Wednesday, June 11, 2014

Common Core Geometry 2.0

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

Friday, August 16, 2013

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.

Saturday, August 3, 2013

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.


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.

Friday, August 2, 2013

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.


  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.

Wednesday, July 31, 2013

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…

AndreasIn 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 FriedmanWhat 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='' type='text/javascript'> </script>

A few tests:


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

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.

Monday, July 22, 2013

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.