Common Core Geometry 2.0

I've nearly come to the end of a complete revision of my geometry course. Take a look.

The Common Core Standards for Geometry: Resources

I'm a Common Core convert. Here in Indiana, they're a real improvement over the standards they replace. I've compiled a list of resources for Common Core geometry. All are of use to a teacher as she plans her CC geometry. If you know of something not on my list, please let me know.

• Tools for the Common Core Standards. Bill McCallum, math team coordinator of the Common Core State Standards, provides expert answers to questions about the implementation of the mathematics standards. If only the responses were a bit more timely.
• Progressions Documents for the Common Core State Standards. As of 8/10/2013, there's no high school geometry document, but one has been promised soon.
• Teaching Geometry According to the Common Core Standards. Dr. Hung-Hsi Wu provides detailed guidance on the implementation of the CCSS. The document will be an intellectual work-out for most teachers of high school geometry, but it is the best single resource as of late 2013. 
• Beauty, Rigor, Surprise: Elementary Euclidean Geometry. A complete CCSS-aligned course in geometry. Classroom-ready PowerPoints and worksheets.
• Illustrative Mathematics. An initiative of the Institute for Mathematics & Education funded by the Bill & Melinda Gates Foundation. A source for CCSS-aligned tasks, organized by standard. Sometimes classroom-ready, sometimes not.
• Mathematics Vision Project. A set of CCSS-aligned texts released under the Creative Commons License. I'm deeply impressed (and I don't impress easily).

Parallels: The Triangle Exterior Angle Inequality

The Triangle Exterior Angle Inequality (TEAI) does much of the work in the proofs to come. Let us prove it first.

Triangle ABC is given. We extend side AC through C and mark point D on the extension. Angle DCB is by definition an exterior angle for triangle ABC. The angles of ABC to which it is not adjacent - angles A and B - are its remotes.

We will prove that the exterior angle DCB is greater than the remote angle B.

Mark the midpoint of BC; name it M. Construct the segment AN through M such that AM = MN. Connect N to C.

Angles BMA and NMC are vertical and thus congruent. M is the midpoint of segment BC and so BM = MC. AN was constructed so that AM = MN. Thus triangles BMA and CMN are congruent by SAS.

From this it follows that angles ABM and NCM are congruent. But angle DCB is the sum of angles DCN and NCM, and so angle DCB is greater than angle NCM. Thus angle DCB is greater than angle ABM, as was to be shown.

This technique can be used to prove that any exterior angle of any triangle is greater than its two remotes.


Commentary

The result proven holds in both Euclidean and hyperbolic geometry. But it does not hold in elliptic geometry. Where does the proof break down in elliptic geometry? We assumed that the segment from N to C is unique and that it lies in the interior of angle DCB. This cannot be assumed in elliptic geometry.

Parallels: Assumptions

I've had the intent for some time now to do a little non-Euclidean geometry with my students.

I didn't want it to be mere history. A bit of history is fine, but mostly it should be mathematics. This means that we should prove a set of results.

But what results? I'd like to shock a bit, and so I decided to seek out a proof that, in the variety of non-Euclidean geometry that is called hyperbolic, the sum of the angles of a triangle is less than 180 degrees.

Of course the proof must be elementary. My students are quite bright, but they're only beginners. They have only the resources of that part of Euclidean geometry that we've developed. Thus my task was to find such a proof.

I think I have it. The technique comes from Saccheri. In a set of posts titled Parallels, I'll outline the proof.

Today I'll list those assumptions on which I'll draw. Some are definitions, some are postulates, some are theorems. Which is which is irrelevant. All that matters is that they'll be in place when on that day late in the second semester I begin.

I wish my assumptions to be, as it were, geometry-neutral. I wish them to hold in both Euclidean and hyperbolic, non-Euclidean geometry. Thus I do not include the Parallel Postulate (or any proposition equivalent to it) in the list.

The concept of congruence is key, and so I'll begin there.


Assumptions

1. Vertical angles are congruent
2. In congruent polygons, sides and angles can be paired up in such a way that sides which correspond and angles which correspond have the same measure.
3. In congruent triangles, side which correspond lie opposite angles which correspond.
4. SAS. If two sides and an included angle of one triangle are congruent to two sides and an included angle of a second triangle, then those triangles are congruent.
5. SSS. If the three sides of one triangle are congruent to the three sides of another, then those triangles are congruent.
6. ASA. If two angles and an included side of one triangle are congruent to two angles and an included side of a second triangle, then those triangles are congruent.
7. HL. If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg, respectively, of a second right triangle, then those triangles are congruent.
8. Through a pair of given points a line may be constructed. It is unique.
9. Lines are infinite in extent.

