Thursday, July 14, 2011

Grade Inflation

I make a most solemn pledge not to inflate grades. It seems that this will put me in a minority.

Below are the right standards:
A: Mastery
B: Solid grasp of almost all material.
C: Solid grasp of most material; shaky grasp of the remainder.
D: Shaky grasp of most material. No grasp of the remainder.
F: No grasp of most material.
I don't scale. Instead I design tests that distinguish between students based on the standards above.

I know what students need to know. I know how to teach it. I know how to test it. If no one deserves an A, then so be it. An A is for mastery, and if no one has that, then no one gets an A.

D's and F's are for those who have little or no grasp of the material. If that's you, you get a D or an F. If that's a third of the class, so be it.

Oh, and it's test where you display your knowledge. Homework is practice. Don't expect that you can half-ass a semester's work of homework, or copy it from someone else, and thereby help your grade. It won't work. You half-ass or copy, you fail the tests, you fail the class.

Monday, July 11, 2011

Something Big

Our country needs a task. Something  big, something that we can all get behind. This is the sort of thing that I mean. This too.

At present, we're fractured and afraid. We blame others for our all-to-real problems, and we suspect that we've gone into decline.

Our children are given no goal other than narrow self-interest. They need something more than that. They must come to see themselves as integral parts of this great nation, and that nation must undertake some great task of tasks that will capture the imagination of its young people.

Want our children to excel in school? Want them to welcome the rigor of mathematics and the sciences? Give them a reason! A real reason, a reason that fires the heart. Don't insinuate that the only reason is wealth. That's a individual motive, a purely selfish motive. We need goals that extend past the boundaries of the self. We need goals that are at least national if not universal in scope.

What are we to tell our children? What goal do we given them? The details are of little importance. Tell that in 10 years we will have permanent colonies on Mars. Tell them that in 10 years we will have weaned ourselves off fossil fuels. But no matter what you tell them, tell them something big.

The only way for this to happen is for a leader to emerge who relentlessly pushes for something big. National purpose does not emerge bottom-up. From the bottom we only get a cacophony of voices, each of which advocates for its narrow self-interest alone. From the top, we have the potential for a single vision that can focus the energies of an entire people.

This is my hope, indeed my only hope. I hope that such a leader will emerge. If one does not, decline is I think inevitable.

(Cross-posted at The Philosphical Midwife.)

Wednesday, July 6, 2011

Test the Teachers: Geometry

Do our teachers know their subjects well? Are they genuine experts? Some are, some are not.

Almost all know it better than their students. Almost all know it well enough so that they can stand in front of a class and teach.

But one can do these things and not know a subject well.

I do realize that expertise is difficult to acquire if one teaches. One has precious little time to dig deep into a subject. I do not come to blame, then. But I do call for change. When a teacher has a moment for study, it should be devoted to the subject he teaches. We should demand real mastery from our teachers, and we should give them the time and resources to achieve it.

Many teachers learned what they know from the texts they teach. Thus they know the text and nothing more. Given how bad texts have become, this means that they really know very little about what they teach.

One might reply that, since our teachers all have degrees, then of course all our experts in the subjects they teach. I say that this just isn't so. Some didn't have any courses in the subjects they now teach, and of those who did, many had only a few or one. In the case of geometry, the subject that I know best, one is it; and one isn't enough for mastery.

I know elementary Euclidean geometry (eEg for short) much better than when I began. (No doubt I'll continue to learn. eEg is rich, very rich.) This has radically changed how I teach. I know the history of the development of eEg, and I know how its ideas hang together; and I now structure my class so that I can convey something of this. Before it was an endless procession of isolated little atoms of information. I apologize to those classes.

One might object that mastery isn't necessary to teach at the primary or secondary level. I admit that this is so. Many teach without mastery. But their students suffer. Of course I don't mean to say that a teacher should attempt to teach all that he knows, or that his students should become as expert as he is. This is of course impossible. But mastery of a subject changes how you teach. What you teach something you have mastered, you teach better even when you teach to beginners. For example, the connections between ideas, connections that sometimes are not at all obvious, become crystal clear, and one then teaches so that these connections are brought to light.

