Lesson 3-2: Types of Angles (Day 32)

Today I subbed in a high school ROP digital film class. I wonder whether this counts as another sub revisit, since I was in another digital film class at the same school back on February 23rd. (As usual, you can refer to my February 23rd post for more information about that day.)

But this class appears to be different from the February visit. The teacher for this class is different (but the teacher from February is no longer listed on the school website.) Moreover, in second period, the students create the announcements video (to air during third period), while in February, no video was created for morning announcements.

This is the same school where I subbed for art two weeks ago, and the schedules for both the art and film teachers are similar. Both teachers have their most advanced class during second period, then the next advanced class during third period, and then the regular classes from fourth period on. Of course I won't do "A Day in the Life" for today, but I will describe some of the classes.

Naturally the two best classes of the day are the advanced classes. I actually put third period a little ahead of second period. In this class, one senior girl takes over as a student TA. Sometimes, even having a senior TA is as helpful as having another teacher in charge. She knows exactly what the students should be working on in third period, and so she redirects them back on task.

This reminds me much of the project that the coding teacher tried to do with the -- oops, ixnay on the arterchay odingcay! (Wow, so many days of subbing remind me of the year that I promised not to mention during the Eugenia Cheng reading!)

Meanwhile, In the regular classes, the students must study quietly for 15 minutes, and then I get to show them a movie of my choosing. Three of the four regular classes have TA's, but in this situation they are mostly irrelevant. This marks my only real classroom management of the day. I decide that sixth period is the best class of the day. This class is right after lunch, and so I must enforce my no restroom rule until midway through the period. (I send my attendance roster with the TA, so there's no loophole here.) But other than that, these students do work silently for the 15 minutes. A few students do try to whisper or talk in the other classes.

In seventh period two girls decide to study Geometry. They are completing a worksheet based on Lesson 2-4 of the Glencoe text, "Deductive Reasoning." This lesson fits with the logic that we're currently reading about in Eugenia Cheng's book.

The regular teacher directs me to access his iTunes account and choose a movie -- the students aren't allowed to choose. I decide to go to the Action/Adventure section and choose a random number (on my TI-83) between 1 and 91, since there are 91 movies listed there. If I choose an R-rated movie, I make another selection so that the rating is PG-13 or lower.

Here are the movies I randomly select today:

4th period: Dawn of the Planet of the Apes (PG-13)
5th period: Now You See Me (PG-13)
6th period: Chronicles of Narnia: The Lion, the Witch, and the Wardrobe (PG)
7th period: Kong: Skull Island (PG-13)

Chapter 10 of Eugenia Cheng's The Art of Logic in an Illogical World, "Where Logic Can't Help Us," begins as follows:

"Cardiac surgeon Stephen Westaby writes in Fragile Lives about the fact that if the heart stops, the brain and nervous system will be damaged in less than five minutes. So he often had five minutes or less to decide how to perform surgery."

As the title implies, there are just some situations where logic just can't help us. Clearly, surgeons don't always have enough time to do a full logical analysis. Cheng writes:

"It's important to understand how far logic gets us and where emotions have to help, rather than pretend that logic can get us all the way there. But we'll start by thinking about where logic can start to kick in, which is only after some help with finding starting points."

Cheng's first example involves language. She points out that there's no logical reason, for example, why the word for "cat" is, well, "cat." Indeed, she contrasts this with her native language:

"'Cat' in Cantonese is a high-pitched 'mow' (rhyming with cow), which does sound quite like the sound a cat makes. More than 'cat' does, anyway."

Of course, English speakers would render the sound a cat makes as "meow." Anyway, young infants learning how to speak the language must figure out for themselves what a "cat" is, since for most words there's no logical connection between the word and the object -- again, except for certain imitative words like English "cuckoo" and Chinese "mow."

The author moves on to discuss flashes of inspiration. She writes:

"You might argue whether they really exist or not, but I have definitely had moments that I would describe it like that. Perhaps it would be less melodramatic to call them an 'idea.'"

Cheng tells us that art and music are all about such flashes of inspiration. Only occasionally would a musician use "logic" to compose a song:

"Some composers, famously Bach and Schoenberg, use symmetry to transform parts of their composition into new but related music."

And indeed some Common Core Geometry texts mention this in the transformations unit -- in particular, this appears in Lesson 4-8 of the U of Chicago text (Third Edition).

According to the author, even math often begins with flashes of inspiration:

"Once we have had the idea we proceed using logic, but that part comes afterwards, when we test and exhibit the robustness of our idea."

Having shown us where logic begins, Cheng now proceeds to tell us where logic ends. She presents us the following menu:

Marinated Roast Breast of Chicken -- $18.50
Pan Fried Ostrich -- $21.00
Char Grilled Fillet Steak -- $26.95
Smoked Haddock Fish Cake -- $16.95
Roasted Summer Vegetable Tart -- $17.95

Cheng explains:

"Perhaps you've decided that you can't spend more than $20, and also you don't like fish. This logically narrows down your options to the chicken and vegetable tart, but beyond that, logic can't tell you anything."

And of course, there isn't always time to think logically in an emergency:

"If there is a fire you will, I hope, have an instinctive reaction 'I must get out!' If this is an instantaneous reaction, it probably wasn't exactly processed logically."

But Cheng tries to write it out logically anyway, as follows:

A: There is a fire.
X: I must get out.

It might go like this:

A is true (there is a fire).
A implies X (If there is a fire I must get out)
Therefore X (I must get out) by modus ponens.
\
The author also inserts additional steps in case we need to explain this to a child:

Let A = There is a fire.
Let B = I stay here.
Let C = I burn.

Then we have:

A is true.
A and B implies C.
C implies bad.
Therefore I must make sure B is false, i.e. I must get out.

On the other hand I think we can all agree that this would not be a logical deduction:

There is a fire.
I will stay here.

(But replace "fire" with "hurricane" and consider recent news events. Google: refuse to evacuate.)

Cheng continues to write about how insufficient time leads to insufficient information to make a logical decision:

"This can happen in an emergency but it can also happen in sport, where the trajectory of a ball is in principle entirely governed by physics, but we can't take all the necessary measurements in time to do the calculation before needing to hit the ball."

And she adds:

"Any consequences involving human reactions to things are almost certain to be guesses about human behavior rather than logical conclusions."

This applies to both economics and voting.

In fact, there's one very famous conundrum in which insufficient information leads directly to people making an illogical decision -- the prisoner's dilemma. David Kung, in his thirteenth lecture, discusses the prisoner's dilemma, but I only wrote a little about it in January 2016 (only because my post that day was jam-packed with other information).

So I'll try to explain it better using Cheng's book today. Alex and Sam are two prisoners, and the prosecutor asks each of them separately to testify against the other:

• If neither testifies, each get a year in jail.
• If both testify, each gets five years.
• If exactly one testifies, the snitch goes free and the other gets ten years.

Imagine you are Alex. At first you might considering staying quiet:

"What if Sam actually lands you in it and goes free? Then it would be safer for you to testify as well, to protect against that possibility."

Of course, Sam is thinking the same thing. Thus both confess and each gets five years.

Cheng points out that this is counterintuitive. And so she comes up with a clarifying example by assuming that there are more than two people involved. (I've done the same in the past with the Monty Hall paradox -- imagine that there are more than three doors.)

• If no one betrays the group, each gets $500.
• If more than one betrays the group, each traitor gets $0 and everyone else loses $1000.
• If exactly one betrays the group, that traitor gets $1000 and everyone else loses $1000.

The author explains:

"If the only other person involved is your best friend then hopefully you know and trust each other enough to know that you won't denounce each other and you'll go home with $500 each. However, imagine doing this with a group of 100 strangers."

Of course, most people will be afraid that someone will betray the group, and hence they'll betray the group as well. She writes:

"This shows us that in all scenarios for other people's behavior, you get a better outcome if you betray. In game theory this is called a dominant strategy and the logic says that this is the strategy that you should take for the best outcome in either scenario."

(Kung uses another name for "dominant strategy" -- the "Nash equilibrium," named after the mathematician featured in the film A Beautiful Mind, John Nash.)

