The Pentagon is a Pentagon: Against Trivial Application

I teach nothing but geometry. From the start of the day until its end, I'm immersed in it. When I first began to teach, I had little choice but to follow the text. But over time, I've come to doubt the value of the text that was chosen for me, and I ditch it whenever I can. (When I began, the text was Glencoe's Geometry 2007. Today it's the 2011 edition.) I'd like for a moment to focus upon one of its many faults, a fault that it shares with almost all others of its type.

First a bit about geometry. Simply put, geometry is the mathematics of shape, and like all of mathematics, it is abstract. (Hold on to that word. It's central in the argument that follows.) What does this mean? Consider the simplest of polygons, the triangle. (Much of elementary geometry is the geometry of the triangle.) Let us say that we have drawn some conclusion about it, perhaps that the sum of the measures of its angles is 180. This conclusion is not about just this or that triangle. Rather it is about them all. If you like, it is about triangularity as such. Thus we ignore all other properties that a triangular object possesses and consider only its shape, and about that shape we draw our conclusion.

Now of course our conclusion will hold of all triangular objects. But this really is a trivial matter. Of course we must say that if the sum of the measures of the angles of a triangle is 180, then the sum of the measures of the angles of any particular triangular object is 180. But to say this is to say nothing new. It isn't a discovery. It isn't an advance. It is at most trivial application.

Now, what's the relevance of this little foray into the philosophy of mathematics? It's this: the current crop of geometry texts demand near-continual trivial application of elementary geometry.

Here's one of my most despised examples. Glencoe's Geometry gives us a picture of the Pentagon and asks us to determine the sum of the measures of its angles. The task is two part. First we must determine the shape of the given object. Second we must draw a conclusion about that shape. I contend that the first is trivial and hence a waste of time; I contend of the two tasks, only the second is genuinely geometrical. Let me explain.

Here's how we go about our task. First we abstract out the shape, that is we focus upon the shape of the Pentagon and ignore all else. Color, manner of construction (whether brick, stone or concrete), location - all else but its shape is put to the side. After this task is done (and no doubt it is done quickly and effortlessly) one then begins to reason about the shape, and with this one begins to actually do geometry. Nothing is learned in the first step, nothing at all. It requires only that students identify the shape of a thing, and given just how simple the shape is, there's nothing to it.

It's idiocy to have a student already 16 years old pick out pentagons (or triangles, or circles or any other simple geometrical figures). Perhaps we should have them count to ten before we allow them tackle a quadratic, or spell "play" before we have them write an essay about Hamlet. I say that we should dispense with these questions of trivial application and get on to the geometry. That means that we go straight to the shapes themselves. If we wish to reason about pentagons and a picture would be of use, make it as simple as possible. Don't put anything into it that the student will have to immediately abstract away. Don't assume that you've taught students anything that they did not know about the applications of geometry when you present them with pictures of kites, hubcaps, cat whiskers, bobcat tails and all the rest of the clutter you'll find in today's texts. These applications are drop-dead obvious. They're a waste of time.

(A word of caution. I don't mean to say that all applications of elementary geometry are trivial. Some most certainly not. But if it's something that a bright two-year-old could do - and a bright two-year-old could most certainly pick out triangles, circles and other simple geometric shapes - then trivial they are.)

Advice for Parents

I know that one should not be quick to give advice.

I know that I run a risk, the risk of hypocrisy. None of us always do what we know we should. I am no exception. But I still know what I should do, even when I don't do it.

But I'll run that risk. I have advice that I wish to give, advice for the parents of my students.

I'm certain that this advice will offend. I'll be told that the obstacles for some are simply too great. In my defense, let me say that I do know how hard life can be. I know just how unjust - economically unjust - our country has become. But you simply cannot wait for a solution from outside. If the poor do make any gains, it will be only slowly and in tiny increments. There will be no revolution, no quick transition from the gross inequality of the present. If there are to be solutions, solutions for the present, they must come from within your family. They must come from you, the parent. You wish your children to have a decent life. But this is not possible without a decent education. You have to take the reins. You must make your children do.