On one variety of non-Euclidean geometry - elliptic - lines are finite in extent. On the other - hyperbolic - lines are infinite, as they are in Euclidean geometry.

How Should We Teach?

I'd like a short, pithy answer, but I don't have one. The best that I can do is talk around the question. I'll have as much to say about how we shouldn't teach as how we should. When I'm done, I'll quote from a Timothy Gowers' post on the failures of mathematics education. It's goes right to the heart of the matter.

I use the term 'recipe' quite a bit in what follows. By it I mean a mathematical algorithm that if faithfully followed will solve problems within a certain class. The Euclidean algorithm is a good example. The Power Rule is another.

School mathematics often takes this form:

1. Teacher presents mathematical recipe.
2. Teacher solves problems on board by application of recipe.
3. Teacher gives student two dozen or so problems like the one solved.
4. Teacher asks for questions the next day and then takes up students' work.
5. Repeat

I do admit that, at least in outward form, I often run my classes just like this. But where I differ from this sequence is in its first stage. I never (or at least try to never) simply give a recipe. For each new result that we encounter in class, either I prove it (sometimes), students prove it (sometimes), or we prove it together (most often). I never just give a recipe.

Why? Why not just give the recipe? (It's often what students want. It's what they've been trained to want.) First, recipes without explanations - and proofs are the form in which mathematicians give their explanations - aren't mathematics. This isn't ideology. It's history. Second, when an explanation can be given, and it always can, not to give it is intellectually lazy. Third, as Aristotle put it, human beings by nature desire to know, and if we give only the recipe, the better part of what can be known has been left out. Fourth, mathematics is not a loosely connected set of recipes (though if all you've ever know is school mathematics, that's likely just what you'll think). Instead, it is the most systematic of the sciences. What comes after is deduced from what comes before. This is the essence of proof. If we leave out the proofs, we miss the essence of mathematics. Fifth, students are unlikely to remember a recipe unless they know why it works, and if they how why it works, they can reconstruct it for themselves if need be.

I know of only three real objections to what I've said.

1. This demand to do genuine mathematics with students, to actually give the explanations that will satisfy their curiosity, is beyond most of them. It's just too hard.
2. Anyway, students aren't curious. Why give them something they don't want?
3. All that students will ever really need to know - either for standardized tests or for a later course - is mastery of the recipe.

I'll answer in reverse order.This claim about need is very narrow. What students really need is to be stimulated. They need to have good problems set before them and then given the space and time necessary to solve them. They need a teacher who can guide them in this pursuit. To do any less - to teach just recipes because "that's all they need to know" - is to abdicate the role of teacher.

Second, I know that it's false that students don't want to understand. I know this because I've taught thousands, and of those thousands most have been wanted to understand. It does often take some time for students to become accustomed to the idea that in mathematics, it's proper and indeed obligatory to demand explanations. Such is the state of mathematics education today that students often find it strange that there are such things as mathematical explanations. (How often I've been met with "It just is!" or "Because the teacher said so." when I ask for an explanation.) But once that hurdle has been cleared, students take to the mathematics like birds to the air.

Last, it's simply false that genuine mathematics is beyond most students. Again I know this from experience. I do not deny that some take to it more readily that do others. But all have some degree of ability, many a very high degree. (Every year I have students that could become research mathematicians.) The truth here, I suspect, is that to teach the recipe is much easier than to search out the explanation, and teachers often seek the easier path. (Another possibility - one that I think applies only to a minority - is that they believe, mistakenly of course, that the recipe is all there is.)

I'll let that suffice for now. No doubt I'll return to these issues later.

I end with a set of quotes from Timothy Gowers' post What maths A-level doesn't necessarily give you. I begin with Gowers himself. After I take a few lines from the comments. (If you haven't taken Calculus, you won't understand everything that's said. But I'm sure you'll get the points about pedagogy, which are the most important.)

Let’s suppose that your aim is simply to do well at maths A-level and that there are no questions that test your familiarity with the formula for the derivative of an arbitrary (nice) function at an arbitrary point. Which is better?

1. Don’t make any effort to learn and understand the formula, but simply learn a few basic examples of derivatives (polynomials, exponentials, logs, trig functions) and rules for differentiating combinations (linearity, product rule, quotient rule, chain rule) and you should be able to differentiate anything that comes up in the exams.
2.  Learn what the derivative means, derive the formula for the derivative of an arbitrary function at an arbitrary point, calculate a few derivatives from first principles, derive the product rule, quotient rule and chain rule, and then learn how to use them to differentiate combinations.