Cheng explains that climate agreements, such as the Paris Climate Agreement, work similarly to the Prisoner's Dilemma:

• If no one defects, there is some cost but the benefit is global.
• If everyone defects, the effect on the world could be drastic.
• If exactly one country defects, then that country benefits the most -- they reap the benefits of conservation without any of the costs.

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 two weeks and skip all posts that have the "Eugenia Cheng" label.

(This is even though Cheng never names the country which would defect from the agreement.)

"Now according to the logic of the prisoner's dilemma, we should expect everyone to defect. It is perhaps heartening that this isn't universally the case."

As the author explains, it's all about trust:

"I think what this is actually saying is that if a community is infused with enough trust to act as a coherent whole rather than as a collection of selfish individuals, then the logic of the situation changes, and becomes one that can benefit everyone rather than everyone suffering as a result of a few selfish individuals."

This is what little I wrote in January 2016 about the Prisoner's Dilemma (including a link):

Prisoner's Dilemma

I claim that the Prisoner's Dilemma is one of the most commonly played games by the teenagers in our classes -- except it's the Lover's Dilemma:

• If I like my friend, and my friend likes me back, then we can start a relationship.
• If I don't like my friend that way, nor does my friend like me, then we don't start a relationship, but at least we can remain friends.
• If I like my friend, but my friend doesn't like me back, then our friendship is ruined. My (former) friend gets to laugh at me -- the best of all situations for my (ex-)friend, the worst for me.

Then the Nash equilibrium is for neither lover to confess his/her true feelings. This explains why two teens may like each other, but so often neither one admits it. As Kung points out at the end of the lecture, these simple games tell us a little more about how the world works.

And Cheng concludes the chapter similarly:

"We have seen that logic cannot explain and decide everything in the world, so we are going to have to do something when it runs out. We should not pretend that those non-logical things are logical, but we should also not assume that those non-logical things are bad."

I've decided to add the "traditionalists" label to this thread. This is because traditionalist Barry Garelick posted today. Even though this post isn't directly about the Barbara Oakley article, Garelick mentions Oakley in passing, so this post can be considered part of the extended Oakley discussion:

https://traditionalmath.wordpress.com/2018/09/29/3099/

(Again, notice that this post is dated September 29th, yet it was visible on my computer in time for my own post dated the 28th. And this is despite Garelick's living right here in California. I'm wondering whether the Wordpress clock is now suddenly defaulting to Greenwich.)

The Kansas City Star published a column in defense of Common Core’s math standards, containing the usual rhetoric–to wit:

I was recently part of a conversation about education. It was a social media conversation, intended to bash the alternative strategies of teaching math.

The strategies have been caught up in the term “common core,” but are actually teaching methods designed to help kids reach common standards.

I offered an alternative viewpoint to the woman’s outrage. I was once told by the faculty at my kids’ school that these learning strategies aren’t designed just to teach the material, they teach the kids to learn. How to analyze. How to understand why math works, not just how to solve a problem.

And Garelick responds to Emily Parnell (the KC Star columnist) by citing Oakley (and linking to her infamous article):

Barbara Oakley, a professor of engineering at Oakland University in Michigan, wrote a book called “Learning how to Learn”  in which she describes techniques one can use to succeed in difficult subjects. She does not hold memorization in disdain, nor learning facts, nor practice. Recently she wrote an op-ed that appeared in the New York Times about the value of practice and memorization in becoming proficient in math–and was castigated in comments that followed as well as in blogs for what was characterized as narrow-mindedness and resistance to innovation in education. The criticisms bore a resemblance to the Kansas City Star columnist’s view of having to deal with the great unwashed.

Pot, kettle, black -- to me the ones who don't want to deal with "the great unwashed" are, of course, the traditionalists. Which side, after all, is promoting tracking to keep "the great unwashed" out of the highest tracks?

Once again, I agree that traditional memorization and p-sets can be beneficial, provided that the students actually do the memorization or the p-sets. Even professional mathematician Eugenia Cheng doesn't like to memorize -- how much less, then, do the students in our classes like to memorize?

I'm glad that I labeled this post "traditionalists," because today's Lesson 3-2 will contain an activity that's definitely an alternative to traditional p-sets.

Lesson 3-2 of the U of Chicago text is called "Types of Angles." In this chapter, students learn about zero, acute, right, obtuse, straight, complementary, supplementary, adjacent, and vertical angles.

In the new Third Edition of the text, this actually corresponds to Lesson 3-3. But the definitions of acute, right, and obtuse are actually combined with yesterday's Lesson 3-1. Only the last four definitions (mainly adjacent and vertical angles) remain in the new Lesson 3-3.

(By the way, Lesson 1-4 of the Glencoe text, which I covered in the special ed class that I subbed for last Friday, is most analogous to Lesson 3-1 of the U of Chicago Third Edition. Last week's students learned all about angles, including acute, right, and obtuse angles as well as bisectors, in that lesson.)

In between these, in the new Lesson 3-2, are rotations. I've mentioned before how strange is this that both the old and new editions define a rotation as the composite of two reflections in intersecting lines, yet the new edition has a section on rotations before defining reflections! The U of Chicago most likely placed this section here so that in introducing rotations, students become more familiar with angles. (Again, I point out that Hung-Hsi Wu of Berkeley, in his recommendations for Common Core Geometry, teaches rotations before reflections, but he defines rotations differently. His lessons have nothing to do with the new Lesson 3-2.)

In fact, Jackie Stone -- a Blaugust participant last month -- also introduces rotations when teaching her students about angles, just like the U of Chicago text:

https://mathedjax.wordpress.com/2017/08/23/what-is-that-how-do-you-use-it-blaugust/

What was intended to be a five minute “review” of these skills to launch into the real lesson activity of the day turned into a much more in depth “teaching” of how to use this tool.  Although they might NEVER use a protractor outside of my class again I do find the task of measuring something using a tool useful. The task also spoke to the CCSS Math Practice Standards of attending to precision and using tools strategically. It is so challenging (especially at the beginning of the year) to determine what are appropriate scaffolds to help students work on a task. Moving forward, I plan to assume less which is actually a good thing because then we can talk about refined meanings of things.  For instance, because of their lack of background we were able to really talk about that the measurement in degrees was actually a measurement of a rotation. I think next year my approach might be different.

This is what I wrote last year on this lesson:

Section 3-2 of the U of Chicago text discusses the various types of angles. It covers both the classification of angles by their measures -- acute, right, and obtuse -- as well as related angles such as vertical angles and those that form a linear pair. Complementary and supplementary angles also appear in this lesson.

As usual, I like to look for other online resources by other geometry teachers. And I found the following blog post from Lisa Bejarano, a high school geometry teacher from Colorado who calls herself the "crazy math teacher lady":

http://crazymathteacherlady.wordpress.com/2014/08/23/

Bejarano writes that in her geometry class, she "starts with students defining many key terms so that we can use this vocabulary as we work through the content." And this current lesson, Section 3-2, contains many vocabulary terms. So this lesson is the perfect time to follow Bejarano's suggestion.

Now two textbooks are mentioned at the above link. One is published by Kagan, and unfortunately I'm not familiar with this book. But I definitely know about the other one -- indeed, I actually mentioned it in my very last post -- Michael Serra's Discovering Geometry.

The cornerstone to Bejarano's lesson is the concept of a Frayer model. Named after the late 20th-century Wisconsin educator Dorothy Frayer, the model directs students to distinguish between the examples and the non-examples of a vocabulary word.

Strictly speaking, I included a Frayer-like model in last week's Section 2-7. This is because the U of Chicago text places a strong emphasis on what is and isn't a polygon. But Serra's text uses examples and non-examples for many terms in its Section 2-3, which corresponds roughly to Section 3-2 in the U of Chicago text.

Here are steps used in Bejarano's implementation of this lesson:

1. I used the widget example from Discovering Geometry (chapter 1). It shows strange blobs and says “these are widgets”, then there is another group of strange blobs and it says “these are not widgets”. I have students define widgets in their groups. Then they read their definition and we try to draw a counterexample. Then we discussed what makes a good definition and we were ready to go!