I propose a test for teachers of eEg. It is below.

It is a test for mastery. If you teach eEG, you should know all of this. Each occupies a central place in eEg. If you know only a little of this, you have work to do; at the end, I have a few book suggestions.

Questions are in no particular order. The test is not exhaustive.

The Teacher's Test
1. You teach eEg. Discuss what is meant by "Euclidean" in this context.
2. Every system of eEg includes some form of the Parallel Postulate. State at least two forms of this postulate.
3. Prove that an exterior angle of a triangle is greater than either of its remotes. Make sure that in your proof you don't assume the Parallel Postulate (or any result that can be traced back to it).
4. Prove that the sum of the angles of a triangle is 180 degrees. Make certain that in your proof you discuss the relation of this theorem to the Parallel Postulate.
5. David Hilbert made SAS triangle congruence a postulate of his formulation of eEg and proved the other triangle congruent principles, namely SSS and ASA, on its basis. Many later authors followed Hilbert. Assume SAS, and from it prove SSS and ASA.
6. Prove the SSS and SAS triangle similarity principles. (Yes, I expect you to assume AA similarity.)
7. Provide at least two proofs of the Pythagorean Theorem.
8. Prove the converse of the Pythagorean theorem.
9. Prove the Pythagorean inequalities.
10. Prove the Hinge Theorem (also called the SAS triangle inequality) and its converse.
11. Derive the Law of Sines and the Law of Cosines.
12. Discuss the so-called "Ambiguous Case" of the Law of Sines.
13. Prove the inscribed angle theorem. (Expect no credit if you prove it for only one special case.)
14. Prove that an angle inscribed into a semicircle is right. Do it in more than one way.
15. Prove that a quadrilateral in inscribable into a circle if and only if its opposite angles are supplementary.
16. Prove that the medians, the angle bisectors, the perpendicular bisectors and the altitudes of a triangle are concurrent.
17. Derive the sphere surface area and volume formulas. Do it as it would have been done before the invention of the calculus.
18. Derive the pyramid volume formula. (Shame on you if you don't know where the one-third comes from.)
19. In the 19th century, mathematicians came to realize that one could build an internally consistent geometry in which the Parallel Postulate was denied. What figures were involved in this development? Outline the two broad categories of geometry that they developed. (Hint: each corresponds to one of the ways in which the Parallel Postulate may be denied.)

By the time I'm done with them, my Honors students can answer each of these questions. Can you?

If you didn't pass the test, you need to go back and study. Here are some texts to help you along.

Kiselev's Planimetry and Stereometry.
Jacob's Geometry: Seeing, Doing, Understanding.
Hartshorne's Geometry: Euclid and Beyond.

Just work through them.

Thursday, June 30, 2011

The Danger of Happiness

Here's a piece from the Atlantic about the dangers of the pursuit of happiness. If that's all we desire for our children and if we always strive to secure it for them, then paradoxically they often won't have it.

Aristotle knew this. Happiness shouldn't be a goal in itself. Rather we should pursue success. Learn to do a thing well and happiness might come as a consequence. But pursue the happiness itself and likely it will elude you.

Choose a task. Devote yourself to success within in. The sum of your successes is the value of your life. If you achieve some measure of happiness along the way, feel fortunate. But do not mistake that happiness for the purpose of your life. Your purpose is success.

Wednesday, June 29, 2011


I've poked around on YouTube recently in hopes that I would find a few proofs from geometry.

I was deeply disappointed. YouTube is a geometrical wasteland. Most of what's there is either wrong in one way or another or just plain trivial. Go on a hunt for, say, a proof of the special properties of parallelograms and all you're likely to find is demonstration after demonstration of how to find the area of a parallelogram. What a bore.

I mean to fix that. Tell me what you think.

Tuesday, June 28, 2011

It's Society, Stupid

A conviction has begun to grow in me over the last few years.

