Table of Contents
1. Introduction
2. Yule Blog Prompt #2: A Challenge I Faced in 2021
3. Cheng's Art of Logic in an Illogical World, Chapter 4
4. Conclusion
Introduction
I'm naming each of these winter break posts after something special that happens that day. Today is known as Go Caroling Day, presumably because many people go Christmas caroling that day. No, I didn't go caroling today -- indeed, it's been months since I've sung anything in public. (And of course, the first winter break post was my Super Saturday post, named after one of the biggest shopping days of the year -- but no, I didn't actually go shopping that day either.)
In fact, I once often sang math songs in the classroom. But this year, I'm not. So I've really fallen out of the habit of singing.
Yule Blog Prompt #2: A Challenge I Faced in 2021
It goes without saying that a new teaching job brings forth new challenges. Of all the classes I'm teaching this year, the most challenging is my AP Calculus course. And the reason for this is obvious -- I, as the teacher, can define what success is for all my other classes, but for Calculus, success is defined only by passing the AP exam. Nothing I do in that class matters if I can't get my students to pass the big test coming up in May.
It's not as if I have a large class. In fact, the class has only six students -- four girls and two guys. But it's making sure that these kids are well-prepared for the exam that's challenging. I must cover the first seven chapters of our text for the AP exam. So far I taught the first three chapters, leaving me with Chapters 4-7 for the second semester.
One girl in particular was concerned with the way that I was teaching the class. This was due to a sequence of events, starting back with Chapter 2 on limits. I didn't give enough examples for the students to understand and do well on the Chapter 2 Test. While all of this was going on, this girl was absent for the assessment, and so she made it up on the day the others had their retake -- but then she struggled and wanted a retake herself, which I didn't give her as I really wanted to move on.
My school is tiny -- in fact, I'm one of only two math teachers at the school. The other math teacher covers Integrated Math and Pre-Calculus. She's been at our magnet since it opened five or six years ago, which automatically makes her the Department Chair. More often than not, I see the two of us as partners, as we're working together to educate all of our students in math.
But there's a much more experienced math teacher in our district -- and he teaches Calculus to boot. He has worked at the main high school of our district for over two decades. And so, after receiving complaints from my students about my Calculus class, the principal sent me to the flagship high school on November 17th (Day 67) to observe how this veteran teacher works with his students.
The experienced teacher runs his class much differently from me. He starts his class by either giving a weekly quiz, or going over such a quiz. Then he spends plenty of time going over homework -- his assignments are very long (50+ problems), and they aren't due until he's answered all of the students questions (which could take all week). Then he gives the main lesson, and finally there might be some sort of group discussion. During a 90-minute block period, he often spends a half-hour each on the homework and the main lesson.
Some of these ideas are radically different from the way I teach. But the idea of asking each student to name a problem he or she needs help on and then spending lots of time going over those problems -- hey, that makes sense in any math class. And so I'm definitely considering adopting this idea in my own Calculus class going forward.
Last week I gave the final exam of 30 multiple-choice questions -- twenty without a calculator, and ten calculator active. This is to simulate the real AP exam. Unfortunately, the most common score was 19 out of 30. Two of my kids earned higher scores -- one guy who struggled for most of the semester somehow got one more question right, 20 out of 30, while one girl who was absent for a long time and asked for extra help while another class was taking its final scored 24 out of 30.
Many math students have trouble remembering stuff for the final, so even students who get A's and B's throughout the year might end up with D's on the final exam. But this isn't acceptable in an AP class, where students must remember material for May. If I can't get my students ready to take exams like this, then those D's will turn into 1's and 2's on the AP exam.
And so working with my Calculus students was my biggest challenge for 2021 -- and it will continue to be a challenge through May.