(Note: Bejarano writes that she found the "widget" example in Chapter 1 of Serra's text, but I found it in Section 2-3 of my copy. Of course, I don't know how old my text is compared to Bejarano's -- mine is the Second Edition of the text, dated 1997. Hers could be the Third Edition or later.)

Notice that "widget" here is a non-mathematical (and indeed, a hypothetical) example that Bejarano uses as an Anticipatory Set. As we've seen throughout Chapter 2, many texts use non-mathematical examples to motivate the students.

3. Students worked in small groups with their 3 terms copying the examples & non-examples, then writing good definitions for each term. I set a timer for 10 minutes. (Yes, I'm skipping her #2.)

Now I don't necessarily want to include this as a group activity the way Bejarano does here. After all, I included the Daffynition Game in my last post as a group project [...]

And let's stop right here, because today's an activity day, and I haven't posted that Daffynition game yet this year -- so let me post it today! This is what I wrote last year about the activity. (Oh, and if you thought we were done with Serra's text after finishing Chapter 0 last month, think again!):

It's tough trying to find activities that fit this chapter. One source that I like to use for activities is Michael Serra's textbook, Discovering Geometry. Just like most other geometry texts, in Chapter 2 he discusses the concept of definition (Section 2-3, "What is a Widget?") Then the text introduces a project, "the Daffynition Game," where the students take turns making up definitions to real words.

A few comments I'd like to make about the game as introduced by Serra. Step 3 reads:

3. To begin a round, the selector finds a strange new word in the dictionary. It must be a word that nobody in the group knows. (If you know the word, you should say so. The selector should then pick a new word.)

The problem is that this depends on the honor system -- how do we know that a student who knows the word will actually admit it? Rather than depend on the students' honesty, why not make knowing the word an actual strategic move? That student will then earn a point for knowing the word -- and the student can still make up a fake definition in order to earn even more points? This means that the selector must be very careful to choose a word that isn't in the dictionary.

Another question is, what affect would this project have on the English learners? I'd say that this would be a great project for them, since they can learn both English and math in this lesson. But English learners might be at a disadvantage in this game, since if they choose a word that a native English speaker knows, the English speaker would earn a point (since I'm not counting on the honor system here). One debate that always comes up in a group activity is whether to group homogeneously or heterogeneously. For this project, it may be best to group homogeneously, but by English, rather than mathematical, ability.

Finally, this project requires students to look up words in a dictionary -- but what dictionary? I threw out the problems in earlier sections that depended on the availability of a dictionary. Perhaps the night before this activity, part of the homework assignment could be to look up a word in the dictionary and write down its definition -- but that assumes that the students will actually do the homework, and besides, there's no guarantee that the students have access to a dictionary at home (or online) either.

My solution is for the teacher to have enough index cards with words and definitions on them. Therefore the selector chooses an index card, not a word from the dictionary. Indeed, the teacher can give an index card to each student even before dividing the class into groups! But the selector should still follow the other steps as originally written in the Serra text.

OK, so let me post the worksheets. I decided to post only the first page of Lesson 3-2 (Lisa Bejarano's lesson) and then go directly to the Daffynition Game.

In cutting Bejarano's second page, I'm dropping some terms that don't appear until later in Chapter 3, but I also dropped "vertical angles" and "angle bisector," which do appear in Lesson 3-2. Teachers can either make sure to write those two dropped terms on index cards in the Daffynition Game, or else go full Bejarano and use the Frayer models as a full group project, just as the Colorado teacher originally intended.

[2018 update: Jackie Stone still posts weekly on her blog. Lisa Bejarano hasn't posted since April.]

Lesson 3-1: Angles and Their Measures (Day 31)

Today on her Mathematics Calendar 2018, Theoni Pappas writes:

Determine Triangle BCD's area.

(Here is some given info from the diagram: ABD and BCD are right angles, AC = 4, BC = 6.)

Here we use the Right Triangle Altitude Theorem of Lesson 14-2:

In a right triangle:
a. the altitude to the hypotenuse is the geometric mean of the segments into which it divides the hypotenuse.
(We don't need part b here.)

This tells us that BC is the geometric mean between AC and DC:

BC = sqrt(AC * DC)
6 = sqrt(4DC)
36 = 4DC
DC = 9.

Now DC is the base of Triangle BCD while BC = 6 is its height. From Lesson 8-5:

A = (1/2)hb
A = (1/2)(6)(9)
A = 3(9)
A = 27.

Therefore the area of Triangle BCD is 27 square units -- and of course, today's date is the 27th.

Chapter 9 of Eugenia Cheng's The Art of Logic in an Illogical World is called "Paradoxes." Here's how it begins:

"I am an avid writer of to do lists. I find it an excellent way to procrastinate in a mildly useful way. Sometimes if I'm feeling particularly tired or stressed I will put some very easy things on my to do list so that I can easily declare that I've achieved something."

This chapter is all about paradoxes. Recall that we've already looked at paradoxes during a previous side-along reading (or side-along DVD viewing) -- David Kung's lectures on paradoxes. This was back in January 2016. So I might be referring back to posts from that month in today's blog entry.

What does Cheng's to do list have to do with paradoxes? Well, Cheng wonders what would happen if one entry on her to do list is "Do something on this list." Can she then immediately cross it off? She also mention another real paradox in her life -- to apply for a visa application, she had to enter her name exactly as written on her passport, but the online application won't accept her hyphenated middle name. So of the two commands "fill in your name exactly as written on your passport" and "only alphabetic characters are allowed," she can obey one or the other, but not both. She writes:

"I think of these loops and contradictions as paradoxes of life. Paradoxes occur when logic contradicts itself or when logic contradicts intuition."

Cheng begins with the liar paradox -- "I'm lying!" It can also be written as:

1. The following statement is true.
2. The previous statement is false.

David Kung also mentions the liar paradox in his very first lecture. Back in my New Year's Day 2016 post, I wrote:

-- The Liar Paradox: "This sentence is false."

And indeed, the title of that first lecture is "Everything in this lecture is false." Here is another example given by Cheng:

Cette phrase en francais est difficile a traduire en anglais.

which can be translated literally as:

This sentence in French is difficult to translate into English.

but it no longer makes sense.

Cheng's next paradox is called Carroll's paradox, named after British author Lewis Carroll. I don't think that Kung ever mentions Carroll's paradox in his lectures -- I've mentioned Carroll on the blog before (most recently in my Thanksgiving post), but never in connection to Kung. So this is a new paradox for us.

Carroll's story is called "What the Tortoise Said to Achilles." The titular tortoise asks the ancient Greek hero to show that a triangle is isosceles by measuring his sides, and he does:

A: Both sides of the triangle equal the length of 5 cm.

Z: Both sides of the triangle equal each other.

The tortoise asks "Does Z follow from A?" Achilles replies "Yes," so the tortoise adds it to the list, since it's a new assumption that we're making:

A: Both sides of the triangle equal the length of 5 cm.

B: A implies Z.

Z: Both sides of the triangle equal each other.

The tortoise asks "Do A and B imply Z?" Achilles replies "Yes," so the tortoise adds it to the list, since it's a new assumption that we're making:

A: Both sides of the triangle equal the length of 5 cm.

B: A implies Z.

C: A and B imply Z.

Z: Both sides of the triangle equal each other.

The tortoise asks "Do A, B, and C imply Z?" Achilles replies "Yes," so the tortoise adds it to the list, since it's a new assumption that we're making:

A: Both sides of the triangle equal the length of 5 cm.

B: A implies Z.

C: A and B imply Z.

D: A, B, and C together imply Z.

Z: Both sides of the triangle equal each other.

And the tortoise tortures Achilles with these statements ad infinitum. Cheng tells us that the only real escape is to use the rule of inference, modus ponens. It's modus ponens that allows us to conclude Z from the earlier statements.

The author repeats her example from earlier about breaking glass:

A: I dropped the glass.

B: A implies Z because the glass was too fragile.