It has grown as I have become more confident of my abilities. I know my subject-matter. I know it well. Moreover I know how to teach. Many of my students leave my class with a deep knowledge of elementary geometry. This is rare today. The quality of geometry instruction in the U.S. is quite low. I am an exception. (Think that I am overly confident? That I praise myself too much? Engage me about elementary geometry. You'll find that I know it and know how to teach it. I have made that my sole study for four years.)

I do love geometry. I do love to teach it. But still, every semester, many of my students leave my class with little or nothing to show, and the fault isn't mine. Many students are superb, but more are quite poor. I fault the society around them.

Let me quote here a commentator on a recent New York Times article on teacher evaluation. She makes the point well.

My husband used to teach in a low-performing public school in Maryland. It nearly killed him - waking up at 5am, coming home at 6pm, working at home until almost midnight, and then grading papers and writing lab and lesson plans all weekend and during most of winter and summer breaks. He was a highly rated teacher (and deservedly so) but the fact is that much of his time, when not under observation, was devoted to keeping fights from breaking out in his classroom, taking phones and ipods away from kids who texted or listened to music during class, and disciplining students, since the school administration had informed teachers that sending kids to the principal for discipline was a failure of teaching and was unacceptable.

The same kids who routinely slept and fought in class would aggressively petition him at grading time, urging him to "drop a D on that b****" in lieu of a failing grade. And the administration implemented byzantine procedures for failing a student, including a requirement that the teacher successfully make contact with the parent several times to discuss the student's problems. In some cases, the student provided a false telephone number for the parent at registration time, so there was no reaching parent. In most other cases, the parent either was unreachable or did not respond. Nonetheless, the same students who would have failed if not for these procedures eagerly anticipated attending college, which they predicted would be easier than high school.

In his second year, after surviving a round of teacher firings, my husband quit mid-year and went back to practicing law, where he makes several times the salary for less than half the effort. All the evaluations in the world aren't going to fix this problem.
I attended a highly ranked public high school in an affluent part of the midwest. As good as my teachers were, I have no doubt that each of them would fail if reassigned to my husband's school.

This is exactly right. We have a bit of a problem with bad teachers. But it's dwarfed by the problem with bad students. Lazy students. Disrespectful students. Lackadaisical students. Students who care little (or none at all) about their education.

(Don't think that I mean all students. Of course I do not. I have many superb students. But I have more that exemplify these traits. Recall that I have said that we have one country but two cultures. One values education. One desires to learn. The other places not value on it, or in some cases is openly hostile to it.)

How did this come to be? Its cause is the wider society in which the school is embedded. Students bring the culture outside the classroom into the classroom, and that culture often thinks education worthless.

Student quality is a reflection of culture quality, and that has been in decline for decades now. But few will say this, because it requires that we look at ourselves and what we have let our culture become. We would rather blame our problems on others.

As I have said before, the problems of the classroom don't have their origin in the classroom and thus cannot be solved there. Culture must be restored. We must all begin to value education, and we must show that we do so everywhere - in our homes, in our media, in our places of business, in our churches, and everywhere that we congregate. Let us begin now.

Sunday, June 26, 2011

The Khan Academy on Centroids

Here's a short little video from the Khan Academy on triangle centroids.

(Not know the Khan Academy? Not heard all the chatter about it? Try here.)

What you have here is typical of the sort of thing you find in the current crop of texts. What's called a proof is not really a proof at all. Lots is loaded in that is unproven. What's worse, it's not even noted that there are gaps in the proof. What's a perceptive student to think? That she's stupid because things that the speaker seems just to assume aren't obvious to her? Don't call something a proof if it's not. You do students a disservice.

Here are my objections in detail. Read them after you've watched the video.

1. It's never explained why the medians are concurrent, that is why they all come together at a common point. It will seem utterly mysterious to students why this is so. The concurrency proofs are some of the most beautiful in elementary Euclidean geometry. Why pass over them? Why not even mention that a proof is necessary? Inexcusable.

