Lesson 2-4: Converses (Day 24)

Today I subbed in a middle school English -- you guessed it! -- special ed class. I visited this class a couple of times last year -- most recently in my December 5th post. Three classes have an aide and the other two have a co-teacher (albeit another sub), and so once again there's no "A Day in the Life."

Of the three classes with an aide, two are seventh grade and one is eighth grade. And yes, some of the eighth graders remember me from last year. All classes are working on an article for which they must read, answer questions, and do a current event.

In fact, it's the same article on bullying that I mentioned in my December 5th post. There's one part that strikes me in this article (which is written by a high school student):

"I have seen physical and verbal bullying. When there is physical bullying at my school, the kid is taken out of school for three days. Then he or she comes back and starts bullying again."

Here in California, there is a big debate regarding whether there should be suspensions for "willful defiance" (that is, for bullying the teacher). The pro-suspension side argues that any punishment other than a suspension would be just a slap on the wrist. But, this girl writes that an actual three-day suspension would itself be just a slap on the wrist, since it doesn't deter bullies at all! In other words, she agrees with the pro-punishment side, but to her a suspension doesn't go far enough. (But she never states in the article what exactly she would consider to be an effective punishment.)

By the way, in one of these classes I get to sing a song. Of course, I choose the corrected version of "Meet Me in Pomona," since I wish to sing it before the fair closes this weekend.

Meanwhile, the two co-teaching classes are both eighth grade. This class answers vocab questions on words describing types of government (democracy, monarchy, and so on). This sets them up for a video to watch, called 2081. Since it's on YouTube, I might as well link to it here:

The title refers to a year in the future -- a dystopian future. In this world, the US Constitution (how ironic is it that this video is being played today, September 17th -- Constitution Day) now has over two hundred amendments. Under these new amendments, equality is guaranteed. But equality is enforced by a Handicapper-General, who places handicaps on all individuals so that no one is more intelligent, athletic, or attractive than anyone else. The main character is Harrison Bergeron -- a fugitive who tries to escape being handicapped by the Handicapper.

The concept of equality vs. freedom is a deeply political one. It cuts into other issues currently in the news, including discrimination in New York magnet schools as well as the Ivy League. And we can even tie it to the issues of bullying and suspension. (For example, some people are bullied for being different, so should we enforce 2081-style equality to prevent bullying?) Indeed, I do want to discuss some of this in my next traditionalists' post (scheduled for next week).

Instead, let's hurry and get to Lesson 2-4 on converses. In past years, I would post a few political examples of conditionals and converses. Last year, I started reading Eugenia Cheng's third book on logic during our reading of Lesson 2-4, which emboldened me to include more political examples.

On one hand, going into today I wanted to avoid some of those political (and racial) examples, since I'm supposed to avoid political posts during the school year (except for traditionalists' posts). But who am I kidding? This post has already become political when I mentioned first bullying/suspensions and then equality/2081.

Therefore I'm preserving the political examples (from either our text or Cheng's book) that I included in my Lesson 2-4 post from last year.

Lesson 2-4 of the U of Chicago text is called "Converses." (It appears as Lesson 2-3 in the modern edition of the text.)

This is what I wrote last year about today's lesson. 

There is a little bit of politics near the end of this post, because I'd perceived one of the examples of fallacious reasoning (assuming that a statement and its converse are equivalent) as one often committed by Republicans. So I added a similar fallacy made by Democrats in order for this post to remain politically balanced -- the point being made that both parties are prone to making logical fallacies. But only the example from the text actually appears on the worksheet:

When I wrote this a while ago, I was worried posting too much about politics. But the examples I gave in this post are tame compared to what Cheng writes in Chapter 2 of her book! It all goes to show that the easiest examples of logical vs. illogical statements to find are political.

Lesson 2-4 of the U of Chicago text deals with an important concept in mathematical logic -- converses. We know that every conditional statement has a converse, found by switching the hypothesis and conclusion of that conditional.

The conditional "if a pencil is in my right hand, then it is yellow" has a converse, namely -- "if a pencil is yellow, then it is in my right hand." The original conditional may be true -- suppose all the pencils in my right hand happen to be yellow -- but the converse is false, unless every single yellow pencil in the world happens to be in my right hand. A counterexample to the converse would be a yellow pencil that's on the teacher's desk, or in a student's backpack, or even in my left hand -- anywhere other than my right hand.

If converting statements into if-then form can be confusing for English learners, then writing their converses is even more so. Here's an example from the text:

• Every one of my [Mrs. Wilson's] children shall receive ten percent of my estate.

Converting this into if-then form, it becomes:

• If someone is Mrs. Wilson's child, then he or she shall receive ten percent of the estate.

Now if a student -- especially an English learner -- blindly switches the hypothesis and the conclusion, then the following sentence will result:

• If he or she shall receive ten percent of the estate, then someone is Mrs. Wilson's child.

But this is how the book actually writes the converse:

• If someone receives ten percent of the estate, then that person is Mrs. Wilson's child.

In particular, we must consider the grammatical use of nouns and pronouns. In English, we ordinarily give a noun first, and only then can we have a pronoun referring to that noun. Therefore the hypothesis usually contains a noun, and the conclusion usually contains a pronoun. (Notice that grammarians sometimes refer to the noun to which a pronoun refers as its antecedent -- and of course, the text refers to the hypothesis of a conditional as its antecedent. So the rule is, the antecedent must contain the antecedent.)