C: A and B imply Z because the floor was too hard.

D: A, B, and C together imply Z because gravity intervened.

E: A, B, C, and D together imply Z because I didn't catch the glass.

F: A, B, C, D, and E together imply Z because nobody else caught the glass.

G: A, B, C, D, E, and F together imply Z because...

Z: The glass broke.

Cheng tells us that Carroll's choice of a tortoise and Achilles as characters goes back to more famous paradoxes -- Zeno's paradoxes. Kung mentions Zeno's paradoxes in his fifth lecture. Cheng writes about the race between the tortoise and Achilles, in which the reptile gets a head start:

"But then Zeno argues like this: by the time Achilles gets to the place where the tortoise started, the tortoise will have moved forwards a bit, say to point B. By the time Achilles gets to point B, the tortoise will have moved forwards a bit, say to point C."

Hey, what am I doing typing out Zeno's paradoxes in full? This is what I wrote in January 2016 about Zeno's paradoxes -- including a link to all three paradoxes:

Zeno's Paradoxes are so well-known that it's easy to find links to them. The following link mentions the first two of them:

http://platonicrealms.com/encyclopedia/zenos-paradox-of-the-tortoise-and-achilles

When Kung gives the story of Achilles and the tortoise, he doesn't give any specific numbers, but the above link does. We say that Achilles is running at 10 m/s (about the same speed as an Olympian sprinter like Usain Bolt) and the tortoise can only walk at 1 m/s. So how long does it take for Achilles to catch up to the tortoise? At first the answer may appear to be one second since that's how long it takes for Achilles to run 10 meters, but in that second, the tortoise has moved up one meter. And then Achilles can cover that meter in 0.1 second -- but by then, the tortoise has moved another 10 cm. And so on -- and that is Zeno's first paradox.

Returning to Cheng, she writes that a falsidical paradox is one where a fault of logic has been hidden in the argument:

"Zeno's paradoxes are falsidical paradoxes: the error is in the logic, not in our intuition about the world. The error is very subtle though, and it took mathematicians a couple of thousand years to work out how to correct it."

And as Kung tells us in his lectures, those corrections are now known as calculus.

Cheng's next example is related to the infinite sum

1 + 2 + 3 + ...

She tells us that according to a Numberphile video, the sum is -1/12. I've linked to Numberphile myself on the blog in the past, and so it's no effort for me to bring up the relevant video:

Cheng explains:

"I hope you feel that the end result is absurd, not least because all the numbers we're adding get bigger and bigger to infinity. Indeed this is why the infinite sum cannot be said to have a sensible answer without substantial qualification."

And she adds:

"Unfortunately, the video fooled millions millions of people, partly because of the good reputation of Numberphile videos in general. It is perhaps a case in point about memes being popular and believable even if they contradict both logic and intuition."

Back on January 13th, Mathologer refutes the Numberphile video. I assume Cheng would approve of the contents of this video:

But again, Numberphile videos are usually spot on. Indeed, here's a reputable Numberphile video about Zeno's paradoxes:

Cheng's next paradox is Hilbert's paradox. I don't need to type this us again -- Kung mentions it in his sixth lecture, and Cheng herself wrote about it in her second book Beyond Infinity. Let me cut-and-paste from the Kung posts of January 2016, which in turn contains a link to the full paradox

Kung begins his lecture by discussing the Hotel Infinity -- a hotel with infinitely many rooms. He states that this is often called Hilbert's Hotel, named after David Hilbert -- the mathematician who first came up with this analogy. (Yes, this is the same Hilbert who came up with a rigorous formulation of Euclidean geometry.) Hilbert's Hotel is so famous that it's easy to find a link to a description of the hotel, such as this one:

https://nrich.maths.org/5788

Cheng writes:

"It warns us that we can't just extend our intuition about finite numbers to infinite numbers, because strange things start happening. Those things aren't wrong, they're just different."

The author tells us that this is related to the current issue of Internet piracy. How is it possible to steal digital media when there can be potentially infinitely many copies of a file? She explains:

"Indeed, the theory of infinitely developed following Hilbert's paradox tells us that subtracting one from infinitely still leaves infinite. The math can't tell us what to do about these moral issues, but it can give us clearer terms in which to discuss them."

Cheng's next paradox is Godel's paradox, which Kung mentions in his first lecture. She recommends another book, Godel, Escher, Bach by Douglas Hofstadter, who explains the paradox more elegantly than any of us can:

"In it Hofstadter elucidates not only the incompleteness theorem but all sorts of fascinating links between logical structures and abstract structures in the music of Bach and the prints of Escher, both of whose works are deeply mathematical while also being immensely artistically satisfying."

In a nutshell, Godel's paradox is the following statement:

This statement is unprovable.

This is what I wrote back in my New Year's Day 2016 post:

On the other hand, Godel's statement "This statement is not provable" can be written as a mathematical statement -- but you have to be as smart as Godel (the Austrian mathematician Kurt Godel) to figure out how. The conclusion is that there exists statements that are true, yet not provable in mathematics.

And Cheng adds:

"This is an example of the fact that even in the logical world of mathematics if a conclusion feels wrong there are mathematicians who refuse to believe it although they can't find anything logically wrong with the proof."

In the next section, Cheng writes about Russell's paradox. She begins:

"When I meet people and say I'm a mathematician I often get slightly strange responses. It's funny how some people immediately boast about how bad they are at math, but other people immediately try to show off how knowledgeable they are."

Replace "mathematician" with "math teacher," and recall that Fawn Nguyen basically wrote about the former response ("I'm not a math person!") in a summer post. Anyway, Cheng tells this story because one person replied with, "Doesn't Russell's paradox show that math is a failure?"

Kung describes Russell's paradox in his ninth lecture, but he also mentions a simpler version of it, the barber paradox, in his first lecture. It's about the barber who shave anyone who doesn't shave himself, which Cheng illustrates as follows:

• If person A shaves person A, then the barber doesn't shave person A.
• If person A does not shave person A, the barber shaves person A.

This results in a problem if person A is the barber:

• If the barber shaves the barber, then the barber doesn't shave the barber.
• If the barber does not shave the barber, then the barber shaves the barber.

Each of these statements produces a contradiction. This is Russell's paradox. Cheng writes about how set theorists avoid Russell's paradox:

"The idea is to say that we have different 'levels' of sets, a bit like how we have different 'levels' of logic. Russell's paradox is caused by statements involving sets that loop back on themselves."

Cheng's last example is about tolerance. First, she explains that two "nots" cancel out:

"If I am 'not not hungry' then I am hungry. If we add up the 'nots' we find that 1 'not' plus another 1 'not' makes zero 'nots.'"

Cheng compares this to the structure of the Battenberg cake -- one of her favorite baking analogies that goes all the way back to her first book How to Bake Pi. (Cheng writes that it's because she does love Battenberg.) It's analogous to addition modulo 2, or adding even/odd numbers, or multiplying positive/negative numbers.

She believes it also comes up if we think about tolerance and intolerance:

• if you're tolerant of tolerance then that is tolerance
• if you're intolerant of tolerance then that is intolerance
• if you're tolerant of intolerance then that is intolerance
• if you're intolerant of intolerance then that is tolerance

Cheng adds:

"For me this means that I feel no pressure to be tolerant of hateful, prejudiced, bigoted, or downright harmful people, and moreover, I feel an imperative to stand up to them and let them know that such behavior is unacceptable."

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 two weeks and skip all posts that have the "Eugenia Cheng" label.

(Look at how far we got in the chapter before we needed that disclaimer!)

The author explains how different levels of sets resolves Russell's paradox:

1. Collections of objects, carefully defined. These are called sets.
2. Collections of sets; these are sometimes called large sets.
3. Collections of large sets, which we might call super-large sets.
4. Collections of super-large sets, which we might call super-super-large sets.
5. ...and so on.

We could do this with tolerance as well. We could set up levels like this:

1. Things
2. Ideas about things
3. Ideas about ideas about things; we might call these meta-ideas.
4. Ideas about meta-ideas, which we might call meta-meta-ideas.
5. ...and so on.