2. It's never explained why the coordinates of the centroid will be (a/3, b/3, c/3). Instead it's just assumed. This makes the "proof" circular. When one assumes these coordinates, one has in effect assumed that the centroid lies 2/3rds of the way from vertex to midpoint of opposite side.

3. It's never explained why the centroid represents the center of gravity of a physical triangle. This isn't really very hard. It begins with the claim that a median divides a triangle into subregions of equal area. Why isn't this done? Time? Ignorance? No matter the reason, again it seems inexcusable to me.

I expect that students (the perceptive ones, anyway) will come away with the impression to do mathematics, one must have little mathematical nuggets must rain down from heaven, unmotivated and unexplained. What a terrible impression.

Friday, June 24, 2011

Consumption and Creation

Consumption of technology is easy. Creation is hard.

To consume, all you have to do is learn the interface, whatever it is. How difficult is the Facebook interface really? The interface to the iPhone? These and all the rest are simple. Give a reasonably intelligent person a few hours and they'll have the basics down.

The point? Our task is not to teach students to consume technology. That they will do on their own. Rather our task (at least in part) is to teach them to create those technologies, and that's hard.

We need scientists, we need programmers, we need engineers. These are the creators of our technologies, and they must have a deep grasp of the mathematical and scientific foundations of the technologies that they will create. This requires what it has always required - hard intellectual labor.

The mind must be be trained to carry through lengthy and intricate deductions. This is what it is to think. This is what it is to take a seed of any idea and bring it to fruition. You don't learn this on Facebook. You don't learn it on an iPhone. You learn this today as students always have, with a text, a teacher and time.

Students often have little idea of this distinction between consumption and creation. They think they know the technology, but all they really know is its surface, its interface; and that has been designed for simplicity of use. What lies below the surface is quite extraordinary complex. How will students come to understand that? How will they learn to make something of such complexity? They must know the theory behind it. They must know the science and the mathematics. The traditional course of study isn't made irrelevant by the new technologies. The traditional course of study is responsible for the creation of those technologies. So let us continue to require that students complete rigorous courses of study in science and mathematics. This is hard, I know. Students would much rather just sit back and consume. But they don't know what they need, and we teachers must not shrink from the task appointed to us.

Saturday, June 18, 2011

Could Poverty be the Root Cause?

We have problems here in the U.S., deep problems. Might this little piece get at the root of it?

What I Should Have Said

Ms. Cornelius at A Shrewdness of Apes has given us a fine read. Read this. Read it from start to finish. I beautifully captures what I've tried to say here many times.

Here are a few of the best passages:

If one listens to all of the cant coming out of the talking heads who purport to be educational experts, especially those who claim to be experts, you will notice that the dominant assumption regarding students is that they are acquiescent, empty vessels waiting to be filled. A whole passel of those alleged "reformers" like to use the "consumer" paradigm when describing how to fix American public schools. Students and their families are depicted as "consumers" of educational services. The problem with this stereotype is the absolute passivity of consumers in our consumption culture. The deluge of advertising and its claims that consumption can be transformative is probably THE seductive lie of the 20th century in terms of the lives of the common people.

I can assure you that many students in public high schools also are disinclined to value their educations since it always emphasized that this education is free. Unfortunately, they also interpret that word to mean "requires no real effort." Schools often abet this notion by lowering standards and removing consequences for failure to master concepts. However, even in the face of this trend, I do want to say there are more than a few of us in the classroom who are swimming against that tide, who seek to maintain and enforce high standards and rigor. We ARE out there, banging our heads against the wall daily for the sake of our students. We do it because we KNOW that our students CAN do the work, CAN learn the concepts and skills needed. They just have to be pushed into it.

That's just right. The primary problem in our schools today is student passivity. So very many don't really care about their education at all. At most, they grade-grub a bit and hope to get by with the minimum possible work. (Of course, many won't do even this. Such laziness there is!)

This problem is deeply ingrained, and no mere change in curriculum or teacher technique will fix it. The culture must be transformed. We must begin to value academic achievement. Indeed it must become our primary value.

Imagine the transformation that would occur in the classroom if academic achievement became as important as athletic achievement. Imagine what would happen if academic achievement became the primary goal of students and their parents.

