Teacher Introduction

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 what I've structured it as I have.

The Purpose of the Course

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 "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; all 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 its 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.

Postulate and Theorem

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 come C then. 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 were 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, then we'd an explanation of nothing.

So in the chain of derivations we have to come to an end. Moreover, that end point must be something itself that is 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.

What It Means To Think

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 that way 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.

What Wasn't Said

Note that I did not say about the purpose of the class. I did not say that it was to learn geometry. (unfinished)

Typical Yet Atypical

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:

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