Don't Cry for Me

My love of math and science is no secret to those who know me. Friends call me up at all hours, asking me to answer questions or settle bets regarding all manner of things. I am the Phone a Friend. At least, that's how it works with those who have come to love me. The reaction from strangers is quite different.

I've noticed lately, and especially at work, that the discovery of my mathematical predilections is normally accompanied by a wince and/or head tilt. As soon as I mention that I am majoring in math and physics, I am treated as though I've announced a death in the family. "Oh, I'm sorry," they say. Or my personal favorite, "So you're one of those." Usually, I laugh it off, and use it as an opportunity for research. I ask why they feel that way. Why is they're fear and loathing of math so complete that someone else's involvement causes them pain? I find the conversations fruitful, if not more than a bit repetitious.

The respondent almost always remembers loving math as a small child. They can usually pinpoint an exact year or teacher which soured them on their studies. Often times, they remember being told by a teacher that math was simply not for them. It is at that point that I am able to commiserate. In my junior year of high school, after having taken all honors math classes, my teacher told me one day that I simply lacked the "flare for math." It galls me that people who would say such things are allowed to teach any subject at all, let alone such a notoriously tricky one.

I know now, and I continuously attempt to impart to my friends, that math class is not terribly different from shop class. Both are all about tools and toolboxes. Math class is no more about mathematics than wood shop is about craftsmanship and design. Just because you can hammer a nail does not make you an architect, nor does hating long division mean you are cosmically predestined to avoid math. This is a fact that is lost on most students, and too many teachers, and it is one that bears constant reminder. Students must be given a glimpse of the horizon so that they have something to journey toward. Otherwise, we are asking them to practice for a championship game that will never come.

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.

Love,
Your big brother

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.

Cross-Purposes and Collegiate Complaints

As I struggle to find where exactly on the political scale of math education I reside, I have to constantly question many things that I once took for granted. One of those things relates to the transition from secondary to post-secondary education. One of the latest tactics that the traditionalists are employing against the reform math programs involves pressure from college professors and deans. They insist that these new programs are not preparing students adequately for the collegiate curriculum. The latest article comes out of Pennsylvania, where a new integrated math program in the public schools is forcing many colleges to adapt their methods.

The first step of problem-solving is to actually determine if a problem exists. In this situation, Pennsylvanians have to determine what they feel the primary purposes of education are, and then see how high college preparation is on the list. Conveniently, their state code lists a Purpose of Education section, and nowhere in it is college even mentioned. It does say that

Public education prepares students for adult life by attending to their intellectual and developmental needs and challenging them to achieve at their highest level possible. In conjunction with families and other community institutions, public education prepares students to become self-directed, life-long learners and responsible, involved citizens.

It wasn't that long ago when most students did not go on to college. University was for the rich white men and everyone else either got a job or got a husband. Fortunately, those days are gone. There is much more equality of opportunity in education and everywhere else. But the rise in post-secondary enrollment creates new issues and debates. Here we see colleges parroting similar complaints of the business community, that essentially the public schools are not operating with their particular agendas in mind. Frankly, I don't see a problem here.

Education is important. That much is clear. It is difficult to get ahead without obtaining new knowledge and skills. Learning is necessary to grow and adapt with a system. But I believe that burden falls largely on the individual. It is each person's choice whether or not to educate themselves. No teacher can teach a student that refuses to learn, and many great men and women have been self-taught. Although there is no "right to an education" in our federal constitution, the Framers certainly realized that an informed citizenry is a prerequisite of democracy. They supported and encouraged each state to develop public education guidelines, so that every child wanting to learn had an opportunity to do so. Not everyone took advantage of the programs, but enough did to make it worth the expense. And there is quite a bit of expense. We spend as much on education in the US as we do on defense, believe it or not. Unfortunately, the complicated administrative bureaucracies swallow most of that funding before it can trickle down to the actual classrooms. Now we are stuck with a system that is hugely inefficient, stupendously expensive, and perpetually maligned.

I firmly believe that we are trying to do way too much. There are so many goals that we aren't really reaching any of them. Let's put aside cultural integration, job training, and even college prep until we can get the simpler goal of "self-directed, life-long learners and responsible, involved citizens." To that end, I think that the two goals of any curriculum, mathematical or otherwise, need 1) teaching students how to teach themselves, and 2)providing them with skills required to be a good citizen. That may very well mean something other than what is considered a "traditional" math curriculum. For example, when in the voting booth or in the jury box, what field must you better understand, statistics or calculus? Yet stat is usually taught as an afterthought in most Algebra II classes.

Colleges can complain all they want. There objections/recommendations are duly noted. But there mission statements are fundamentally different than those of public education, and vice versa. Even now when most students will go on to some form of post-secondary program, the focus of K-12 must remain on civic responsibility rather than collegiate complaints.