"But the schools, Dr. Mason! The schools are no damn good!" I say that they're better than many folks believe. The primary problem here isn't the facilities or the teachers. (I don't say that they're perfect, or that we shouldn't concern ourselves with their improvement. Of course we should. I just say that they're not the primary problem.) The primary problem is the students. Too many don't value their education. Too many don't do their work. Now, my purpose here isn't to condemn. Instead it is to point a way out. I don't much care how we got into the fix we're in. (I have a theory about that; I suspect you do too.) I care about how we get out. I'm convinced that the solution must be bottom-up. It must come from within the family, and it must be initiated, guided and brought to completion by the parents.

I do know that no one can follow this advice always and everywhere. But in your home you should establish a pattern, a norm. This advice should be followed whenever possible, and that of course is most of the time.

Now for the advice.

1. Do you have a conversation with your children at the end of each school day? Do you ask them what they learned? Be ready to question them at length. Demand that they answer.

2. Is homework your child's first priority once they arrive home? Play is for later. Homework comes first. It must always be completed. It must always be turned in.

3. Does your child have a quiet place to work? Turn off the television. Put away the cell. Spread out on a table. Get quiet. Get to work.

4. Do you help where you can? Stay close by as your child does her homework. Answer questions when you think it appropriate, but refuse to do the work when you know your child can do it.

5. Do you review the homework? Even if you don't know yourself how to do the work, you will of course know when the work is shoddy and incomplete and when it is not. Demand that your child redo what was not done well. (The first time you do this there will be wails of anguish. If you don't cave in, there likely won't need to be a second time.)

6. Do you contact the teacher whenever you have questions about your child's work? We're really quite delighted when parents get in touch. For most of us, it's a rare event.

7. Do you share a meal with your children? Is that meal nutritious? Your child should not have free access to crap. Better yet, don't keep crap. Don't pretend you don't know what the crap is. I know that it's easy to give in and let the kids have their crap. Don't.

8. Is your child in bed at a decent hour? I'll be blunt: no child should have a TV in her room. The temptation is simply to great to turn it on and watch. Children have little impulse control. Your task is to control their impulses for them. Do you know that the cell is off when your child goes to bed? You can find out; I know you can. If you suspect that your child uses her cell when she should be asleep, I have a simple solution: just take it away.

9. Do you wake your child so that she has enough time to eat, bathe and get to school on time? If your child is chronically tardy, the fault is yours. Fix it.

10. Do you reward your child for work well done, but refuse rewards when the don't do well? Rewards that are given come what may are worth nothing. They serve only to spoil a child. Rewards given only when rewards are due? Now that's something that will be cherished.

11. Do you punish a child when their work isn't up to their ability? If you don't, your child has no motivation to do well. Children simply do not think about the long-term consequences of their actions. You have to do that for them. If they screw up and you don't punish them for it, you haven't done your job. Take something away that they value. Don't let them out of the house when they're not in school. Do something they'll really dislike. Really punish them. (That word "punish" doesn't have the currency that it used to. We need to bring it back into fashion.)

I also know that if you begin to do the things I've advised you to do, your child is likely to throw a long nasty fit. So be it. You can weather the storm. There can be no real change without real pain. Be firm. Stay strong.

Studious Silence

I plan to make my classroom a place of studious silence.

This runs counter to much of the advice given to teachers today. We are told that our classes should be filled with the sound of conversation. Work should be done in groups, not alone. Ideas should be tossed about, criticized and then refined.

Now, I don't doubt that a bit of this is good. (When it's done, it must have direction and that must be provided by the teacher. Students, if left on their own, will often allow themselves to put the work of the day aside and simply socialize.) But more than a bit is not good at all. To learn mathematics, one must be able to think. One must have the time, and the quiet, in which to step through a sequence of ideas. The continual chatter of others makes this difficult if not impossible.

Thus as I said I plan to clamp down. I plan to enforce silence.

Customer/Business or Master/Apprentice?

Here's what happens when we treat our students as if they are customers and education as if it is a business. Instead of actually educate them, we strive to please them; and to please them, we make them work little but create the illusion that they excel. Thus the demands that we place on them decrease and yet the grades we give them increase.

It isn't a simple matter of grade inflation. No, it's much worse than that. Grades are inflated and rigor is abandoned.

We should ditch the customer/business model in education. We teachers don't run a business. Our students are not our customers. What model should be put in its place? I propose a model of great antiquity, the model that prevailed in Europe in the United States for centuries, both within education and without. Call it the Master/Apprentice Model. We teachers know something of great value, something that our students wish to learn. So they come to us and place themselves under our authority. We define success. We pass judgment. Our students are not our equals. We are above them. They are below us. (Of course I mean this to hold only within the classroom. Qua human all are equal.) We instruct. They obey. (This model still holds sway within sport, as of course it should.)

