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.

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.)

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.

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.

Wasteland

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.

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.

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.