Friday, April 1, 2016

Andrew Hacker and the Case for and Against Algebra

Today I decided to quit my job as a substitute teacher. Rather than continue to pursue my goal to become a math teacher, I will join the military instead. But I will continue to post to the Common Core Geometry blog anyway, whenever I have the chance, since I will bring the U of Chicago text with me. My hope, of course, is that I won't be in the middle of posting to the blog when I am captured...


Yes, today is April Fool's Day. Nothing in the first paragraph of this post is actually true. In fact, today is the last day of spring break. But last year, April 1st was a school day -- and I actually did have an April Fool's Day joke in one of the classes. This is what I wrote last year about the prank:

In the Earth Science classes, students were to watch a video on earthquakes. I decided to pull an April Fool's Day prank on the students -- instead of saying "Today there will be an earthquake video," I said, "Today there will be an earthquake drill." Many students were fooled and were expecting to evacuate the classroom. The only thing that would have made this prank better was if there had been a sudden announcement on the intercom right at that moment -- even if the announcement had nothing to do with earthquakes, a student might have jumped under the desk anyway! (I know, this may be confusing my readers outside of California who don't know about earthquake drills.)

There are a few things I want to discuss in this special spring break post. First of all, I'm finally going to discuss the Andrew Hacker article. It almost seems as if I was already pulling an early April Fool's Day joke -- I keep saying I'd write about the article, then I end up not writing about it.

Well, here is the article:

This article is written by Dr. Keith Devlin, a Stanford mathematician. I won't repeat the article here, instead I'll just summarize it. (Click on the link above if you want the whole article.)

-- In paragraph 1, he explains that Hacker published The Math Myth: And Other STEM Delusions.
-- In paragraph 2, Devlin refers to a 2012 New York Times article, "Is Algebra Necessary?" In both the article and the new book, Hacker promotes the removal of Algebra as a required course in K-12.
-- In paragraph 3, Devlin points out that high schools merely "introduce" the subject that was first invented by medieval Muslims -- algebra was named after the Arabic word al-jabr. What Hacker is really criticizing is "school algebra," not real algebra.

This reminds me a little of Dr. Hung-Hsi Wu -- sorry, Dr. Devlin, I know you're from Stanford, yet now I'm quoting a Berkeley mathematician. Dr. Wu uses the term "Textbook School Mathematics" to refer to how math is taught in schools today:

Devlin writes, "If Hacker had instead used his NYT connection to argue for a major make-over of 'school algebra' (as I think we should call the object of his criticisms), he would have garnered massive support from the pros, including me" [and Wu, presumably -- dw].

Perhaps the best way to summarize the rest of Devlin's article is for me to quote the one commenter who posted on this article, Kay Cornwell Romer. Notice that Romer only wrote her comment this week (even though Devlin's article is a month old).

Romer writes:

This article demonstrates exactly why math is incomprehensible to so many in school. Those who understand it can't explain it, and those who teach it don't understand it.

When they were confused by a particular math lesson, I always told me kids to ask their math teachers in what real-life situation could any given algebraic concept be used. I figured that would make the concepts easier to understand if they knew how it could be applied in their world. Their teachers never could give an answer, and my kids were pegged as troublemakers for asking a perfectly legitimate question.

Blame the teaching, not the students. You have generations of poor mathematical instruction to fix. Good luck with that.

So Romer's children asked the age-old question, "When will we use this in real life?" but their teachers never gave them a straight answer.

Let's go back to the main article. In paragraph 4, Hacker lamented that there was no good reason to require students to master, among other topics, polynomial functions. And in paragraph 5, Devlin responds by writing, "There is good reasons to study both polynomial functions and equations..." -- but of course, he never states what these reasons are. So Devlin never answers the questions of Romer's children either.

Notice what Romer writes again -- she has her children question their math teachers "when they were confused [emphasis mine -- dw] by a particular math lesson." This is so important that it bears repeating -- the age-old question "When will we use this in real life?" is asked only when students are confused by a lesson -- that is, when they find it hard or boring. The question "When will we use this in real life?" never occurs when a lesson is easy or fun -- no matter how useless the lesson actually is in real life.