Cheng's Art of Logic in an Illogical World, Chapter 4
Chapter 4 of Eugenia Cheng's The Art of Logic in an Illogical World, "Opposites and Falsehoods," begins as follows:
"There are only two debates I remember from the debating club at school. One was 'This house believes that Margaret Thatcher should go," which was particularly memorable because she actually resigned the morning of the debate. The other one was 'The house believes that strawberries are better than raspberries.'"
Again, I remind you that Cheng writes about race and politics throughout her book. If you prefer not to read this, then I suggest that you avoid this blog for the next three weeks and skip all posts that have the "Eugenia Cheng" label.
Debating is all about proving an opponent to be wrong. Of this, Cheng writes:
"Logic, mathematics and science are all ways of finding out what is true. But they are also ways of finding out what is not true. Negation is how we argue against things."
The author tells us that there are two ways to argue against the view that the Asian education system is better than the British (or American) one:
- Measured and calm: I don't think the Asian education system is better.
- Extreme and excited: No way! The British education system is better!
According to Cheng, we might think of the second statement as the "opposite" of the original claim (which some traditionalists have argued for in the past). But the first statement is more like the "negation" of the original. To form a negative, we simply declare that it is not true. Here are some more of her examples:
Original statement: I think the EU is fantastic.
Opposite: I think the EU is terrible.
Negation: I do not think the EU is fantastic. (It could be terrible, or it could be in between.)
Original statement: Margaret Thatcher was the greatest Prime Minister.
Opposite: Margaret Thatcher was the worst Prime Minister.
Negation: Margaret Thatcher was not the greatest Prime Minister. (She could be in between.)
Original statement: Climate change is definitely real.
Opposite: Climate change is definitely fake.
Negation: Climate change is not definitely real. (But it's the mainstream scientific view.)
Original statement: Sugar is good for you.
Opposite: Sugar is bad for you.
Negation: Sugar is not good for you. (It could be OK in small amounts.)
Original statement: I am male.
Opposite: I am female.
Negation: I am not male. (I could be intersex.)
Original statement: Barack Obama is black.
Opposite: Barack Obama is white.
Negation: Barack Obama is not black. (He is mixed-race.)
Cheng tells us that considering opposites rather than negations ignores the gray areas in life:
"People are not very good at dealing with gray areas, and in fact nor is logic. We'll come back to this later, but for now it's important to note that the gray area should be included somewhere, otherwise we're just ignoring part of reality."
According to the author, ignoring the gray area is formally called the Law of the Excluded Middle:
"It doesn't mean we've excluded the middle in the sense of throwing it away or ignoring it, it just means we've included it with one side or another so that there is effectively no middle any more."
At this point, Cheng draws some diagrams. She shows us a continuum of colors from white to black and asks us where to draw the line. We could draw the line between "black" and "not black," or between "white" and "not white." There could be three sections "white," "black," and "lost middle," or even two equal halves "white(ish)" and "black(ish)." Which division we choose depends on the context -- especially if "white" and "black" aren't merely colors, but races.
Cheng tells us that if we divide the continuum into "white," "gray," and "black," then there are two dividing lines that we draw:
"This is to some degree what happens with the terminology of 'heterosexual,' 'homosexual,' and 'bisexual.'"
The author will return to this idea later, but she warns us:
"Absorbing the gray area into one side or the other is a simplification, but at least not incorrect or contradictory. Whereas denying its existence altogether is where black and white thinking usually goes wrong."
Now Cheng draws some Venn diagrams, which I must describe to you. In her first diagram, the universe of people contains only one circle, representing white people. Then the complement of this set consists of all non-white people. In general, the region where A is not true is the outside of the circle representing A.
The whole point here is that it's possible for show a gray area in these diagrams. The interior of circle A is white and its distant exterior is black, but there is a ring of gray surrounding the white, thereby separating it from the black.
The author proceeds to describe truth values -- here 0 means "false" and 1 means "true":
"You might think mathematicians just love turning things into numbers, but remember, math isn't just about numbers, it's about many other things too. However, numbers are so familiar and easy to reason with that if we can turn a situation into some numbers it can be very helpful."
Cheng now presents the truth table for "not" or negation, which is easy to render in ASCII:
A not A
1 0
0 1
But, as she reminds us, there are some statements whose truth value is currently unknown:
- The universe is finite.
- One day we will be able to cure all cancer.
- A meteor caused the extinction of the dinosaurs.
Cheng asks, how do we negate an implication? So far, we don't know any way to do so other than just to say "does not imply." In her example, the negation of "If you have privilege then you are white" is "Having privilege does not imply that you are white." In symbols we just sort of cross out the implication arrow, but this looks a bit awkward in ASCII, where we must write:
A =/> B (or maybe A =/=> B).
Cheng proceeds with faulty implications. Here is her first example:
"Some black people are better off than me, therefore I don't have white privilege."
And here is a proposed proof:
- Some black people are better off than me even though I am white.
- If some black people are better off than you then you don't have white privilege.
- Therefore I do not have white privilege.
Cheng refers to this as modus ponens:
1. A is true.
2. A implies B.
3. Therefore B is true.
Actually, modus ponens appears in some Geometry texts, including some where it's known by a non-Latin name, "Law of Detachment." To refute a proof that uses modus ponens, we must refute either step 1 or step 2, since as soon as we have both, 3 follows. In this case, Cheng refutes step 2 as a straw man argument.
Cheng now discusses the contrapositive. Here are her examples -- from the above straw man argument, we have:
"If some black people are better off than a white person then that person does not have white privilege."
"If you have white privilege than you are better off than all black people."
If you travel abroad you must have a passport.
If you don't have a passport you can't travel abroad.
These are contrapositive pairs. A statement and its contrapositive are logically equivalent:
A => B
B is false => A is false (contrapositive)
B => A (converse)
A is false => B is false (contrapositive of the converse -- called "inverse")
Cheng shows all the possibilities in a chart, which I don't reproduce here. (Again, her book shows all the combinations involving converses, inverses, and contrapositives.)
Confusing a statement with its inverse is a common fallacy:
If you are a US citizen then you can legally live in the US.
If you are not a US citizen then you can't legally live in the US (not equivalent).
Also, a contrapositive is not to be confused with the negation:
A =/> B
Cheng writes about evidence. For example, some people might believe:
Being of Chinese origin implies being good at math.
and claim every Chinese-looking math person (like Cheng herself) as evidence. But this statement is equivalent to its contrapositive:
Not being good at math implies not being Chinese.
and every non-Chinese non-mathematician qualifies as evidence -- for example, a Canadian goose or an American singer.
Cheng's final example takes us back to science. Back in her own school days, her class performed an experiment to test Hooke's law of springs. The hypothesis is: the extension of a spring is proportional to the load that is hanging from it. Given the right kind of data, we might conclude:
There is sufficient evidence to suggest that this hypothesis is true, to within 95 percent certainty (for example).
(And yes, I've told my Stats students about how 95% is the gold standard -- anything falling outside qualifies as unusual.)
And if we don't get the correct date, we conclude its negation:
There is not sufficient evidence to suggest that this hypothesis is true, to within 95 percent certainty.
So we might refine the hypothesis as follows:
The extension of a spring is proportional to the load hanging from it, within a maximum load limit.
And this is Hooke's law of springs. Cheng points out that a "law" is something that has been determined to be probably true, to within levels of certainty accepted by science.
This leads to Cheng's conclusion of the chapter -- a glib view of scientists and statisticians:
"My excellent math teacher, Mr. Muddle, taught us that when you work as a professional statistician, if you do not have the right data to support your hypothesis the correct negation is 'There is insufficient evidence to support this hypothesis and therefore we need more funding in order to pursue the matter further.'"
Conclusion
Well, that's enough for this post. Hmm, maybe I should go Christmas caroling on this special day -- or maybe I'll just sing around the house. I definitely won't be singing in the classroom at all.
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