And so when we write the converse of a statement, the hypothesis must still contain the noun -- even though the new hypothesis may be the old conclusion that contained a pronoun. So the converse of another conditional from the book:

• If a man has blue eyes, then he weighs over 150 lb.

is:

• If a man weighs over 150 lb., then he has blue eyes.

Saying the converse so that it's grammatical may be natural to a native English speaker, but may be confusing to an English learner.

Now let's look at the questions to see which ones are viable exercises for my image upload. An interesting one is Question 9:

A, B, and C are collinear points.
p: AB + BC = AC
q: B is between A and C

and the students are directed to determine whether p=>q and q=>p are true or false. Now notice that the conditional q=>p is what this text calls the Betweenness Theorem (and what other books call the Segment Addition Postulate). But p=>q is the converse of the Betweenness Theorem -- and the whole point of this chapter is that just because a conditional is true, the converse need not be true. (Notice that many texts that call this the Segment Addition Postulate simply add another postulate stating that the converse is true.)

Now the U of Chicago text does present the converse of the Betweenness Theorem as a theorem, but the problem is that it appears in Lesson 1-9, which we skipped because we wanted to delay the Triangle Inequality until it can be proved.

Here, I'll discuss the proof of this converse. I think that it's important to show the proof on this blog because, as it turns out, many proofs of converses will following the same pattern. As it turns out, one way to prove the converse of a theorem is to combine the forward theorem with a uniqueness statement. After all, the truth of both a statement and its converse often imply uniqueness. Consider the following true statement:

• Donald Trump is currently the President of the United States.

We can write this as a true conditional:

• If a person is Donald Trump, then he is currently the President of the United States.

The converse of this conditional:

• If a person is currently the President of the United States, then he is Donald Trump.

This converse is clearly true as well. By claiming the truth of both the conditional and its converse, we're making a uniqueness statement -- namely that Trump is the only person who is currently the President of the United States.

[2019 update: Currently President Trump is in right here in Southern California. So I can argue that this mention here is a current event, not a pointless political reference.]
So let's prove the converse of the Betweenness Theorem. The converse is written as:

• If A, B, and C are distinct points and AB + BC = AC, then B is on AC.

(I explained why segment AC has a strikethrough back in Lesson 1-8.)

Proof:
Let's let AB = x and BC = y, so that AC = x + y. To do this, we begin with a line and mark off three points on it so that B is between A and C, with AB = x and BC = y. This is possible by the Point-Line-Plane Postulate (the Ruler Postulate). By the forward Betweenness Theorem, AC = x + y.

Now we want to show uniqueness -- that is, that B is the only point that is exactly x units from A and y units from C. We let D be another point that is x units from A, other than B. By the Ruler Postulate again, D can't lie on AC -- it can only be on the same line but the opposite side of A (so that A is between D and C), or else off the line entirely (so that ACD is a triangle).

In the former case, the forward Betweenness Theorem gives us AC + AD = DC. Then the Substitution and Property of Equality give us DC = (x + y) + x or 2x + y, which isn't equal to y (unless 2x = 0 or x = 0, making A and D the same point when they're supposed to be distinct).

In the latter case, with ACD a triangle, we use the Triangle Inequality to derive AD + DC >AC. Then the Substitution Property gives us x + CD > x + y, and then the Subtraction Property of Inequality gives us the statement CD > y, so DC still is not equal to y.

So B is the only point that makes BC equal to y -- and it lies on AC. QED

(The explanation in the book is similar, but this is more formal.) Later on, we're able to use this trick to prove converses of other theorems. So the converse of the Pythagorean Theorem will be proved using the forward Pythagorean Theorem plus a uniqueness statement -- and that statement is SSS, which tells us that there is at most one unique triangle with side lengths a, b, and c up to isometry. And, more importantly, the converse to the what the text calls the Corresponding Angles Postulate (the forward postulate is a Parallel Test, while the converse is a Parallel Consequence) can be proved using forward postulate plus a uniqueness statement. The uniqueness statement turns out to be the uniqueness of a line through a point parallel to a line -- in other words, Playfair's Parallel Postulate. This explains why the forward statement can be proved without Playfair, yet the converse requires it.

In the end, I decided to throw out Question 9 -- I don't want to confuse students by having a statement whose converse requires the Triangle Inequality that we skipped (and the students would likely just assume that the converse is true without proving it) -- and used Question 10 instead.

But now we get to Question 13 -- and here's where the controversy begins. I like to include examples in this lesson that aren't mathematical -- doing so might engage students who are turned off by doing nothing but lifeless math the entire period. This chapter on mathematical logic naturally lends itself to using example outside of mathematics. But the problem is that Question 13 is highly political. Written in conditional form, Question 13 is:

• If a country has communist, then it has socialized medicine.

and its converse is:

• If a country has socialized medicine, then it is communist.

In the end, I will include Question 13 on the worksheet. But I left a space so that if a teacher feels that the question is politically slanted, then he or she could add another question to balance it. For example, that teacher can add a fallacious argument often made by Democrats, for example:

• If a white person is racist, then he or she opposes Obama.
• If a white person opposes Obama, then he or she is racist.

Of course, this question adds a new layer of controversy (race) to the mix. Teachers who want to add a balancing question should just write in their own question, or just throw out the question about socialized medicine altogether.

In the review section, I'd have loved to include Question 15, a review of the last lesson on programming (and of course changed it to TI-BASIC). But in deference to those classes that skipped the lesson because not every student has a graphing calculator, I've thrown it out and included only questions from the fully covered Lessons 2-2, 1-8, and 1-6.