img.latex_eq { padding: 0; margin: 0; border: 0; }

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.