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.
(Honesty demands that I note two exceptions to this "everything must be explained" rule. The first are the trigonometric functions. I simply give them a table and do not attempt to explain how it's compiled. Of course they can understand much about it - why, for instance, the sine of 23° is the same for all right triangles, why the sine and cosine of an angle cannot exceed 1, why the sin of 23° and the cosine of 67° must be the same, why for any angle the square of the sine plus the square of the cosine is 1, etc. But as to why sin 23° is approximately 0.391, that is a matter that I tell them will be taken up in their Calculus class. The second possible exception is the two sections, both about circles, where I make use of the concept of a limit. I don't of course make this fully rigorous - delta-epsilon proofs have no place in a course such as this. Nonetheless, I do think students will get the genuine flavor of how the proofs go.)
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.
This course is for my Honors Geometry. Most are 9th graders. Most come in well-prepared. They've had Algebra I, and they can read and understand the prose I present to them.
This is a proof course. So the teacher must be experienced in that sort of mathematical work. I assume the teacher will be if they have an undergraduate major in mathematics.
I assume as well that the teacher takes joy in the attempt to find solutions to hard problems, and that the teacher is able to give clear and rigorous proofs.
The topics covered are the usual. Glance at the page list on the sidebar. But let me make two points.
First, little analytic geometry is found outside Chapter 12. Thus the geometry of Chapters 1 - 11 is almost all 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 (with the sole exception of the last two sections of the Right Triangles chapter), 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.
For each section, I provide links to a lecture, a PowerPoint and a worksheet.
In the lecture, I talk through the PowerPoint.
On the worksheets, I don't often simply have students rehearse techniques demonstrated in the lecture. I always do a bit of this. Practice is indeed important in mathematics as it is elsewhere. But on the worksheets I almost always have students go beyond what was said in the lecture. Mathematics is a creative enterprise, just as much as the arts! My aim is to draw upon student's native creative abilities. (It seems appropriate here to recommend Paul Lockhart's A Mathematician's Lament. It's not the only book I've ever read about classroom mathematics that's been of any real help to me.)
I should also note that, as a result of this aim to elicit in students novel insights, the worksheets often contain blocks of text that students absolutely must read. The purpose of this is two-fold. First, it is of course necessary if I wish them to consider problems not explicitly addressed in the lecture. Second, I hope to inculcate in students the belief that they are able to read, comprehend and then apply technical material. This is a skill that all students must learn, but sadly students are often given little opportunity to practice it.
You'll find that when I first introduce proof, I structure them as a table with three columns. The columns are labeled "Statement", "From" and "Reason". The proof itself consists in the successive rows, each of which is numbered. The first statement is the Given of the proof (or a part of it), and the last is the Conclusion. In each row, the Reason entry contains the definition, postulate or theorem that allows us to infer the statement that begins the row; and in each row, the From column contains the previous line or lines to which the reason was applied.
Of course table-style proofs are a bit unnatural. The natural and no doubt better way to structure a proof is as mathematicians typically do - build it up sentence by sentence, paragraph by paragraph. I do transition to the more natural style as time goes on, but I've learned that, when students are first introduced to proof, tables are superior. Why? They force students to give a reason for what they say, and they force them to cite the previous statements to which their reason is applied. Tables thus instills in students a necessary habit of a mathematician - we don't just say what we believe is true but instead give the reasons why we believe it.
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
Selected Proofs
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.)
In Selected Proofs, I've done a few core proofs for each chapter. My hope is that this will help students learn good proof form.
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.
I mean for this course to be (if I may borrow a phrase from the software world) open source. You may use and modify the material here as you wish. All that I ask is that you attribute it to its source and that you drop me a note to let me know. (I'm fcmasonjr@gmail.com.)
Of course, it would be helpful, if decide to use some or all of my materials, to have the original documents. If you look on the sidebar, you'll see a page titled "The Course Zipped". You'll find (at least) two versions. In the summer of 2025, I undertook major revisions to the course, and I've included both the pre-revision and post-revision versions of the course.
I know me. I know that, no matter how hard I try, errors in the material will remain. If you find any, I'd be delighted if you'd let me know. Drop me an email at fcmasonjr@gmail.com.
If you've enjoyed Think Logical! and would like more from its author, let me recommend Think Functional!. It's computer science taught from a point of view very much like that which animates Think Logical!.