Does this mean that our power is absolute, that no one ever has any right to pass judgment on us? Of course not. But students do not have that right. Only our peers, the other experts, have that right. Let us do away with student evaluation of their teachers. It has degraded the quality of the education we give. We strive to flatter our students, not challenge them. Over time, rigor falls off and then is lost.

Let there be a return of rigor. Let us tighten the screws. Let education be a crucible in which our students really are tested instead of flattered.

Carrot and Stick

Students have a habit, a very bad habit. But the fault isn't their own (at least not wholly). Teachers are to blame. We've inculcated the habit.

What is this habit? To work only when there's an immediate prospect of reward or punishment.

Thus students believe that the only point of homework is the grade that it will receive. To learn the day's lesson? To prepare for the test to come? To lay a foundation for future work? Perhaps even genuine interest? None of that seems to matter at all.

The result is this. Students work only when that very assignment will receive a grade. Tell them that it won't receive a grade and they blow it off. (Of course not all will. But many do. In my experience, the percentage is well over 50.)

I hate this. I have that everything we do in class must have its own carrot and stick. I do love what I teach. Geometry is very rich ground in which to till, and the skills that its study imparts have application far outside mathematics. Would that I could convey this to my students. (Lord knows I try. Sometimes I burn so brightly in my classes that I end the day completely exhausted.) Would that work would be done because it was thought important, not because it would receive a grade.

The Long, Slow Slide

I teach only geometry.

I've thrown myself into the subject over the past four years. Only now do I believe that I've begun to understand what it is and how it should be taught. (Of course the former should determine the latter. Pedagogy must follow the contours of content. Geometry is often ruined by the insistence that it be taught in accordance with one or another pedagogical theory.) What is it? What is this thing that we call "Geometry"? We must say first that it is a branch of mathematics. It is, if you like, the mathematics of shape. But what then is mathematics? I don't pretend to have a complete answer. But this I know: mathematics is of its essence systematic and deductive. (The application of mathematics need have neither of these qualities, and unfortunately what often goes by the name of "mathematics" in our schools is simply unsystematic application.) By "systematic" I mean that it begins in a small set of first principles and that all else grows out of them. By "deductive" I mean that all of its conclusions must be true - not just might be, but absolutely must be true - if the first principles are assumed.

The core concepts within mathematics - and so in geometry too - are axiom, proof and theorem. The axioms are the first principles, those fundamental assumptions, with which we begin. From them, we construct proofs of certain propositions that, once proven, are no less certain than the axioms on which they are based. A conclusion once proven comes to bear the name "theorem".

Since this is what mathematics in general and geometry in particular are, this is how geometry must be taught. It is a proof course. It is, moreover, the very first (and alas likely the very last) proof course that students will ever take. In this lies the importance of geometry. We all wish our students to learn how to think. This is a teacher's first goal, the goal that all others serve. But to think is to reason, and to reason is to reason from one thing to another. Thought is a process, a process that begins with what we might call a data set (whatever precise form this might take) and from that data set draws a set of inferences. In geometry, we shed all extraneous concerns and do just this. We think. We reason. Thus in geometry we focus to the exclusion of almost all else on the most important of skills that can be imparted to students.

On this basis I form my judgment of texts. Is a geometry text primarily about application of theorems whose proofs are at best an afterthought? I despise such books. Of these there are many. It seems that this is the only sort of text that the major publishers - Pearson, Glencoe and all the rest - can produce.

Over these past four years, I've built up a collection of texts, some old some new. The good ones are the ones that teach proof and teach it well. The bad ones are the ones that only occasionally touch on proof and teach it poorly when they do. Here's a sample:

1. Euclid's Elements. A reviewer at Amazon called it "the best book ever written by a human being". An overstatement, perhaps. But I share the reviewer's enthusiasm. It always has been, and always will be, the model of how mathematics should be done. More than any other book every published, it has shown us how to think.

2. Kiselev's Geometry. Published in the late 19th century, this was the standard text in Russia for almost 100 years. I finally yielded to a more Soviet text but was still widely used. Verdict: absolutely superb. I'd be delighted if I could use it.