For example, many math teachers point out that many lessons in history or literature classes aren't very useful in real life at all, yet the question "When will we use this in real life?" occurs more often in math classes, not history or literature. We now see the reason for this -- history and literature lessons are either fun or easy -- at least easy compared to math lessons. If math teachers don't want to answer the question "When will we use this in real life?" we should have lessons that are either easy or fun dominate the school curriculum, and the question will vanish.

It's difficult for me to tell what Romer is saying here. Again she writes, "You have generations of poor mathematical instruction to fix." But I can't tell how she suggests we fix math instruction.

Earlier she writes, "Those who understand it can't explain it, and those who teach it don't understand it" -- which implies that if we could somehow find teachers who both completely understand math and can make it crystal-clear to students, then math instruction would be fixed.

But Romer could also be implying that finding such teachers is impossible, and that math instruction should be fixed by dropping the hard and boring subjects -- namely Algebra -- altogether. If this is true, then she would agree with Hacker.

In the original 2012 article, Hacker recommends replacing algebra with a new class, which he calls "citizen statistics." He writes, "Instead, it would familiarize students with the kinds of numbers that describe and delineate our personal and public lives." To be fair, let me link to the 2012 article that launched this whole discussion:

I've been thinking about this for awhile. My own opinion is that Hacker somewhat has a point here, and perhaps we should take another look at what math classes students should take:

-- to prepare for a STEM major in college
-- to prepare for a non-STEM major in college
-- to graduate from high school without college

Let's look at the various classes taught in high school, and for whom I believe -- based on what I learned from the Hacker and Devlin articles -- the classes should be taught:

-- AP Calculus: This class should obviously be for prospective STEM majors only.
-- Pre-Calculus: As the name of this class implies, Pre-Calculus prepares students for Calculus. As I say that Calculus should be for STEM only, Pre-Calculus should be for STEM only as well.
-- Algebra II: This one is tricky. Colleges require Algebra II for all majors, STEM or no. While I'm sympathetic to the idea that maybe non-STEM majors shouldn't have to take Algebra II in order to enter college, this idea may be too radical to convince most universities to go along it. So I must maintain that prospective college students should take Algebra II.
-- Algebra III/Governor's proposed math course: Juniors who complete Algebra II should still take a math course as seniors before entering the college. This class would therefore be for non-STEM, as STEM students should be in Pre-Calc.
-- Geometry: Perhaps I'm biased on this is Geometry blog, but I actually would set Geometry as the last course required for high school graduation, college or no. Notice that, as we see from the Hacker article, most of the problems students have are with Algebra, not Geometry.
-- Algebra I: If we require Geometry for high school graduation, then of course Algebra I must also be required for graduation. But I write this with a caveat -- we protect non-college or non-STEM students from higher levels of algebra by delaying certain topics to later math courses.

In particular, polynomials, factoring, and the Quadratic Formula could all be delayed until Algebra II, thereby freeing students from having to learn these to graduate. (Sometimes I'm also sympathetic to the idea of the Quadratic Formula as the last topic required for Algebra I.) Other topics like polynomial division and rational functions could be delayed until Pre-Calc, thereby freeing students from having to learn these to enter college.

At this point, you may ask, where does Integrated Math fit into all of this? Notice that Integrated Math I contains mostly linear equations only -- polynomials and quadratic equations are already delayed to Integrated Math II. So Math I already contains all the algebra I say that high school grads really need -- and there's also a healthy amount of geometry included as well. So I can easily defend the idea that Math I should be the only class required for graduation.

But then again, I've posted that there's not much difference between Common Core Math 8 and Integrated Math I -- so this would imply that CCSS Math 8 should be the last required class! This idea now suddenly approaches Hacker's -- a completely math-free high school experience!

