On the off chance that another teacher of geometry might come across Think Logical!, I thought it might be helpful to talk for a few moments about why I've structured it as I have.
On the first day of class, I make a most solemn promise to students. I promise them that they'll never have to take anything on faith. That they'll never to have to "just trust me". That nothing will be justified by "the teacher said so".
I promise them that they themselves will understand, as well as anyone every has, why what we say is true.
This is the real purpose of the class, the one to which all others are subordinate.
Students are often shocked by this. Most if not all of their teachers simply tell them what is true and make little attempt to explain why it is true. (I don't blame their teachers. This is how they were taught.) Students thus come to believe, even if only implicitly, that no explanation is possible; they've come to believe that it all just is. (I've heard it put just this way many times. I sometimes ask on the first day of class why the Pythagorean Theorem is true. Students find the question strange, as if it's never occurred to them. They say it "just is" or that it's true "because the teacher said so".) I'm an enemy of this, and I intend to make my students enemies of it too. They deserve more. They crave more. They want to know all the why's; and since we can give it to them, we must.
So, the purpose of the class is to explain. But what does it mean to explain? My answer: to explain is to derive (or, if you like, deduce). Derive from what? From that which is already known.
This is the key insight: explaining is deriving; and from this insight, we can begin to tease out the one, inevitable structure that the course must have. Let's say we want to know why A is true. (Let it be the Pythagorean Theorem if you like.) Well, it is true because of some B (which of course might be multi-part, which in the case of the PT it most certainly is). Now what of B? Why is it true? We can't say A, for if we do we're in a tight circle and nothing has been explained. So it has to be some C. So we have A because of B because of C.
No doubt you see the threat here. If were never to bottom out, if instead we forever chase some new "because of", nothing would ever have been explained. If we get A from B, B from C, C from D and so on ad infinitum, we'd an explanation of precisely nothing.
So in the chain of derivations we must come to an end. Moreover, that end point must be something that is itself known; for if not, its uncertainty infects everything derived from it.
Those endpoints, those foundations if you like, are not known through something else. Instead they are known through themselves; or, as the philosophers say, they are self-evident. In mathematics, we call these self-evident foundations "postulates"; and of course we call what's derived from them "theorems".
A system such as this - one comprised of postulates and theorems - is typically called a deductive system.
(For those who know a bit about the history of mathematics, you'll recognize that I take a pre-modern view of mathematical proof. You won't find many modern mathematicians who say that postulates are self-evidently true. Instead, they'll put the question of their truth to the side, and concern themselves solely with what can be derived from them; and if the system they develop is consistent, they're perfectly happy. I admit that I have a certain affection for this view. I am a great admirer of Cantor, and he believed that nothing more than consistency is required to render a mathematical system legitimate. Yet when one teaches geometry to students in their teens, I think it best to adopt the old view and say that the postulates must be self-evidently true.)
Here's an audacious claim: to think is to inquire into, and discover, the relations between things.
But I told you above that, in this class, we build a system in which what comes later is related to what came before in a certain way; and I told your that the nature of that relation is proof.
These two claims together imply that, in this class, we learn how to think. This I think is the real purpose of the class. It isn't to learn geometry. It's to learn how to think.
Yes, yes. We do learn quite a bit of content. The sum of the angles of a triangle is 180°. The sum of the squares on the legs of a right triangle equals the square on the hypotenuse. Etc., etc. But what's most important here is not the theorems themselves but how we arrive at the theorems, and I firmly believe that the skills necessary to arrive at them can transfer to subject areas outside mathematics.
So, what we do in this class is math properly so-called. It isn't applied math. It isn't algorithm rehearsal. It's proof. It's what professional mathematicians do.
To my delight, I discovered some years ago that students in their mid-teens are perfectly capable of proper math. Thus I think that it's our obligation to give it to them.
This course is, judged from a certain point of view (or points of view), atypical. Yet judged from another it is typical.
How is it atypical? It is atypical if compared to the recent crop of textbooks, e.g. those from Pearson and McGraw-Hill. How so? This course (which I'll call "TL") is relentlessly systematic. Here's what I mean by this:
Every proposition that is part of the system built in TL is either a definition, a postulate or a theorem.
Every definition, postulate or theorem is explicitly labeled as such; no doubt is ever left about the role that any statement that is part of the system plays within the system.
No theorem is ever proposed to students if it is not, at the moment at which it is proposed, proven.
Proofs are rigorous; that is, every step in every proof can be justified by a definition, a postulate or a theorem previously proven.