3. Bartoo and Osborn's Plane Geometry. This was the text used by my grandmother when she first began to teach in the 30's. (It has her notes scribbled in the margins.) The date of publication was 1939. I know little about its authors. I don't know how widely it was used. But I do know the quality of the text. It isn't as good as the Kiselev, but it is still excellent.

4. Lewis' Geometry: A Contemporary Course. First published in 1964, the text maintains a standard of rigor that is matched by very, very few texts in publication today. I find the discussion sometimes less than crystal clear, but the problem sets are quite good. They're proof after proof after proof, just they they should be.

5. Harold Jacobs' Geometry: Seeing, Doing, Understanding. First published in 1974, this text is now in its third edition. I've had a fascination with it for some time. To my dismay, it was not the one chosen by my corporation. It is by far the best text currently in use. If I were king, it would be the only text ever used. Explanations are always clear, but Jacobs truly excels in the problem sets. They are always creative, always a challenge for students, and always fun. (I grin through every problem set that I do.) Moreover, Jacobs knows his history. The authors of most geometry textbooks seem to have a knowledge of the subject that extends back only to the prior edition of their text. Not so Jacobs. Jacobs' text seems to have gone out of print. It seems that demand was not great. Indeed it was I gather the least used of the texts currently available. Only a brave few took its challenge. Verdict: an extraordinary achievement and by far the best of its era.

6. Glencoe's Geometry 2004 . We now have a text written by committee. Coherence has been sacrificed. We do not have a single vision throughout. We have multiple visions. This creates a kind of content schizophrenia. Upon occasion it's good, but the quality is never sustained. The good and the bad (and the completely irrelevant) coexist in a random mix. Proof is done upon occasion, but problem sets are dominated by simple-minded application of results whose proofs are sometimes given, sometimes not.Verdict: mediocre at best.

7. Glencoe's Geometry 2011. Here the bad has become worse. The book is a travesty. Many of the decent problems from the 2004 edition - ones that had the potential to challenge students - have been sacrificed. All that's left is an endless profusion of problems that require little more than simple-minded application of basic results. Verdict: a multicolor abomination.

8. McDougal's . Equally as bad as the 2011 edition of the Glencoe text. Unfortunately the McDougal and the Glencoe texts are in wide use. Most of our students learn their "geometry" from texts such as these.

Note that my list was chronological. Moreover, each is representative of the era in which it appeared. A certain conclusion appears inevitable: we're on a long, slow slide downhill. Content has been, and continues to be, watered down. Hard problems have been cut. Postulate sets have become bloated. Proof is often an afterthought. Irrelevant little side-topics abound. The systematic construction of a system of geometry has been sacrificed. Most of what remains requires only the mindless application of basic algebra to a set of monotonous problems.

Your children have been cheated. Their course in geometry (unless that are lucky enough to find themselves with a teacher who can correct these many problems) has been so dumbed down that it has become a waste of their time. If I were a parent of a child in such a course, I'd be as mad as hell.

What to do about this sad state of affairs? I say that we need a return to rigor, that we should make our classes much, much harder than they are at present. We need a return to a genuine geometry, the sort given to us by Euclid, Kiselev and Jacobs. The new breed of text produced by corporate behemoths like Glencoe and McDougal should be consigned to the flames. I do not doubt that more students will fail. But so be it. Some will rise to the challenge; those that do will have learned something worthwhile.

The New Economy and the Corruption of the Classroom

Here's the best piece about the U.S. economy that I've read in some time. It's author is Andy Grove, co-founder of Intel. He has the courage to state the obvious: the U.S. no longer creates the good jobs that it once did. The reason? We've shipped so much overseas - so many jobs, so much technological know-how - that we've largely lost the ability either to create new technologies or to scale-up their production and thus create good jobs. Instead we chased short-term profit and so sacrificed the long-term stability of the U.S. economy.

What consequence does this have for my classroom? Why does this matter to a teacher of mathematics? Our politicians and our administrators know perfectly well that good factory jobs are a thing of the past. Thus they force every student onto the college track. I do understand the motive. Without a college degree, a student will likely fare quite poorly in this new economy of ours. With a college degree, there is a possibility of decent pay; without it, there is little or none.

But here's the problem. Many of my students should not be on the college-track. Many don't want to be. Many don't have the emotional maturity for it. Many aren't prepared for a rigorous course of study. Many simply lack the ability.