Perhaps it might be better to say that if a student earns a grade of C in Math 8, the student should move on to Math I -- and if they continue to earn a C, then that Math I class can be the last math class they need to graduate. Students who earn an A or B in Math I should move on to Math II -- and those who earn an A or B in Math 8 should go directly into Math II as freshmen (as these are likely the STEM students working towards Calculus). It's also possible to add Hacker's "Citizen Stats" class to this -- students who earn a C in any course could be placed directly into Hacker's class, as these are the ones who might struggle to proceed along the Calculus track. (Those who earn a D or F in Math 8 could go directly into Hacker as freshmen, instead of suffering through Math I.) Then the Hacker class would be the last class they need to graduate.

All of this recalls an idea I once mentioned last year -- the concept of Dickens age. This comes from the idea that in other countries, students don't have to stay in school until age 18 or completing the equivalent of 12th grade. Some people advocate that perhaps in the U.S., students should be allowed to leave school earlier and seek out employment. Then this instantly protects them from having to take advanced math courses in high school, as they wouldn't be taking any classes at all. Then we would keep Algebra II fully rigorous, since the students who can't handle Algebra II would likely have already left school.

In the earlier posts, I set the Dickens age to be 15, or the completion of ninth grade. I came up with the name "Dickens age" because of my fear that if we set it too young, students might find themselves working in factories at a young age, which would be Dickensian. My setting of the Dickens age at 15 is a compromise between the leaving age of 16 that's common to many European countries, and a commonly proposed Dickens age of 14 (or the end of eighth grade, as this represents a clean break between middle and high school).

Implementing a Dickens age may be too radical for American schools at this time. But in this post, I'm essentially proposing that students should only have to take math up until the Dickens age.

So far, I have yet to mention any traditionalists in this post. But we can imagine what traditionalists would say about this proposal -- they don't like it one bit. The traditionalists want to see as many students taking AP Calculus as possible, and they want students to begin preparing for Calculus as early as eighth grade -- when the students should be taking Algebra I, not Integrated anything!

Let's get back to what I wrote about the question "When will we ever use this in real life?" I said that this question is only asked when a subject is hard or boring -- not when it's easy or fun. I mentioned last month about the traditionalist who calls himself "Niels Henrik Abel" -- we saw Abel criticize tessellations as a useless topic that high school students don't need to learn. I suspect that Abel, on the other hand, would consider logarithms as a useful subject that students should learn. Yet I wrote that students are more likely to ask "When will we ever use this in real life?" during the unit on logs, not on tessellations, even though logs are more useful. The reason is that logs are hard and boring, while tessellations are easy and fun. So we see that the question "When will we ever use this in real life?" is a proxy for "This is hard and boring!" even within math class, and not just in math vis-a-vis other high school classes.

Here's a traditionalist solution to the Hacker dilemma -- students should be better prepared for Algebra I by using a more traditional pedagogy in early elementary school -- the cornerstone of which is the memorization of arithmetic facts. Such students will find themselves more ready for Algebra I, and so are more likely to find Algebra I easy, not hard. Then the question "When will we ever use this in real life?" doesn't come up, since the students would find Algebra I easy -- they wouldn't ask the question that's reserved for subjects that are hard.

Indeed, we can only imagine what the traditionalist Barry Garelick -- the other author who published a book about math education last month -- would write. Since I don't have Garelick's book yet, I will mention that back in February, Garelick tweeted that he agreed with the following letter that was submitted to the NYT to rebut Hacker:

Here are some excerpts from the letter:

As a high school student, I strongly disagree [with Hacker -- dw].

While your job may never require you to know the difference between a postulate and a theorem, it will almost certainly require other math-based skills, like how to prove something or how to understand a graph. If nothing else, people need math to understand finance, which is a part of everyone's life.

Algebra and geometry have a place in the classroom. If students are failing, then the way math is taught may need to change. But what is taught needs no alteration.

