fully expect that phase transition to manifest itself here. As you know, I will be working full-time while carrying 16 hours my first year. I will be busy to say the least, and many of you have expressed understanding, should the frequency of my posts diminish. I thank you, but I doubt that is going to happen. The primary reason that I began blogging was to get a jump on my own education. I have learned so much already from teachers like Dan, Dave, and IB, that despite my real life instructors, I can't imagine cutting myself off from the free communal knowledge I can get online.

I am, however, going to structure my posts a bit differently. In addition to my random musings, I am going to add two regular weekly elements. Once a week, I am going to give a lesson based on some concept I am covering in school. I want to practice explaining math skills to others and now is as good a time as any to start. Please be critical of them, so that I can learn from your experience. Another weekly theme will be a kind of meta-analysis. While I am in class, I will not only be listening to the teachers, but also studying the other students. I will be in there with a bunch of 18-20 year old kids. This group isn't too much older than the students I plan on teaching, and I feel that a little sociological journaling might help me prepare for my own classroom. I will change names when necessary, but I plan to write once a week about how my younger classmates are responding to the teaching methods of the instructors.

You'll still get a healthy dose of the random; I can hardly help that. But since you've been at my side for the journey so far, I thought I ought to bring you with me on the next step.

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.

Tuesday, July 31, 2007

Conservatives Sleep with the Lights On

This one goes out to all my right-wing pals out there, and you know who you are. If you're tired of the inflammatory Liberal rhetoric embedded in your child's favorite picture book, then I have the thing for you.

Help! Mom! There Are Liberals Under My Bed is the book conservative parents have been seeking. This illustrated book the first in the "Help! Mom!" series from Kids Ahead is perfect for parents who seek to share their traditional values with their children, as well as adults who wish to give a humorous gift to a friend.

Hailed as "the answer to a baseball mom's prayers" by talk radio host Melanie Morgan, Liberals Under My Bed has already been the subject of coverage in The Wall Street Journal and Harper's magazine. Written by a self-proclaimed "Security Mom for Bush" and featuring hilarious full-color illustrations by a Reuben Award winning artist, it is certain to be one of the most talked about children's books of the year.

Regardless of where you fall on the political continuum, the idea of Ted Kennedy under your bed should scare the bejesus out of you.