Even before the advent of Common Core, there has been much debate about what to include in a geometry text, and ultimately a geometry course. Here are some of my favorite sites discussing geometry and how it should be taught. I will refer to these sites often throughout this blog.
On this page, David Joyce, a professor at Clark University in Massachusetts, gives a review for a certain geometry text. It is not my U of Chicago textbook, but a book published by Prentice-Hall in 1998. I am a math tutor who has taught several students whose classes use this text, so I am familiar with it.
Joyce criticizes the Prentice-Hall text very harshly, and I agree with some of what he writes. In particular, he laments how late the text covers parallel lines (Chapter 7) and congruent triangles (Chapter 8). Most books I’ve seen cover these topic in Chapters 3-4, and so I also wonder why these are mentioned so late in the Prentice-Hall text.
But there are several topics that Joyce argues should be left out of the course (like trig for example). Even though this webpage was written before the Common Core Standards, many state standards (including my home state of California) required trig in the geometry courses. So Prentice-Hall includes topics that Joyce finds extraneous because the states required them
The cornerstone of Joyce’s critique involves logic -- in particular, postulates and theorems. I bet that most geometry students don’t know the difference between a postulate and a theorem. Why, for example, are SSS, SAS, and ASA called “postulates,” while AAS and HL are called “theorems”? The reason is that the first three are assumed, while the last two are proved in the text. But most students don’t learn the proofs of AAS and HL, and most teachers don’t teach these proofs either. “Postulate” and “theorem” end up being a distinction that Joyce and most textbooks make, yet most actual students and teachers in the classroom ignore this distinction.
Indeed, we see how important the difference between “postulate” and “theorem” is to Joyce in his discussion of Chapter 1:
"Chapter 1 introduces postulates on page 14 as accepted statements of facts. The four postulates stated there involve points, lines, and planes. Unfortunately, the first two are redundant. Postulate 1-1 says 'through any two points there is exactly one line,' and postulate 1-2 says 'if two lines intersect, then they intersect in exactly one point.' The second one should not be a postulate, but a theorem, since it easily follows from the first. And what better time to introduce logic than at the beginning of the course. No statement should be taken as a postulate when it can be proved, especially when it can be easily proved."
Think about it for a moment -- suppose Postulate 1-2 were false and two lines (say m and n) could really intersect in two points (say P and Q). Then through two points (P and Q) there would be two lines (m and n) when there's only supposed to be one line through the points, by Postulate 1-1. So this is a contradiction, and so Postulate 1-2 is true. But this constitutes a proof of Postulate 1-2, which would make it a theorem, not a postulate!
So why doesn't Prentice-Hall label this a "theorem"? I suspect it's because this is an indirect proof (or a proof by contradiction), which Prentice-Hall (along with most other texts) cover late. The authors probably didn't want to bring up indirect proofs in Chapter 1. (Interestingly enough, the U of Chicago text does call this a theorem, the "Line Interesection Theorem.")
On this blog, I'll classify proofs by different levels. A high-level proof is what most students think about when they consider proofs in geometry -- given two triangles, with some of the pairs of corresponding sides and angles congruent (either given or easily derived using vertical angles, the Reflexive Property, etc.), prove the triangles congruent using SSS, SAS, etc., then use CPCTC to derive another pair of congruent sides or angles. In other words, they are exercises or problems to do in the homework.
A medium-level proof is the proof of a well-known theorem. Most textbooks give the proofs or leave them as exercises, but students and teachers skip over them. Students typically think of theorems as something to use in high-level proofs, not something to be proved themselves. But since this is a Common Core blog, we must refer to the Common Core Standards, and the standards do require students to prove these theorems:
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
A low-level proof is a proof of a theorem, but it isn't as well known. Most books simply label the statement as a "postulate" rather than a theorem and omit the proof. An example of this is given at the Joyce link:
"Chapter 8 finally begins the basic theory of triangles at page 406, almost two-thirds of the way through the book. This chapter suffers from one of the same problems as the last, namely, too many postulates. The three congruence theorems for triangles, SSS, SAS, and ASA, are all taken as postulates. One is enough. The other two should be theorems."