I don't mean to denigrate my students, for I value much besides academic achievement. Love of family, commitment to work, church and community - these and many others are among the greatest of goods, and a college degree isn't necessary (or sufficient) for the achievement of any of them. But I do bemoan the fate of those of my students who cannot succeed in college. Through no fault of their own, they find themselves at a time and place where simple virtue is not by itself sufficient to escape poverty. They are forced onto a path for which they are ill-suited and thus fail.

What is the consequence of this in the classroom? Perhaps as many as half of my students should not be in my class. They are not prepared or able to do the work. One of two results is inevitable: either I dumb-down the course, or I fail them.

The reality of course is that I and my colleagues do a bit of both. Out of compassion, and a belief that if we were to fail a significant percentage of our students this would reflect badly on us, we make our courses easier; and we still fail quite a few. Thus our classrooms are corrupted. Good students are forced to endure dumbed-down courses; bad students are forced to take classes they never would have found themselves in before. Few students are well-served. (In my cases, only those who take Honors courses escape this fate, but at times I fear that even they too suffer. Once standards are allowed to slip anywhere, they tend to slip everywhere.)

The pessimistic that a solution to the problem I've described will be found. Indeed I suspect that it will only grow worse in the near future. The Obama administration's Race to the Top punishes schools if any of their students show themselves unready for college. This is surely folly.

A Dirty Little Secret

Here's a dirty little secret for you, one that many know but few have the courage to speak.

Many students do little or no work.

No, I don't mean all of them or even most. Some work. But of those who fail, the great majority don't; and their lack of work is the cause of their failure.

Now, I don't mean to blame my students. For most, the fault is the result of forces outside their control. The culture around them - parent, school, media and all the rest - has failed them. But though most students are victims here, still we must acknowledge the problem with them.

Some no doubt will say that the blame is really mine, that I do my job poorly and then shift the blame. In my defense, I say that I know my imperfections well and that this is not one of them. How do I know the fault isn't mine? (I suspect that my answer here is the typical teacher answer.) I am quite clear about my assignments. (They're spoken, written on the board, and published online.) Moreover, students know that I always give an assignment. Thus students have no excuse not to know what they were to do.

But with a consistency that is near perfect, a significant portion of students simply do not do the day's assignment; and of those who do attempt it, an even larger portion do only part and do that part poorly.

On most assignments, the total percentage of half-doers and non-doers is 50 or above.

Perhaps you will suspect that I am a bore in front of a class and have little ability to inspire. You'd be wrong. I put great thought into what I teach, and I work hard in front of my classes. When the final bell rings, I'm worn out. Moreover, I know that I inspire students, for in the past I have inspired many. I'm not the one at fault. My students' failure is not my failure. It is theirs. I do love what I teach. Students occasionally poke fun at me about my passion. They find it deeply strange that anyone would care so much about geometry. But they do see that I love it, and they know that I can teach it.

Let me say too that I'm not in any way special. All of my colleagues work at what they do. Of course some work more, some less. But of this I am convinced: if the average student were to work as hard as the average teacher, average student performance would increase dramatically.

So we come to this conclusion: a poor work ethic is among the primary causes of student failure, and this failure should be blamed upon the culture in which students have been brought up.

Don't expect a politician to ever say this. It would be political suicide. You can't blame those who would elect you for the failures of their children. Instead you must scapegoat. You must find a minority on whom to pin the sins of the majority. Teachers are the obvious choice. Thus teacher's are continually subjected to the censure of politicians and are continually buffeted by attempts to remake education. (The Obama administration is typical. Culture isn't blamed for their children's' failure. Teachers alone are blamed.)

But I beg you to consider another possibility. Please recognize the corrosive influence of society for what it is and do all that you can to fight it.

Truth

Truth is rare.

We care about much besides truth, and thus we often sacrifice it. We wish to spare another pain and thus sacrifice the truth. We wish others to think well of us and thus sacrifice the truth. We wish to manipulate another and thus sacrifice the truth. Truth is sacrificed for these and a myriad of other reasons. We will never lack reason to sacrifice the truth.

I will attempt not to do this. I will attempt to tell the truth in all I say. (Anonymity of course makes this much easier.)

This will be difficult. I feel the same temptation to bend the truth as do you.

About what will I tell the truth? About me and my performance. About my students and the quality of their work. About my colleagues and their behavior. About administration and its management of my school corporation. About lawmakers and the policies they enact.

Let us begin.