Lucy Brownstein

Unfortunately for the traditionalists, most high school students are not like Lucy Brownstein. When most students see something that they perceive not to be useful, they ask questions like "When will we ever use this in real life?" This is especially so if they perceive the subject to be hard and boring, as many Algebra topics tend to be. If we math teachers were guaranteed to have 30 Brownsteins sitting in our classes, then we can teach the math way the traditionalists want us to. Instead, we must teach math to satisfy and educate the "When will we ever use this in real life?" crowd who is actually sitting in our classes.

Actually, let me amend what I said earlier a little. I said that students have no trouble with lessons that are easy or fun. But there is a third motivator for learning -- high status.

Some traditionalists point out that there are many tasks that are hard and boring, yet students have no trouble with completing them. For example, students need to do repetitive drills to prepare for sports, or repeat scales to learn how to play an instrument. Running 30 laps or practicing a song 30 times is just as hard and boring as answering 30 math problems. The traditionalists argue that just as sports drills are necessary to make someone an excellent athlete, and music drills are necessary to make someone an excellent musician, math drills are necessary to make an excellent mathematician. Yet the progressives who oppose math drills have no problem with sports or music drills, they complain.

My theory is that a high status is the third motivator for learning. Athletes and musicians enjoy a high status on many high school campuses, and so students are willing to endure hard and boring drills in order to attain that high status. An easy way to determine whether a person is high-status is the ease with which that person can attract mates (i.e., boyfriends or girlfriends). Athletes and musicians are high-status because they can attract mates easily.

Meanwhile, we expect students who are good at math to be high-status since, after all, math prepares students for high-paying STEM careers and, as we know, those with six-figure incomes have an easier time attracting mates...

...yes, that's an April Fool if you ever saw one. The very existence of the word nerd in our language tells us that the exact opposite is true. Students who are great at math are considered "nerds," and nerds have a harder, not easier, time attracting mates.

This should answer the question for the traditionalists. In order to motivate students to work, a subject must be easy or fun -- or leads to a high status. Students are willing to work at hard, boring drills for sports or music because excellence at them leads to a high status. But not only is math -- especially Algebra -- considered hard and boring, but it leads to a low status. This explains why traditionalism in high school math doesn't work. If the traditionalists want to promote their pedagogy in high school math, they must find a way to make it either easy, fun, or high-status, period.

It's time to finish Rucker. Chapter 11 of Rudy Rucker's The Fourth Dimension: Toward a Geometry of Higher Reality asks, "What is Reality?" Yes, this final chapter of Rucker is the perfect chapter to read today, since we're always trying to figure out what's real and what's not on April Fool's Day.

Rucker begins by asking:

Starting with no preconceptions at all, what is the most reasonable model of the world that we can build up?

I'll let you think about that as we discuss the three puzzles for this chapter:

Puzzle 11.1:
Suppose that space and time are really just mental constructs. In general, the only reason we have for saying that one state of mind B comes after another state of mind A is that B includes a memory of state A, but A does not include a member of state B. Under this definition of "before and after," would a person's perceptions necessarily fit into a linear time sequence?

Answer 11.1:
No. There are probably some sections of your life that do not involve any thoughts at all about the other sections. In "A New Refutation of Time," [Argentinian writer Jorge Luis] Borges argues that any state of mind that recurs in your life is actually the same event.

Puzzle 11.2:
The British writer J. W. Dunne felt that our dreams are made up of impressions taken equally from past and future. He claims that in his 1927 An Experiment with Time that the dreaming mind is able to rise out of spacetime and see what lies ahead. This seems to lead to a sort of paradox: Suppose that I am to catch a plane Tuesday and that, unknown to me, the place is going to crash. Monday night, my dreaming mind sees into the future, and I have a horribly vivid dream of dying in a plane crash. Tuesday morning I am so badly shaken that I decide to postpone the trip. Tuesday evening, I watch the news and see that my intended plane did indeed crash, killing all aboard. The paradox is this: Since I did not in fact experience the plane crash, how could I have seen it as part of my future? Dunne's way out of this hinges on the claim that there is a second dimension of time. Can you fill in the details of his argument?