If you poke about in most recent textbooks, you'll find multiple violations of these rules. For instance, I looked through the table of contents of the Big Ideas textbook (which bills itself as "Common Core"). There I found that the Distance Formula is handed to students in Chapter 1. How terrible! The Distance Formula is simply the Pythagorean Theorem in the coordinate plane, and so should come only after (a) the Pythagorean Theorem has been proven, and (b) the coordinate plane has been constructed.
Numerous other equally atrocious examples are easy to find. Indeed, if you have not done so, I suggest that you sit down with a mass-market geometry text and subject everything to the test outlined above. That is, for any assertion that is made, ask yourself whether it's identified as a definition, a postulate or a theorem; and for those that are identified as theorems, ask yourself whether a rigorous proof has been provided. I expect you find many assertions that do not fit this scheme.
So, if judged by the standard set by (most) modern textbooks, TL is atypical. Yet judged in the light of the entire history of geometry - a history which stretches back to at least 300 BC - the class is really very typical. For most of its history, geometry was a proof class. I hope that I can do my part to return geometry to its roots.
The topics covered are the usual. Glance at the page list on the side bar. But let me make two points.
First, no analytic geometry is found outside Chapter 12. Thus the geometry of Chapters 1 - 11 is purely synthetic; that is, it's of the definition-postulate-proof type found in Euclid's Elements. I do make use of number in a way that Euclid did not, for I assign to both angles and segments reals that give their measures. But prior to Chapter 12, I do not coordinatize points.
Second, this course does not ever stray into the third dimension. It is on the contrary a class in planar geometry alone. Why is that? Planar geometry offers more than enough content for a year. Moreover, it is no trivial matter to develop a geometry of the third dimension; and an attempt to do so would no doubt result in too much content for a yearlong course.
If you glance at the sidebar, you'll see a number of links that don't take you to a chapter. They include:
Final Reviews
Definitions, Postulates, Theorems, Formulas
The Myriad
Triangle Solver
See below for a discussion of The Myriad. (It's really important!)
Triangle Solver is a short Python script that will solve for unknown sides and angles in triangles. It's of great use in the study of right triangles.
Final Reviews is self-explanatory.
Definitions, Postulates, Theorems, Formulas gathers together all instances of those four types found in the course. I think that the most important of these is Theorems. It not only collects every theorem of consequence proven but also lists them in the order proven. That order is important! If a student ever wonders whether result A can be used in the proof of result B, it's always safe to do so if A comes before B on Theorems. (You see the potential logical pitfall here, don't you? If you use A to prove B but B was previously used in the proof of A, you have yourself a tight little logical circle in which nothing has actually been proven.)
I assume a two semester year.
Semester One
Chapter 1. Objects, Names, Diagrams
Chapter 2. Inference
Chapter 3. Proof
(optional) Euclid's Theorem and Pythagoras' Theorem
Chapter 4. Congruence
Chapter 5. Parallels
Introduce The Myriad.
Chapter 6. Inequalities
Semester Two
(optional) Algebra Review: Roots, Radicals, Ratios, Proportions (found at the start of Chapter 7)
Chapter 7. Similarity
Chapter 8. Right Triangles
Chapter 9. Circles
Chapter 10. Area
Chapter 11. Concurrency
(optional) Chapter 12. Analytic Geometry
A few notes:
Euclid's Theorem states that the number of primes is infinite. Pythagoras' Theorem (distinguished of course from The Pythagorean Theorem) states that the square root of two is irrational. (These are my names. They're not standard.) I include them after Chapter 3, because in Chapter 3 we study indirect proof and these two theorems are wonderful examples of the indirect method. Typically I present these two results after the Chapter 3 exam, and then have a little quiz where I have the students recount one of the two.
Chapter 7, where the topic is similarity, is preceded by an optional algebra review. I've found that students are often a bit rusty with the algebra that inevitably arises in a study of similarity and could do with a bit of review.
I rarely do Chapter 12 myself. It's a core topic in algebra classes, and I'm more than happy to let the teachers of algebra have at it!
Please take a moment to take a look at The Myriad. Over the years, its importance has grown in my mind. Indeed at present I'm of the opinion that it, more than the content of the individual chapters, is the real point of the class. It's where students are given the opportunity to put the tools they've acquired to work in the solution of genuinely difficult and (I think) fun problems. Please, please, please do not omit The Myriad!
I do recognize that The Myriad makes heavy demands on both students and their teacher. But so be it. This class should be a heavy lift. My suggestion is that you devote at least a few weeks to The Myriad before you assign it to students. Do the problems! By the time you're done, I'm almost certain you'll be a much better geometer than you were before.