So we can be tolerant of people's ideas, but not their meta-ideas. Intolerance is such a meta-idea.

Cheng also shows us that we can do the same thing with knowledge:

1. Things
2. Knowledge about things
3. Knowledge about knowledge about things; we might call this meta-knowledge.
4. Knowledge about meta-knowledge, which we might call meta-meta-knowledge.
5. ...and so on.

She tells us that this arises when allegations of sexual harassment emerge, especially when it's against a well-known figure. (Hmm, wasn't there some sort of hearing in Washington DC today?) She writes:

"This is one of the reasons the aggressors try to prevent communication between victims, with threats and abuses of power, or even a settlement and non-disclosure clause, or other forms of payment."

Cheng concludes the chapter by clarifying this surprising link between logic and the real world:

"But to me this is just part of the fact that logical thinking helps us in all aspects of life, even in our personal interactions with illogical humans."

Lesson 3-1 of the U of Chicago text is called "Angles and Their Measures." I didn't write much about this lesson two years ago due to a subbing day, so instead we'll go back three years.

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

I don't have much to say about the book's treatment of angles. This text is unusual in that it includes a zero angle -- an angle measured zero degrees. Then again, in Common Core we may need to discuss the rotation of zero degrees -- the identity function.

The key to angle measure is what this text calls the Angle Measure Postulate -- so this is the second major postulate included on this blog. Many texts call this the Protractor Postulate -- since protractors measure angles the same way that rulers measure length for the Ruler Postulate. The last part of the postulate, the Angle Addition Property, is often called the Angle Addition Postulate. Notice that unlike the Segment Addition Postulate -- which this text calls the Betweenness Theorem -- the text makes no attempt to prove the Angle Addition Postulate the same way. Notice that Dr. Franklin Mason's Protractor Postulate -- in his Section 1-6, has angle measures going up to 360 rather than 180.

So let me include a few more things in this post. First, here's another relevant video from Square One TV -- the "Angle Dance":

By the way, last year I taught angles to my seventh grade...

...oops, ixnay on the arterchay athmay! Let me just cut out what I wrote last year here and skip directly to the worksheet.

Chapter 2 Test (Day 30)

Today I subbed in a seventh grade science class. In my old district, today is Day 30 -- which would be the midpoint of the first trimester (or end of the first "hexter,"), except that it's a high school district with no schools on a trimester schedule.

In my new district, where it is only Day 21 -- which would be near the end of the first quaver, except that today I was at a middle school where trimesters and hexters are actually relevant. So in other words, today is nothing -- except Back to School Night, that is. Thus it's a minimum day, with students released at 12:35.

I once thought about my old idea of handing out free pencils on special days. The first day of school is a special day, and so is Halloween (since there are holiday-themed pencils). But with schools starting earlier on the calendar than they did when I was a young student, Halloween is more than two months after the first day of school. That's a long time between pencil giveaways.

And so to my old idea, I add an additional pencil giveaway day about halfway between the first day of school and Halloween. Back to School Night (or during classes that morning) is just about the right time in many districts, including this new district. It also should have been a good time to give away pencils at my old char-... oops, ixnay on the arterchay! (I promised that I wouldn't mention that topic during the Eugenia Cheng reading!)

This is the first minimum day of the year, and since sixth grade is still elementary school, this is the first minimum day that these seventh graders have experienced at their new school. And so as I've done in seventh grade classes throughout September, I help guide the students as they figure out what classes to go to and when. There is no lunch after school, and snack is halfway through the day (right before fifth period). Fortunately, no one asks for a restroom pass in fifth period. One student is tardy to third period, but I cut him some slack -- especially since the minimum day bells ring manually and are slightly off (leaving the kids with only 2 1/2 minutes to make it to class).

In today's science classes, students are watching a video. A British woman named Jane visits the African wilderness to meet -- no, not Tarzan. Actually, it's Jane Goodall who visits Tanzania to live Among the Wild Chimpanzees. This was back in 1960. (As it turns out, Goodall is still alive -- she's currently 84 years old.)

The regular teacher has been out all week. Yesterday's sub finished the video, but today's assignment is for students to add details that they missed the first time to their video notes (in red ink).

Since most of today is just playing a video, I won't write out a full "Day in the Life." But I will write about the management issues that I'm hoping to address now. Again, note that this is a minimum day with the middle school rotation beginning with second period. Sixth period is conference, while all other classes are Science 7. (There are no honors classes, as opposed to last Thursday.)

When each class enters the room, I try to keep the students quiet right away. I tell the students to take out their journals and a red pen. Therefore students know what is expected right away, rather than assume from the start that I'm a pushover.

Also, I decide to reprimand students for talking. It would have been too easy just to create a good list and a bad list of students, with the criteria for making the lists "how much red ink I see." But as I figured out last week, this sends the wrong message. Hardworking students who simply lack a red pen would end up on the bad list, while talkative students who don't pay attention to the film and write random crap on their paper in red ink would end up on the good list.

And so I don't even consider red ink at all. I go around to each student and ask (not tell) them to take out a red pen, but more importantly, I tell (not ask) them to stop talking.

I name fourth period the best class of the day, followed by third period. But I wonder whether I'm able to keep it up in fifth period, which is slightly more talkative throughout the movie. (I give the students a break if they laugh at something funny the chimps do on the screen.)

Meanwhile, I worry about first period (the last class of the day) from the get-go, because yesterday's sub labeled this class merely "OK" rather than "good class." Like fifth period, first period seems to talk throughout the movie. In the end, I decide that first period made a "slight improvement" compared to the note from yesterday's sub -- and indeed, it means that I myself made a slight improvement compared to last week's management.

But still, I wonder whether I could have done better. At this school, homeroom consists of the first period students. It's possible that both yesterday's sub and I had trouble keeping these kids quiet during the morning homeroom announcements, and this extends into first period, regardless of when first period actually meets in the day. (Yesterday the rotation started with first period right after homeroom, and today it ends with first period.)

The fact that the students are watching an excellent video again makes me lament what could have been back in my own -- oops, ixnay on the arterchay iencescay! Let's just jump into Cheng's book.

Chapter 8 of Eugenia Cheng's The Art of Logic in an Illogical World is "Truth and Humans." It's the first chapter of Part II, "The Limits of Logic." Here's how it begins:

"We have seen the power of logic in producing rigorous unambiguous justifications. Now we are going to address the limits of that power."

Cheng warns us logic has its limits -- it works well in the mathematical world, but it doesn't always work in the real world. In some ways, logic is a social construct. She writes:

"This might seem to fly in the face of everything I've said about mathematics being completely rooted in logic, but the situation is more subtle than that."

The author compares this to a trial by jury. Sometimes we lack time to make a logical argument:

"But another way in which logic isn't powerful enough is when we need to convince someone else of our argument. Logic turns out to be a good way to verify truth, but this is not the same as convincing others of truth."

Cheng writes that one way to make an argument is divide it into parts, and then subdivide each part of your argument into parts. She compares this to something we've seen before -- fractals:

"A fractal is a mathematical object that resembles itself at all scales, so that if you zoom in on a small part of it, that small part looks like the whole thing."

Three years ago, our side-along reading book was Benoit Mandelbrot's book on fractals -- and of course, Mandelbrot was the creator of fractal theory. And another side-along reading author, Wickelgren, also compared problem-solving to a fractal-like tree. Cheng draws such a tree in her book -- back in April, I wrote about how to program a computer to draw the tree. She explains:

"This tree represents how finding a proof works, in my head. The base is the thing you're trying to show is true, a but like in our diagrams of causation in Chapter 5. The two branches going into it are the main factors that logically imply it."

Fractal trees are infinite, but proofs are finite. So when do proofs stop? Cheng answers:

"In arguments in real life, we should keep going until the other person is convinced, or until we realize that our fundamental starting points are so different that we will never be able to convince them unless we can change their fundamental beliefs."