Wednesday, June 15, 2011

Better Today?

Yes, we have a wide variety of technologies available to us today. Has this made us better teachers? Do students learn more quickly because of it? Do they better understand what they are taught?

If we restrict our attention to mathematics, I suspect that the answer to each is "No".

In geometry, software such Geometer's Sketchpad does help a bit. We can easily construct diagrams and easily transform them. This allows for quicker generation of conjectures and quicker refutation of false conjectures.

But for the most part my class is conducted as it would have been 100, or 1000, years ago. What we develop is the ability to reason well, and for this all we need is pencil and paper. Those simple tools, and the simple static diagrams we produce with them, were sufficient for Archimedes and Descartes. They are sufficient for us too.

Technology is often a crutch, and a distraction. We think that it can overcome the problems of the classroom. It cannot. Those problems are not ones of proper pedagogical technique. They are problems of culture. We live at a time and place when many place little value upon academic achievement, indeed when many hold such a thing in disdain. That is our problem, and technology can't fix it.

Postulate Set for Elementary Euclidean Geometry

I teach only geometry. That's where my head is all the time.

Geometry is systematic. It is deductive. It begins with a set of postulates that explicitly formulate the assumptions on which it based (at least those of a geometrical nature), and from those it deductively derives its results.

I've given quite a bit of thought to my postulate set. When I first began to teach, I simply relied upon the postulate set provided by my text (a Glencoe monstrosity), but over time I became dissatisfied with it. It sometimes obscures issues of great importance (it makes a real mess of the Parallel Postulate), and it is quite bloated (each of triangle congruence principles SAS, SSS, ASA and AAS are treated as postulates).

I resolved to do better. A bit of research led me to the School Mathematics Study Group and its postulate set. (The SMSG postulate set was constructed expressly for use in the secondary classroom. But apparently textbook publishers didn't think it simple enough, for much that could be deduced from it they added as postulates.) Not content to simply take it over, I began to fiddle with it. Four years in the classroom have (I hope) given me some insight into the best way in which express our fundamental assumptions.

No doubt I'll continue to fiddle in the future, but I will share what I've done so far. Below is my (tentative) postulate set.

Postulate Set for Elementary Euclidean Geometry

P1 Line Uniqueness Postulate Through any two points there is one and only one line.

P2 Ruler Postulate We may divide a line's length into a sequence of congruent segments. We may then assign integer values to the endpoints in such a way that, for a given choice of direction along the line, each next point is assigned the next integer. Each point between these endpoints is assigned a real in such a way that:

a. Distinct points are assigned distinct reals.
b. If, for our given choice of direction along the line, point Q comes after point P, then the real assigned to Q is greater than the real assigned to P.

On any such assignment, the real that corresponds to a point is its coordinate and the distance between points is the absolute value of the difference of their coordinates.

P3 Segment Addition Postulate Point T lies between points A and B just if the length of AB is the sum of the lengths AT and BT. P4 Shortest Path Postulate Of all possible paths that begin in one point and end in another, the shortest is the line segment.

P5 Point Existence Postulate Every line contains at least two points. Every plane contains at least three non-collinear points. Space contains at least four non-coplanar points.

P6 Line Containment Postulate If two points lie in a plane, then the line through those points is contained wholly in that plane.

P7 Plane Uniqueness Postulate Any three points lie in at least one plane, and any three non-collinear points lie in exactly one plane.

P8 Plane Intersection Postulate The intersection of two planes is one line.

P9 Plane Separation Postulate Given a line and a plane in which in lies, the points of the plane that do not lie on the line form two sets such that:

a. each of the sets is convex, and
b. if P is in one set and Q is in the other, then the segment PQ intersects the line.

P10 Space Separation Postulate The points of space that do not lie in a given plane form two sets such that:

a. each of the sets is convex, and
b. if P is in one set and Q is in the other, then the segment PQ intersects the plane.