And I myself had long thought of SSS, SAS, and ASA as postulates, not theorems. Almost every text I'd seen presented these three as postulates, so I was shocked to read this. For explanations on how to prove these, we turn to the author of the world's first geometry text -- the ancient Greek mathematician Euclid, who lived about 2,300 years ago. Joyce gives a link to Euclid's famous text Elements, which contains basically nothing but proofs:
Here, Euclid's Proposition 26 contains proofs of both ASA and AAS. Both of them use Proposition 4 (SAS) and are indirect proofs (which explains why most books don't include these proofs). Also, Proposition 8, which is SSS, is also dependent on Proposition 4. It appears that Joyce suggests having only SAS be a postulate, and both SSS and ASA proved as theorems. Once again, I point out that students think of SSS and ASA as statements to use in proofs as opposed to being proved themselves.
Ironically, Proposition 4 also has a proof in Euclid:
Euclid, Proposition 4:
Let ABC and DEF be two triangles having the two sides AB and AC equal to the two sides DE and DF respectively, namely AB equal to DE and AC equal to DF, and the angle BAC equal to the angle EDF.
I say that the base BC also equals the base EF, the triangle ABC equals the triangle DEF, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides, that is, the angle ABC equals the angle DEF, and the angle ACB equals the angle DFE.
If the triangle ABC is superposed on the triangle DEF, and if the point A is placed on the point D and the straight line AB on DE, then the point B also coincides with E, because AB equals DE.
Again, AB coinciding with DE, the straight line AC also coincides with DF, because the angle BAC equals the angle EDF. Hence the point C also coincides with the point F, because AC again equals DF.
But B also coincides with E, hence the base BC coincides with the base EF and equals it.
Thus the whole triangle ABC coincides with the whole triangle DEF and equals it.
And the remaining angles also coincide with the remaining angles and equal them, the angle ABC equals the angle DEF, and the angle ACB equals the angle DFE.
Therefore if two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides.
Euclid called this "The Principle of Superposition." But, as it turns out, this is exactly the method that Common Core uses to prove SAS! Common Core uses translations, rotations, and reflections, and notice that we have a translation (A slides to where D is) and then a rotation (keeping A fixed on D while turning so that B ends up where E is). But Euclid does make one oversight here -- the angles BAC and EDF are indeed congruent, but the angles may be on opposite sides of AB and DE. It's possible that a reflection is needed (so that we flip one of the angles so that it's on the same side as the other). Finally, the triangles must coincide (one right on top of the other) and so must be congruent!
Some critics decry Common Core as changing the way that math has been taught for years, but as it turns out, Common Core Geometry (at least regarding SAS) actually makes geometry more, not less, like the way it's been taught since Euclid. This is also how the U of Chicago text presents triangle congruence, and so SSS, SAS, and ASA are all theorems!
In conclusion, let's see what Joyce has to say about transformational geometry, the cornerstone of the Common Core method:
"Chapter 3 is about isometries of the plane. The entire chapter is entirely devoid of logic. How are the theorems proved? 'The Work Together illustrates the two properties summarized in the theorems below. Theorem 3-1: A composition of reflections in two parallel lines is a translation. ...' Moving a bunch of paper figures around in a "work together" does not constitute a justification of a theorem.
"The theorems can be proven once a little actual geometry is presented, but that's not done until the last half of the book. A little honesty is needed here. Why not tell them that the proofs will be postponed until a later chapter? Or that we just don't have time to do the proofs for this chapter. Even better: don't label statements as theorems (like many other unproved statements in the chapter). Like the theorems in chapter 2, those in chapter 3 cannot be proved until after elementary geometry is developed.
"In summary, either this chapter should be inserted in the proper place in the course, or else tossed out entirely. Since there's a lot to learn in geometry, it would be best to toss it out."
When Joyce writes that the theorems of Chapter 3 can't be proved until after geometry is developed, he means after the parallel and congruence results of Chapters 7-8. But Common Core flips this around. Instead of using the traditional parallel and congruence results to prove theorems about reflections and translations, in Common Core we use theorems about reflections and rotations to prove the traditional parallel and congruence results! Joyce suggests that we toss this material out, but Common Core requires that we keep it and treasure it.
And those proofs -- using transformations to prove traditional results -- are the focus of this blog.