According to Cheng, even mathematical proofs are subject to a trial by jury, except that this trial by jury is called "peer review." We've discussed peer review back during another side-along reading book (on Perelman's proof of the Poincare conjecture). And just as we found out, submitting a paper anonymously does little to avoid its acceptance being swayed by human emotions, because the world of mathematicians is a small world after all:

"It's a bit like marking exams anonymously if your class has only three students and you've been working with them closely all year -- whether or not they write their names on the exam paper, you will know exactly which student is which."

The author tells us that to appeal to the human emotional side, research papers include material that isn't strictly logical:

"The help comes in the form of analogies, ideas, informal explanations, pictures, background discussion, test-case examples, and more. None of this is part of the formal proof, but is part of the process of helping mathematicians get their intuition to match the logic of the proof."

Cheng now leaves the world of mathematical papers and enters the world of politics:

"For politicians, the 'peer review' is the election -- it doesn't matter if they're right or not, and it doesn't matter if their if their arguments are sound or not, it just matters if people vote for them or not. Voters do not have to justify their vote either."

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 two weeks and skip all posts that have the "Eugenia Cheng" label.

Politics is all about raising skepticism. She tells us that reasonable skepticism about a mathematical proof can arise in two ways:

1. Someone might think there's a gap or error in your logic.
2. Your conclusion might contradict someone's intuition.

The second type of objection happens all the time in politics. Cheng writes:

"It's why some people still believe that vaccinations cause autism although there is no scientific evidence for it. It's why some people believe the universe is only a few thousand years old or that the earth is flat or that human life did not originate in Africa or that Barack Obama was not born in Hawaii, despite evidence."

When writing a mathematical paper, Cheng tries to anticipate her skeptics in advance:

"We are allowed to imagine that they are intelligent as us, which is why it's call peer review and not idiot review, but I imagine they they are highly skeptical of everything I'm saying, or that they are actively trying to find a mistake in my proof, so that I can find any possible mistakes myself."

At this point, Cheng wishes to point out a major distinction. She writes:

"Whenever there is an article about dogs there is bound to be a thread of comments declaring, apropos of nothing in particular, that 'they eat dogs in China.' This brings us to the difference between truth and illumination."

The statement "they eat dogs in China" is true, but it isn't very illuminating, especially if the current conversation is all about, say, a dog walking app.

Cheng explains that the only equations that are true in first-order logic are the equalities x = x:

1 = 1
2 = 2
3 = 3
4 = 4
...

and so on. These are true, but they aren't illuminating. As for all other equations, Cheng flatly states:

All equations that are illuminating are lies.

For example, she writes:

10 + 1 = 1 + 10

As far as first-order logic is concerned, this statement is a "lie," in the sense that it's not necessarily true on its own. It's true only in a system that has the Commutative Property of Addition -- without that property, the statement could be false. But Cheng explains why this statement is illuminating:

"One children work this out, they can use it to help them add up by counting on, knowing that it will always be easier to start with the larger number in their head and count on by the smaller number."

Notice that this is going to be a traditionalists-labeled post. And here's Cheng writing about things that traditionalists don't like -- adding by "counting on," or any method other than the standard algorithm of addition.

Anyway, Cheng tells us that the equations that really have nothing different about the two sides are the ones of the form:

x = x

and these ones are never useful.

At this point, Cheng repeats what she says about research papers containing stuff other than logic:

"Often in a research paper a logical proof will be accompanied by a description of what 'the idea is,' which is something more informal, not rigorous, but invokes ideas and imagery that might help us to understand the logic."

This leads back to the traditionalists' debate. Cheng tells us that some people -- traditionalists -- believe that rote memorization is necessary:

"But other people, often professional research mathematicians themselves (including me), are convinced that they have never really memorized anything in math. In fact, one of the main reasons I always loved math was exactly the fact that it didn't require memorization, only understanding."

And in fact, Cheng continues:

"However, I am perfectly fine at basic arithmetic and certainly above average compared with the general population, and yet I have never memorized my times tables. I know my times tables by some other, more subtle route that does not involve memorizing."

According to traditionalists, for those who don't memorize their times tables, "it's all over" and "doors are closed to STEM careers." Yet Cheng is a living, breathing counterexample -- she has a PhD in math yet never memorized her times tables.

Cheng wraps up the chapter with some Internet memes. Just like equations, memes can often be illuminating without being true. For example, here is such a meme:

HOW IT SHOULD BE:
Scientists (experts): There's a problem.
Politicians (non-experts): Let's debate a solution.

HOW IT ACTUALLY IS:
Scientists (experts): There's a problem.
Politicians (non-experts): Let's debate if there's a problem.

Cheng herself wants to edit this meme, as follows:

HOW IT SHOULD BE:
Scientists (experts): We think there's a problem. Here's a solution.
Politicians (non-experts): Let's debate funding their solution.

HOW IT ACTUALLY IS:
Scientists (experts): We think there's a problem.
Politicians (non-experts): Let's debate if there's a problem.

And here's another meme:

Funny how no country has ever tried to repeal universal healthcare.
It's almost like it works.

But the author responds:

"However, I'm not sure if the meme is true -- arguably some people have been trying to destroy and privatize the National Health Service in the UK."

Cheng concludes her chapter with another warning:

"We should particularly not pit emotions against logic. They are not opposites, but can work together to make things that are both defensible and believable."

As I wrote earlier, this is my regularly scheduled traditionalists' post. As it turns out, the Barbara Oakley discussion has died down. Once again, I'm late to the party -- by the time I finally agreed to add extra traditionalists' posts to accommodate the growing discussion, everyone else has stopped talking about it.

Instead, here are more recent battles being fought in the traditionalists' debate:

• Rochelle Gutierrez is probably the most controversial anti-traditionalist. So far I've been avoiding articles from either Gutierrez or her opponents like the plague. But in our last traditionalists' post, one blogger linked Gutierrez to Eugenia Cheng. And since we're reading Cheng's book now anyway, I think it won't hurt to look at the Gutierrez debate for today's traditionalists' post only.
• The "0 -> 50" grading controversy has appeared again, this time in Florida.

Let's start with Gutierrez. She is mentioned in this article at the Joanne Jacobs website:

Joy Pullman rants well. In The Federalist, she takes on social justice math.

University of Illinois Professor Rochelle Gutierrez, who teaches future teachers, will “argue for a movement against objects, truths, and knowledge” in a keynote to the Mathematics Education and Society conference in January, writes Pullman.

Let's jump directly to Pullmann's article, linked to above. Here are some key excerpts:

A U.S. professor who teaches future public school teachers will “argue for a movement against objects, truths, and knowledge” in a keynote to the Mathematics Education and Society conference this coming January, says her talk description.

“The relationship between humans, mathematics, and the planet has been one steeped too long in domination and destruction,” the talk summary says. “What are appropriate responses to reverse such a relationship?” We can already guess University of Illinois at Urbana-Champaign professor Rochelle Gutierrez’s answer, from reviewing her published writings and comments. Her plans for “an insurgency by the people” to subvert public institutions and American self-rule through “ethnomathematics” will knock your eyebrows off your face. Let’s take a look.

OK, so far I'm not quite sure what this "ethnomathematics" is, but I hope it's supposed to be math taught in such a way so that students can be successful independently of their ethnicity. Not everyone agrees that this "ethnomathematics" is something worth teaching -- especially not Pullmann. But then the author proceeds:

Gutierrez is an education professor who also teaches in Urbana-Champaign’s “Latino studies” program, of course. Her CV says she helped write federally funded Common Core math tests and has been on a host of taxpayer-funded committees, including several of the National Science Foundation.

She’s also affiliated with the National Council of Teachers of Mathematics, which wrote notoriously terrible curriculum rules that destroyed math instruction in many states before helping form Common Core. She’s helped decide which education professors to grant tenure at more than a dozen public universities, and been given visiting lecture position at Vanderbilt University, which is reputed to have one of the most pre-eminent teaching degree programs.

And here lies the problem. Gutierrez, the "ethnomathematician," is a co-author of Common Core. So opponents of ethnomathematics, such as Pullman, now reasonably concludes that Common Core has something to do with this ethnomathematics. Thus Pullman uses Gutierrez and her ethnomathematics as a reason to oppose the Common Core.