P11 Protractor Postulate We may divide the circumference of a circle into 360 congruent arcs and assign to their endpoints in sequence the values from 0 to 359. We say that the direction along the circle is the direction of increase of these values. Points between these endpoints are assigned reals in such a way that:

a. Distinct points are assigned distinct reals between 0 and 360.
b. If point Q comes after point P for our direction along the circle, then the real assigned to Q is greater than the real assigned to P.

The measure of an arc is then the absolute value of the difference of the values assigned to its endpoints. Arc measure determines angle measure in this way:

d. We construct a circle whose center is the vertex of our angle.
e. We mark the points where the angle's sides intersect the circle. Call these S and T.
f. We then say that the measure of the angle equals the measure of the arc ST that lies in the angle's interior.

P12 Angle Addition Postulate If point T lies in the interior of ∠CAB , then m∠BAT + m∠CAT = m∠CAB . P13 Supplement Postulate If two angles form a linear pair, then they are supplementary.

P14 Side Angle Side Triangle Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent congruence.

P15 Parallel Postulate Through a point not on a given line, there is at most one line parallel to the given line.

P16 Similarity Postulate For any given figure and any real k, there exists a second figure similar to the first such that the scale factor from first to second is k.

P17 Area Existence To every finite planar region there corresponds a unique positive real number that we call its area.

P18 Area Equality If two closed figures are congruent, then the regions enclosed within them have the same area.

P19 Area Addition Suppose that the region R is the union of two regions R1 and R2. If R1 and R2 intersect at most in a finite number of segments and points, then the area of R is the sum of the areas of R1 and R2.

P20 Rectangle Area The area of a rectangle is the product of the length of its base and the length of its altitude.

P21 Volume Existence To every finite spatial region there corresponds a unique positive number that we call its volume.

P22 Prism Volume The volume of a right prism is equal to the product of the length of its altitude and the area of the base.

P23 Cavalieri’s Principle Given two solids and a plane, if for every plane that intersects the solids and is parallel to the given plane the two intersections determine regions that have the same area, then the two solids have the same volume.

Friday, June 10, 2011

Two Cultures

Among those who write about education policy, one of the very few I respect is Diane Ravitch.

Here's a wonderful little piece she wrote on the effects of poverty. It's conclusion is that the failures of education are rooted in poverty and that, if we do not seek its redress, all our educational remedies will fail. I concur.

Doubt it? Doubt that the problems of the classroom have their roots outside the classroom? Look here. The data that seems to prove the mediocrity of U.S. schools is aggregate data. It brings together students from all regions and all economic strata. If we disaggregate and compare, say, those from schools in which the poverty rate is less than 10% to students in countries with a similarly low poverty rate, we find that students here in the U.S. outperform all those in all other countries. The problem isn't the teachers. The problem isn't the schools. The problem is poverty.

"But aren't the schools where the majority are poor inferior?" Yes, of course. But ask yourself what's cause and what's effect. Do inferior schools make their students poor, or does the poverty of the students render the schools inferior? The former contains a bit of truth, the latter more than a bit. It's damn hard to teach at a school whose students live in poverty. Those gigs grind a teacher down. Most of the good ones flee. Most of the ones that remain are disengaged at best.

Take care here. When we speak of poverty and say that it is the cause of the failures of education, we must take this to mean the culture of poverty. We have one country but two cultures. One is a culture of achievement. The other is a culture of failure. One is composed of insiders, of those who exemplify the traits of character necessary for success. The other is composed of outsiders. Some outsiders are ignorant of the means for success - diligence, self-denial, frugality and all the rest. Some know the means but lack the ability to bring them about. Some simply don't care.

The goal, of course, is to bring the outsiders inside, and this means that they must begin to behave as insiders. But here, as it were, we face a challenge before the challenge. Precious few acknowledge the real issue. Some seem to believe that money alone will solve the problem They're wrong. (Note that I don't say that money isn't part of the solution. It is an essential part.) An outsider who becomes wealthy doesn't transform into an insider. All that she becomes is a wealthy outsider; the rest remains the same. Some seem to believe that schools deserve the greater part of the blame. They're wrong too.

What we must do first is confront the problem. We must recognize that the task before us is to change minds and reform characters. This is our challenge. Habits of thought and of behavior and must broken and reformed. Let us begin to apply ourselves to that task.

Wednesday, June 8, 2011

What I Can Do, and What I Can't

I can teach mathematics.

I can't fix a broken child.

I've immersed myself in elementary Euclidean geometry over the past four years. I know its contours well. I know the form in which it first appeared. I know the developments that it underwent it its long history. I know the history of its pedagogy, and I have deeply held opinions about how it should and should not be taught today.

Moreover, I've watched students quite carefully as they struggle through the course. I know the sorts of mistakes they are likely to make, I know the sorts of misconceptions they are likely to harbor. I know what I need to say to correct these. Moreover, I know just how students should be led from idea to idea. I know how to build up the body of geometrical theorems so that it's structure will be pellucid. (Don't think that I claim I'm exceptional in this regard. I know how to do it because I've seen others do it and do it well. Almost all that I know I've shamelessly lifted from minds much better than my own. Alas, but this is the fate of those who progress only because they are led.)

If you're bright and motivated and you put yourself under my authority, then when we're done you will know elementary Euclidean geometry.

But still, many of my students don't know much about geometry when they're finished with my course. (I teach about 250 students in a year. Of these, at most 100 are competent by the end.) Why not? What happened with them? Bright though they might have been, motivated they were not. I can't fix that!

Instruction isn't to blame. Neither is curriculum.I do love what I teach, and I know that I make that love plain. (Students marvel at it. Students quite regularly tease me about it. Here's the proof.) I know that my explanations are clear. I know that my assignments are of the right sort - they review the basics but always end with questions that really do challenge.

Classroom disruption isn't to blame. I'm lucky. I have little problem with it. Others do. (One of the dirty little secrets of education here in the U.S. is just how many of our students are little better than feral. What do you do with a student who stands in the hallway and shouts "Motherfucker" again and again. I chewed him out and then wrote him up. Will that make any real difference in his life? Probably not.)

But if none of these things are to blame, what is? What's the source of the problem? I must say that my ideas are inchoate. At times I simply blame the parents. At others I blame the wider culture. But the fact cannot be denied. Many of our students don't care a bit about their education, and I can't fix that.

Saturday, June 4, 2011

The Common Core Standards: Geometry

I have to admit it. I like the new Common Core Standards (CCS) for geometry. (For those not in the know, these are the new federal standards. As of present, all but a few states have adopted them.) If you haven't read them yet, take a moment to do so.

First I'll run down what I like about them. After I'll give my only objection.

1. The CCS demand that congruence and similarity - the fundamental relations of elementary geometry - be defined in terms of the concept of transformation. (How so? Here are quick and dirty definitions. Two figures are congruent just when one can be carried onto the other by a sequence of rigid transformations, i.e. translations, rotations and reflections. Two figures are similar just when one can be carried onto the other by a sequence of dilations and rigid transformations.)

Implicit here is the concept of superposition. Two figures are congruent just when, by a sequence of rigid transformations, one can be superposed upon the other. Likewise for similarity, with the addition that we may dilate too.

Such a strategy has strong historical roots. Moreover, it conforms to current mathematical practice. Euclid employs it at times, for instance in his proof of side-angle-side triangle (SAS) congruence. Felix Klein generalized it and in the Erlangen program made that generalization the basis for the categorization of the different geometries. Mathematicians seized upon Klein's work. Today his methods are ubiquitous within mathematics, and no doubt the CCS wish to bring secondary geometry in line with this.

Of course one might question the pedagogical value of the twin concepts of superposition and transformation. Charles Dodgson certainly did. But I for one find them quite intuitive. Indeed they seem to me just the right way to motivate SAS, AAS and ASA triangle congruence. (SSS is another matter. The only route to it, I think, is through the Isosceles Triangle Theorem. But still transformation and superposition do play a role.)

2. In the CCS, we find a continual insistence upon the importance of proof. I wholly agree. Only a small part of the value of geometry lies in the particular results at which it arrives. (Students will forget most of them. Most of them they will never use.) The greater part of that value is in the way that it arrives at them. It isn't guesswork. (It sure looks like that angles opposite congruent sides are congruent.) It isn't simple induction. (Every triangle that we've looked at so far has at most one right angle.) Instead it is proof, proof from first principles. Students need to know how to put together a logically watertight argument where their basic assumptions have been made fully explicit. That's a genuinely valuable skill.

3. The CCS has students tackle problems of some difficulty. They thus seem to me like a much-needed return to rigor.

Some examples:

Proof of the triangle congruence principles, i.e. SAS, SSS, ASA and AAS. Many texts make them all postulates. (Gah! I hate that. It leads students to believe that a postulate is some damn thing that someone somewhere just made up. Why in the world wouldn't we take the opportunity to explain why they work?)

Prove angle-angle similarity. (I love this one. AA~ has never been obvious to me. I'm sure that it isn't for students either.)

The derivations of the Law of Sines and the Law of Cosines. Explain the so-called "Ambiguous Case" of the Law of Sines.

The derivations of the sphere volume and surface area formulas. Many texts today don't touch the former. What a terrible, terrible mistake! That derivation is one of the most beautiful proofs in all of elementary geometry. Archimedes was so proud of his discovery of it that he had it engraved on his tombstone! I propose a rule: if a text does not reproduce Archimedes' derivation, that text should be banned from use.

So ends the praise. Now for a bit of criticism. The CCS shifts study of conic sections from Algebra II to Geometry. The problem with this is obvious. Geometry students have had only a year of algebra and simply do not have the requisite algebraic sophistication to take on a study of the conic sections. I hope that a later version of the CCS fixes this problem.

Wednesday, June 1, 2011


As I graded my set of final exams, I realized that failure is not always bad. I had before me the exam of a bright young woman. I knew that she'd studied. But still she missed 12 out of 94. As I so often do, I began to question myself. Wasn't her failure a result of my own? Hadn't I failed her?

But then I realized that I had not. I regret to say that only at the end of my 4th year have I achieved the confidence to draw such a conclusion. I do my job. I do it well. I'd taught her what she needed to know, but the exam really was a challenge. When you really challenge, you must expect failure. Not complete failure, of course, but where there is genuine challenge, there too will be failure to always meet all expectations.

Again I say that failure isn't always bad. For what is the alternative? A course in which most of our students give mostly right answers. How could that come to pass? Only if our questions were always easy. But that - a course whose demands were easily met - would be a mistake. We need to challenge our students. We need to give them material that is difficult for them, and if we do that, some of it they won't get.

Now, I don't suggest that we make the work so difficult that students have little or no hope of success. But we do have to challenge. This means that there's a sweet spot to hit. We have to push them (and expect the inevitable failure). But we push them to do only those things that most of them, given sufficient diligence, can do.

I find the analogy of sport instructive. Should we expect our coaches to give their kids only those tasks that their can easily accomplish? Of course not. Such a coach would never last. Her teams would be eaten up by the competition. But if not this, what do we expect of our coaches? Push their kids, push them hard. The inevitable result is that their kids will fail and fail again. They won't live up to that very high standard that the coach sets. But a good coach will not only push. She will make it possible for her kids to reach the goals she sets if only they work hard enough. The sweet spot here is somewhere in damn hard, just a bit shy of impossible. Now, success will come (of course to be met with a demand for some new damn hard thing), and when it does it really will be worth something.

What of those who don't show the requisite diligence and thus get little or nothing out of our courses? Encourage them. Offer them help. But if they don't take up the challenge, they must be left behind. I will not sacrifice those of my students who do the work asked of them for those who do not. I will not dumb down my course so that everyone can succeed. Let those who will not work fall to the side (but reach out to them if ever they reach out to you). Let those who were undeterred by failure enjoy the success that they have earned.

Friday, May 27, 2011

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.

Sunday, May 15, 2011

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.

Saturday, May 14, 2011

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 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.