The answer is that if you are capable of doing 2, then 2 is far better. And the boy I was talking to was certainly capable of doing 2. Why is it better? Because (and this is something I plan to devote a blog post to at some point) memory works far better when you learn networks of facts rather than facts in isolation.

Gowers is right of course. 2 is far better. Indeed I would claim that if a student isn't capable of 2, there's no reason to have the student do 1. I do find it a bit strange, however, that the reason Gowers gives for the superiority of 2 is that it is an aid to memory. I don't doubt that this is true. (I said it myself above.) But of all the reasons for the superiority of 2, I'd think that that's the least important. Indeed, to give only it makes it seem as if the sole reason for 2 is that it serves the purpose of recipe memorization. I'm sure that Gowers doesn't believe that, but the passage seems to suggest it.

Now for a few responses that caught my attention.

Terence Tao: In general, I think fundamentals are too often given short shrift in order to advance prematurely to applications of said fundamentals. When I teach undergraduate real analysis, for instance, I like to spend a fair amount of time on construction of number systems, before getting to the limits and the deltas and epsilons. Unfortunately, the most basic topics are often the hardest to teach correctly…

Andreas: In Norway . . . the notion of proof has been completely removed from the school curriculum, except for some side remarks in the textbooks. In fact, it is perfectly possible to go through all of school (up to age 19) with top grades in the most advanced mathematics courses, without ever writing down a single proof. Even the formula for solutions of a quadratic equation is normally presented without proof, and to attempt a proof of the chain rule in class is just unthinkable.

As for making choices as a teacher, [the problem is that] we are extremely pressed for time, and choosing to include proofs and proper definitions would mean having to exclude other topics, which, unlike proofs, appear in the all-important end-of-year exam.

So I guess my point is that while it is possible that [Gowers'] young friend has been badly taught, it is more likely that the teacher did what she had to do in order to get the majority of ordinary students through the exam with decent grades. It all comes back to the politics determining the curriculum – who are the people actually making these decisions? Are there any mathematicians involved?

Greg Friedman: What seems to me to be the real problem is that students arrive with the idea that math isn’t something that you think about – it’s something where someone gives you a recipe and you follow it. I try explaining to them that, as pointed out in the post, it’s actually easier to learn something if you understand what’s going on than if you just try to blindly memorize formulas. I also try to instill the idea that if you don’t understand what it is you’re doing, there’s no point learning to do it since you’ll never be able to apply it (though of course to them, the application is the exam).

.

Latex in Blogger

Who doesn't like beautifully formatted mathematics? Latex is the standard. 

The question is how to get Blogger to recognize and then property format a Latex script. Here's a way:

1. Go to the Overview page for your blogger blog.
2. Click Template. It should be found on the right.
3. Choose Edit HTML.
4. Add the code below before the first occurrence of <head>:

<script type='text/x-mathjax-config'> MathJax.Hub.Config({tex2jax: {inlineMath: [[&#39;amp;#39;,&#39;amp;#39;], [&#39;\\(&#39;,&#39;\\)&#39;]]}}); </script>
<script src='http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML' type='text/javascript'> </script>

A few tests:

$Ax^2+Bx+C=0$

$\frac{a}{b} = \frac{c}{d} \Leftrightarrow ad = bc$

Simply enclose your scripts between dollar signs ($). If you need a bit of help to build the script for an expression or equation you wish to display, here's an online TeX editor.

Common Core Geometry

I've been a good little boy this summer. I've been at work on a complete overhaul of my geometry class. It's almost done. It's at Beauty, Rigor, Surprise, under Elementary Euclidean Geometry.

I had two goals in mind:

1. Bring my class in line with the Common Core State Standards for geometry.
2. Incorporate all the necessary changes that I'd mentally logged over the years but never had the time to incorporate into my notes and worksheets.

Nothing was left as it was. I've returned to every PointPoint, to every Word document and either reworked it or deleted it and begun new. All told, it represents six years and perhaps 10,000 hours of work, 500 or so of which were logged in the last two months.

The course is a proof course, from its start until its end. Proofs are given on the first day. Proofs are given on the last. Why such emphasis on proof? A proof is simply an explanation of the truth of a  proposition, and our first and most important function, we teachers of mathematics, is to explain.

Please peruse and give me your comments. If you believe that anything will be of help, please take it and use it. But do attribute it to its source.

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.

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.

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.

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.

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.