(Moreover, Pullman also criticizes the NCTM standards here. The U of Chicago text is based on these NCTM standards, which explains the resemblance of our text to Common Core Geometry.)

And not only does Gutierrez bring up politics and race, but economics as well. Pullman writes:

Gutierrez’s December article is, perhaps not surprisingly, more of a blog post that quotes sources such as Medium.com and activist websites. Rather than being scholarship worthy of the name, it is essentially a political game plan for using math classes and teachers to follow the Marxist political playbook in “[d]ismantling White supremacist capitalist patriarchy.”

So Gutierrez lists capitalism as something that needs to be dismantled, and Pullman responds by calling her a Marxist. And in fact, Pullman continues:

If we don’t preference competence over political correctness, kids lose big. An understanding of basic mathematics is crucial to competence in many lucrative jobs, plus an introduction to one of the great mysteries of the universe, as well as centuries of human inquiry. These are kids’ lives and minds we’re talking about here, which don’t deserve to be pawns in somebody’s ideological war for social engineering. But far too often, that’s what they are, and it’s American education’s many interlocking monopolies and cartels that are chiefly to blame, because cartels inherently prioritize tribalism over excellence.

So Pullmann responds to Gutierrez and her discussion of economics by providing an economic reason for why students are failing (the "monopolies" and "cartels").

I said that I'd compare Gutierrez to Eugenia Cheng, so let me do so. Earlier this week, we saw how Cheng draws a cuboid to represent privilege, with "rich white male" at the top. When Gutierrez refers to "patriarchy," it means "male privilege" in Cheng's terms, and "White supremacist" is a stronger way of saying "white privilege." This leaves "capitalist" to refer to Cheng's third privilege -- in other words, "rich privilege."

But my main takeaway here is that because of Gutierrez is a Common Core author and writes about ethnomathematics and opposing capitalism, her opponents like Pullmann are going to associate the Core with ethnomathematics and capitalism. I believe that the Common Core is race-agnostic and has nothing to do with Marxism, capitalism, or social justice.

Let's return to Jacobs. She continues by writing about another anti-traditionalist, Cacey Wells:

Tracking math students is rooted in “capitalist exploitations and settler colonialism” and leads to academic apartheid, argues a University of Oklahoma math education professor. Cacey Wells is the author of recently published study, writes Toni Airaksinen on PJ Media.

And here's yet another mention of capitalism. Once again, I'm not sure whether capitalism per se is the problem but rather the "rich privilege" of Cheng.

But here again is the problem -- Airaksinen continues:

He suggests placing all students in the same math class, regardless of ability, and replacing “Algebra 1” and “Geometry” with “Math 1” or “Math 2” to reduce the stigma.

The problem is that "Math 1" and "Math 2" sound like Integrated Math 1 and 2. So now just as Gutierrez causes Pullman to associate Common Core with anti-capitalism and social justice, now suddenly Casey causes Airaksinen to associate integrated math with anti-capitalism and social justice.

I strongly disagree that integrated math has anything to do with white privilege. Integrated math is how secondary math is taught in nearly every country in the world -- including those with and without white majorities.

Most of the commenters at the Jacobs website are actually responding to Wells, not Gutierrez. For example, our main traditionalist at the Jacobs website is Bill:

Bill:

Wells sounds like an idiot… Math doesn’t care how someone feels about it and yes we had tracking when I was in middle and high school.

Trying to label Algebra 1 as math 1 so that every one doesn’t have a self esteem issue will only result in the student who doesn’t have the skills to handle the coursework to simply fail.

A student who doesn’t have a solid grasp of addition, subtraction, multiplication, division, percentages and fractions will simply flunk Algebra (or any other higher level math course)…

Moronic psycho babble.

Regarding the great tracking debate, it reminds me of the debate regarding ID's and voting. Both major political parties agree with the following statements:

• Stronger ID laws lead to lower black turnout at the polls.
• Weaker ID laws lead to greater black turnout at the polls.

But there isn't a consensus as to why these statements are true. Both sides get into heated arguments, and questions such as "Do blacks know how to obtain an ID?" get thrown around. All that matters here is that these statements are true.

Each party accuses the other party of using politics to determine whether to strengthen ID laws:

• Members of a party that expects high black support will favor weaker ID laws.
• Members of a party that expects low black support will favor stronger ID laws.

Chances are that both parties are at least partly correct here. Notice that Cheng's "false positives" and "false negatives" are at play here. Here a "false negative" is someone who deserves to vote but can't (a citizen without ID) and a "false positive" is someone who doesn't deserve to vote but tries anyway (a non-citizen).

Ideally, we want to avoid both false negatives and false positives. Ideally, there should be strong ID laws that don't reduce black turnout. But no one from either party knows how to achieve this.

Another commenter, CT, writes:

CT:

“Apartheid” means “separation.” Is Wells seriously arguing that merely separating students into groups in order to teach to their level is somehow equivalent to an official, brutally-enforced national policy of racial segregation? What a joke. Wells’ ideas, carried out, inevitably lead to lower math performance by nearly everyone in public schools because people cannot learn when a discussion is too far out of their zone of proximal development. Has Wells ever actually taught a classroom of children?

No, tracking isn't equivalent to brutal racial segregation. But, just as with voter ID laws, every time a school adopts tracking, most blacks wind up on the lowest track. If it wasn't for this fact, more people would probably be in favor of tracking.

Suppose we divided everyone into two groups -- say those with an odd number of letters in their full name and those with an even number of letters. We'd be surprised if most of the even-numbered people found themselves on the lowest track. I want to live in a world where we'd be surprised if most members of a track were of one race, or one gender, or one income bracket.

But we can't even come up with a strong ID law that is race-independent (that is, without diminishing black turnout). An education system where every student is learning within his/her ZPD, yet remains race-independent, is even more challenging to create.

Repeating Pullmann's words:

If we don’t preference competence over political correctness, kids lose big. An understanding of basic mathematics is crucial to competence in many lucrative jobs

I do know that students stuck on the lowest track aren't being well-prepared for the lucrative jobs mentioned by Pullmann here. That's all the more reason to ensure that "low track" doesn't correlate strongly with a particular race.

Before we leave Jacobs, let's look at one final comment:

Jean @ Howling Frog:

Good old ‘story problems relevant to real life.’ Always the solution, until you get them and find out that real life is BORING. Give me Saxon’s fanciful story problems any day.

This comment refers to word problems in Algebra I/II classes, as well as the common student question "When will we ever use this in real life?" I can't tell what Jean is saying about word problems in most common texts, but I do know about the Saxon series of texts -- text that are commonly recommended by traditionalists. My question is, does the question "When will we ever use this in real life?" frequently arise when working on Saxon word problems. (I mean with the majority of students sitting in math classes, not the traditionalists and their math-bright children.)

Before we leave Joanne Jacobs, there's another issue to which I want to apply Cheng's logic. A frequent commenter at the Jacobs website is lgm -- and lgm often criticizes "full inclusion," especially in elementary schools. For example, we check out the following thread:

https://www.joannejacobs.com/2018/09/when-high-hopes-meet-low-expectations/

lgm:
The big issue is the dumbing down of each elementary school grade level so all are ‘included’….the result is capable students who need remediation because large portions of the grade level material simply were not done. A test prep book isn’t enough to catch them up.

lgm:

The elementary reading here has changed due to full inclusion — there is just not enough funding for enough adult aides to supervise those who cannot work independently while the teacher is giving small group lesson.

So lgm is clearly opposed to this "full inclusion," so maybe lgm supports its negation. Let's use Cheng's quantifiers here to find the negation. So "full inclusion" means something like:

For all elementary students, that student is included (in the gen ed classroom).

The negation of a "for all" statement is a "there exists" statement:

There exists an elementary student who is excluded (from the gen ed classroom).

The negation of "all students are included" is "some students are excluded." So if lgm is opposed to full inclusion, maybe lgm supports the negation, "partial exclusion."

