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.