Thursday, August 21, 2014

Section 2-4: Converses (Day 11)

Section 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.


-- 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 Section 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:

-- Barack Obama is currently the President of the United States.

We can write this as a true conditional:

-- If a person is Barack Obama, 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 Barack Obama.

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 Obama is the only person who is currently the President of the United States.

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 Section 1-8.)

Let's let AB = 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 section 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.

Just like the article at the beginning of the chapter (with discussion of "terrorists" before 9/11), the change in political climate since the publishing of this text has rendered the question controversial. For during the past few years, this country has moved in the direction of socialized medicine with the passage of the Affordable Care Act -- better known by its nickname, "Obamacare."

Now some opponents of Obamacare would argue that moving towards socialized medicine and away from a free-market solution is indeed a step towards socialism or communism. But the intended answer of this question is that this reasoning is incorrect, since it assumes that the converse of a statement is true just because the original conditional is true. But some opponents of Obamacare might argue that the converse is true, anyway. Since defenders of Obamacare tend to be Democrats -- that is, of the same party as the administration that passed the law -- and opponents of the law tend to be Republican, the question may be viewed as having a pro-Democratic slant.

Now I wish this blog to be politically neutral. But unfortunately, the problem is that this is a Common Core blog, and Common Core is itself politically charged. Once again, it was a Democratic administration that passed Common Core, and so once again, many Republicans oppose the standards (indeed, one nickname for the standards is "Obamacore"). One argument against Common Core is that it promotes slanted viewpoints that favor the Democrats -- and by including Question 13 on a blog that purports to be a Common Core blog, I'm lending credence to that argument!

Notice that I have a mixed opinion of the Common Core Standards. I write about CCSS not because I wholeheartedly endorse them, but because I want teachers to know know to teach to them. But I do defend Common Core against invalid arguments. For example, the argument that Common Core endorses a slanted view of history is invalid because the standards cover only English and math (and with the upcoming Next Generation Science Standards, science as well), not history.

Now I don't want Question 13 to convince readers that the Common Core encourages a political view slanted towards the Democrats. (Recall that the question comes from a book that was written years before there even was a Common Core, and that the question has a Chinese immigrant -- not an American Republican -- making the fallacious argument.) It is argued by critics that math classes should be teaching nothing but math, not non-mathematical politics. But I want to include non-mathematical topics because these are more likely to engage the students than teaching nothing but math.

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 section on programming (and of course changed it to TI-BASIC). But in deference to those classes that skipped the section because not every student has a graphing calculator, I've thrown it out and included only questions from the fully covered Sections 2-2, 1-8, and 1-6.

No comments:

Post a Comment