And of course, it's easy to name someone who might be opposed to "partial exclusion" -- the parents of the students who would be excluded. This explains why schools use full inclusion -- otherwise parents of the ones to be excluded might riot, or sue.

OK, let's move on to the "0 -> 50" grading controversy. Apparently, a teacher in Florida was just fired because she disagrees with the "0 -> 50" grading system:

https://www.wftv.com/news/local/teacher-fired-after-refusing-to-abide-by-no-zero-policy-when-students-didnt-hand-in-work/840999722

I first mentioned "0 -> 50" on the blog three years ago, then I alluded to it when I needed to do science grades quickly when -- oops, ixnay on the arterchay iencesay (for the third time today).

So let's focus on what I wrote in 2015. I have much to say about 0 -> 50, but unfortunately this post is already jam-packed with other parts of the traditionalists' debate. (And that was after I pointed out that the Barbara Oakley material has dried up!)

So let me just quote a few parts of that old 2015 post about 0 -> 50. I'll insert a few comments relevant to the Florida situation:

A grade represents a percentage, a scale from 0 to 100. One of the most commonly used grading scales in schools is something like this:

90-100 = A
80-90  = B
70-80  = C
60-70 = D
0-60   = F

Now let's return to school grades. For batting averages .300 is excellent, but in school 30% is surely a failing grade. For winning percentages .600 is one of the best teams in the majors, but in school 60% is the lowest passing grade. All of the grade boundaries -- from B to A, C to B, and so on -- occur in the upper half of the scale. The lower half of the scale -- from 0 to 50 percent -- is irrelevant as far as determining letter grades is concerned.

To the extent that C is average, the average student is earning between 70% and 80%. So we expect the average score in the class to be around the midpoint of this range -- 75%. So the average batter gets one hit every four at-bats, the average team wins two out of every four games, and the average student gets three out of four questions right.

And so, unlike the batter who gets a hit then makes an out and sees that overall his average has risen, the student who gets 100% on one test and 0% the next can never have a higher letter grade -- only a lower letter grade is possible. Therefore, a grade is much more likely to drop very rapidly than it is to rise very rapidly. Since the average grade is around 75%, a student would have to receive 150% on a test in order for the grade to rise as rapidly as a 0% drops it. (And all of this is assuming that the student isn't at one of those schools that has abolished D grades and makes 70% the lowest passing score -- that 10% difference means that a student would have to earn 160% on a test to have the same impact on the grade as a 0%!)

I've seen students come up to me and ask me why their grade has dropped so quickly -- especially when their grades have dropped from B to D seemingly overnight, while no one's grade is rising from D to B as quickly. They think that I'm a mean teacher and a harsh grader -- when the true reason is the nature of the grading scale, where the F grade takes up three-fifths of the scale.

Now it's the end of the quarter, and grades are about to come out. Let's assume for simplicity that there are equal numbers of points possible in the first and second quarters (in reality, the second quarter may have more points because that quarter has a final exam while the first quarter doesn't).

[2018 update: In this particular Florida school district, each quarter is actually worth 40%, and the final is worth 20%.]

Let's say a student is earning 10% at the quarter. Then this student will surely fail the class -- even if 100% is earned the second quarter, the average grade is only 55%, an F. This is akin to [the 2018 Baltimore Orioles, who were mathematically eliminated on August 19th].

A student earning 20% at the quarter can still pass the class -- but only if 100% is earned the second quarter, which isn't very likely. And a student who gets 30% the first quarter would still need to earn 90% (an A) the second quarter to pass the class. The type of student who would earn only 30% in a quarter is not the type of student who would earn an A the second quarter -- which typically covers more difficult material than the first quarter. This is akin to a baseball team who is nine games back with only 10 to play -- although the team hasn't been mathematically eliminated, they arerealistically eliminated from the division. It would take a miracle for the team to come back and claim the division, and a similar miracle is needed for the student to get any semester grade other than F.

In fact, in some ways, any quarter grade below 50% would realistically eliminate a student from receiving a grade higher than F at the semester. A 50% quarter grade would require a second quarter of 70%, a C, to pass the class -- and this might be doable, if the student works a little harder during the second quarter.

(And that, of course, assumes that 60% is the lowest passing grade. If the 50% student is at a no-D school, then that student will need 90%, an A, in the second semester to get any letter other than F on the semester report card -- once again, realistic elimination.)

When I am teaching class, what I want is for as few students as possible to be mathematically or realistically eliminated from getting any semester grade other than F, when there are still plenty of weeks separating them from the end of the semester. I have no sympathy for a student who is realistically eliminated when there are only nine or ten days left in the semester, but nine or ten weeks, that's another matter.

A player on a baseball team that is realistically eliminated from the playoffs may be traded to a team that has a shot of winning [like some Orioles]. And even if the player isn't traded, he will still play out the rest of the season, no matter how little his heart is in it, because he's under contract to do so. Even a college student realistically eliminated from passing a class can drop out and take a W instead. But for high school students, there is no choice but for the student to take the F.

I mentioned last year that one way to avoid realistic elimination would be to allow students to retake the tests on which they received a low score. But in this article, there is mention of two separate -- and I repeat, separate -- grading scales that help students avoid realistic elimination.

[2018 update: only one of these grading scales is relevant to Florida.]

The first is to record scores of zero as equaling 50%. After all, it's the devastating effect of 0% on the grades that eliminate students from passing the course. The average of 0% and 100% is 50%, an F, but the average of 50% and 100% is 75%, a C.

Back when I was an Algebra I student, my teacher did something similar. When she recorded test scores, she only recorded a letter grade, so if one received an F, one can't tell whether the F was really a score of 1% or 59%. Then to determine the average at the quarter or semester, she converted the grades back into percentages by letting each letter represent the midpoint of the interval. So an A was 95%, a B was 85%, a C was 75%, and a D was 65%. But an F was converted into 55%. So this represents a similar scheme to the 0 -> 50 system, since a score of 5% instantly became 55%.

And at this point, I kept writing about the alternative to 0 -> 50. This has nothing to do with Florida, and so I don't repeat it.

Since this post is already so long, let me end it with some illuminating insight from Cheng. Those who support 0 -> 50 (like the Florida district) are afraid of "false negatives," while those who oppose 0 -> 50 (like the fired Florida teacher) are afraid of "false positives."

"False negative" students:
Student misses a few early assignments and gets a zero.
Student studies three hours per night to make up the work.
Student gets 100% on every remaining assignment.
Student deeply cares about learning and getting high grades.
Teacher calculates semester average as 59.99% or less, gives student an F.

"False positive" students:
Student misses a few early assignments and gets 0 -> 50.
Student studies very little each night.
Student determines which assignments to do and which assignments he/she can afford 0 -> 50.
Student doesn't care about learning, only about placating parents with a D.
With 0 -> 50, teacher calculates semester average as 60.00% or more, gives student a D.

The whole 0 -> 50 boils down to the fact that a single digit -- the digit in the tens place of the percentage -- determines the student's letter grade. If the student works hard but that digit is a 5, the teacher must give the student an F. If the student slacks off but that digit is a 6, the teacher must give the student an D. The teacher's hands are tied -- especially since it's probably a computer that's calculating the grades. The teacher must assign the grade that appears on the computer. Basing grades on a single digit is almost like determining whether to scold students based on what's written in red ink in a journal -- which takes us back full circle to my day of subbing.

Yes, writing a name on the bad list for not writing in red ink (when the class is so loud that the student never hears the instruction "write it in red ink") is a "false negative." And writing a name on the good list for copying someone else's paper in red ink (while talking all throughout the lesson) is a "false positive." I avoid both the false negative and the false positive by not using the red ink to determine what names to write on each list.

Likewise, perhaps if we know that a student is making a deep and honest effort to make up the work yet can't mathematically reach a semester 60%, then we show that student mercy with 0 -> 50. On the other hand, if we know the student isn't trying to make up the work at all, then we keep 0 as 0. This might be common sense but would be difficult to justify to parents. (Why does another child get the benefit of 0 -> 50 but not my child?)

Here is the Chapter 2 Test: