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.