Answer 11.2:
Dunne's idea is that at any instant we have a fixed future along a familiar time axis called T_1, but that as a second, higher sort of time, T_2, lapses, our future changes. Our actual time motion is a combination of T_1 motion, which we might think of as moving into the future, and T_2 motion, which we might call moving into alternate worlds. When I see something unpleasant ahead in Monday's T_1 future, I am able to move in the T_2 direction into an alternate spacetime with a different T_1 future.

Puzzle 11.3:
According to quantum mechanics, if you don't keep an eye on a person, he or she soon becomes indeterminate for you: you no longer know what the other person is like. Yet if you ask the person questions, you will find him or her to have definite characteristics, Is there a contradiction here?

Answer 11.3:
No, this is just another example of a region of fact-space having a vague position relative to one axis (your opinion), but a sharp position relative to another axis (the other person's opinion). If you are close enough to a person to be in some sense part of him (or her), then your opinion of him will evolve continuously with him. But if you are separated for a while and then brought back together, the other person's many possible states will seem to collapse down to one or two definite facts. In quantum mechanics this abrupt change is called "the collapse of the wave function." It is worth noting that, in certain related states, you are yourself spread out in fact-space, even relative to your own axes. That is, if you are not presently asking yourself whether you are happy, then there is, at present, no definite answer to the question. Sharp or hazy, reality is just what it seems.

By the way, one possible answer to the question at the start of this chapter can be answered if we think about quantum mechanics. Rucker writes, "But in the 1920's, physicists Werner Heisenberg and Erwin Schrodinger discovered that the best way of interpreting quantum mechanics is to say that particles are patterns in infinity-D Hilbert space."

This is a great place to finish our discussion of Rucker's book as it leads directly into his other mathematical book, Infinity and the Mind. As the title implies, Rucker discusses how our minds, as we interpret his "fact-space," are infinite.

I notice that just last week, Numberphile posted a video about the fourth dimension. In this video, we learn that just as there are five regular polyhedra, or Platonic solids in 3-D, there are six regular polytopes in 4-D -- the 5-cell, 8-cell, 16-cell, 24-cell, 120-cell, and 600-cell:

This is another 4-D video produced by Numberphile and another favorite YouTuber, Vi Hart.

The fourth dimension has made its way into other videos as well. Recently there is a Japanese anime cartoon released in the U.S. called "Dimension W." I don't watch the show, but I know that each episode begins by explaining that there are four coordinate axes -- x, y, z, and w.

The rest of this post will let me prove that on this April Fool's Day, mathematicians really do have a sense of humor.

I've mentioned the website Metamath before. This site gives many proofs in two-column format -- and most of these aren't geometry proofs at all. Here's a link to one page of this website:

The list is always changing -- in fact, last year these proofs appeared on page 138, and now they are on page 180! Anyway, as of today, look at some of the proofs that appear on this page:

-- Fundamental Theorem of Algebra (often mentioned in Algebra II classes)
-- The Basel Problem (The series 1 + 1/4 + 1/9 + 1/16 + ... + 1/n^2 + ... adds up to pi^2 / 6.)
-- Bertrand's "Postulate" (There's a prime between n and 2n for every natural n. This is called a "postulate," but it's actually proved here in the link, so it's a theorem.)
-- Ostrowski's Theorem (a complicated theorem about absolute value)
-- various definitions from ring theory (But no proofs appear until the following page.)

The first three of these are easy to state -- even in a high school class -- but difficult to prove. The proofs that appear on the Metamath page are either covered in upper-division math or not at all. There is even a section for Hilbert spaces, just as Rucker describes in his book.

By the way, one of the most recent Metamath proofs is that of the Konigsberg bridge problem -- the one that we discussed back on the first day of school:

Notice that this proof requires 172 steps! Let's look at the first ten steps to see why so many are needed in this proof. We see that while the U of Chicago text, in Lesson 1-4, uses A, B, C, D to represent the four lands, this proof uses 0, 1, 2, 3. The bridges between the lands are represented as sets, so {0, 1} is the bridge leading from land 0 to land 1:

1. 1 is not equal to 0.
2. 0 is not equal to 1.
3. 0 is a natural number. (N_0 is the set of natural numbers including 0, N is the naturals without 0.)
4. 0 is a set.
5. 1 is a real number.
6. 1 is a set.
7. If 0 is not equal to 1, then the set {0, 1} has two elements.
8. The set {0, 1} has two elements. (This follows from steps 2 and 7. Metamath writes #'({0, 1}) = 2, but we see that # means the same as N from Lesson 6-1 -- the number of elements, cardinality, So the U of Chicago would write N({0, 1}) = 2.)
9. Add 1 to both sides of step 8: N({0, 1}) + 1 = 2 + 1
10. {0, 1} is a finite set.

Now we can see why this proof is so long -- the computer can take nothing for granted! It takes eight steps just to prove that the first bridge {0, 1} has two ends! Of course, proofs in our Geometry classes don't need to include these steps!

Anyway, in between all of these theorems are some April Fool's Day proofs. Two of the proofs involve complex math symbols that end up spelling out phrases. The first spells out "April Fool" -- this was first posted to the Metamath website ten years ago:

And the other one spells out "Hello World." Computer scientists are more likely to get this joke, as the traditional first program that one learns is how to print "Hello World" on the screen:

Two years ago, the April Fool's Day entry was a proof that division by zero is forbidden:

The actual statement that is proved is "(1 / 0) = 0" -- that is, 1 divided 0 is the empty set. Here I use a struck-through 0 to denote the empty set, but the line should be diagonal, not horizontal. When I was in that seventh grade classroom that used the McDougal Littell text -- although that text itself directs students to write "undefined" -- the teacher told the students to write the symbol for empty set, which she pronounced "o-slash." Before learning about o-slash, common errors made by the students include giving the answer to 1 / 0 as 1 or 0 -- that is, they added, subtracted, or multiplied the 1 and 0 when they were asked to divide it. After learning about o-slash, a common error is to state that the answer to 0 / 1 is also o-slash instead of 0.

How does Metamath "prove" that 1 / 0 is the empty set, anyway? We notice that the third line reads:

dom / = (C x (C \ {0}))

Here "dom" is short for "domain." So this line states that the domain of the division function is the set of complex numbers for the dividend and the set of complex numbers except 0 for the divisor. To put it simply, we prove that division by 0 is impossible because we defined division to exclude 0 from its domain (hence the reason that 1 / 0 is "undefined") -- which isn't very enlightening at all. If we ask the TI-83 to find the tangent of 90 degrees, it likewise returns a vague "domain" error, whereas 1 / 0 gives a more descriptive "divide by 0" error.

But then again, this is just an April Fool's Day proof. Here's a more common proof to illuminate the reasons that division by 0 is undefined:

This problem is a classic "proof" that 1 = 2. In this proof, we are given that a = b in Step 1, and in Step 8, we divide by a - b (actually given as a^2 - ab, but factoring gives a - b). As a = ba - b must be zero. So we actually divided by 0 -- and this is why division by zero  must be undefined, since otherwise we could prove that 1 = 2. I once tutored a geometry student (not my current student) whose teacher assigned the classic "proof" of 1 = 2 and the student was asked to find the error. It took a while, but I think the student did eventually figure out that a - b = 0, so that they were dividing by 0.

Let's conclude the April Fool's Day part of this post by giving a link to some math jokes:

Here are just a few of the jokes from this site:

    Top ten excuses for not doing homework:
  • I accidentally divided by zero and my paper burst into flames.
  • Isaac Newton's birthday.
  • I could only get arbitrarily close to my textbook. I couldn't actually reach it.
  • I have the proof, but there isn't room to write it in this margin.
  • I was watching the World Series and got tied up trying to prove that it converged.
  • I have a solar powered calculator and it was cloudy.
  • I locked the paper in my trunk but a four-dimensional dog got in and ate it.
  • I couldn't figure out whether i am the square of negative one or i is the square root of negative one.
    • Warning! It is against the rule to use these excuses in my classes! A. Ch.

Note: I left out the last three excuses because they refer to college-level math. Notice that Isaac Newton was born on Christmas Day -- a day when schools are already closed. The joke about the proof that can't fit in the margin refers to Fermat's Last Theorem. I told the story about Fermat's proof and the margin on this blog back during the first week in September.

And of course, now that we've finished Rudy Rucker's book, we definitely get that 4-D joke now. We know that just as A Cube was able to get A Square out of a locked 2-D room, a 4-D dog can take paper out of a locked 3-D room.

    Two male mathematicians are in a bar. The first one says to the second that the average person knows very little about basic mathematics. The second one disagrees, and claims that most people can cope with a reasonable amount of math.
     The first mathematician goes off to the washroom, and in his absence the second calls over the waitress. He tells her that in a few minutes, after his friend has returned, he will call her over and ask her a question. All she has to do is answer one third x cubed.
     She repeats "one thir -- dex cue"?
    He repeats "one third x cubed".
    Her: `one thir dex cuebd'? Yes, that's right, he says. So she agrees, and goes off mumbling to herself, "one thir dex cuebd...".
     The first guy returns and the second proposes a bet to prove his point, that most people do know something about basic math. He says he will ask the blonde waitress an integral, and the first laughingly agrees. The second man calls over the waitress and asks "what is the integral of x squared?".
    The waitress says "one third x cubed" and while walking away, turns back and says over her shoulder "plus a constant!" 

This is a calculus joke. I mention it only because the "1/3" that shows up in this calculus problem is also the source of the "1/3" that appears in the volume of a pyramid or cone -- if we were to use calculus to derive the formula.

Just above these two, we have:

    The Evolution of Math Teaching
  • 1960s: A peasant sells a bag of potatoes for $10. His costs amount to 4/5 of his selling price. What is his profit?
  • 1970s: A farmer sells a bag of potatoes for $10. His costs amount to 4/5 of his selling price, that is, $8. What is his profit?
  • 1970s (new math): A farmer exchanges a set P of potatoes with set M of money. The cardinality of the set M is equal to 10, and each element of M is worth $1. Draw ten big dots representing the elements of M. The set C of production costs is composed of two big dots less than the set M. Represent C as a subset of M and give the answer to the question: What is the cardinality of the set of profits?
  • 1980s: A farmer sells a bag of potatoes for $10. His production costs are $8, and his profit is $2. Underline the word "potatoes" and discuss with your classmates.
  • 1990s: A farmer sells a bag of potatoes for $10. His or her production costs are 0.80 of his or her revenue. On your calculator, graph revenue vs. costs. Run the POTATO program to determine the profit. Discuss the result with students in your group. Write a brief essay that analyzes this example in the real world of economics.

This joke was written back in 2000. (What was I saying earlier about the Precalculus text that expected students to graph lines on the calculator again?) But some of the comments listed under "1990's" often appear in criticisms of Common Core. The fact that this joke is dated 1990's emphasizes the point that the traditionalists are actually attacking the progressive philosophy of teaching. So it predates Common Core -- but of course, the Common Core is an extension of the progressive philosophy.

An engineer, a physicist and a mathematician are staying in a hotel. 
The engineer wakes up and smells smoke. He goes out into the hallway and sees a fire, so he fills a trash can from his room with water and douses the fire. He goes back to bed. 
Later, the physicist wakes up and smells smoke. He opens his door and sees a fire in the hallway. He walks down the hall to a fire hose and after calculating the flame velocity, distance, water pressure, trajectory, etc. extinguishes the fire with the minimum amount of water and energy needed. 
Later, the mathematician wakes up and smells smoke. He goes to the hall, sees the fire and then the fire hose. He thinks for a moment and then exclaims, "Ah, a solution exists!" and then goes back to bed. 

This joke refers to the fact that many proofs in mathematics are existence proofs -- they don't actually tell how to find any solutions. We don't deal with this that much in Geometry -- but in Algebra II, the aforementioned Fundamental Theorem of Algebra tells us that an n-degree equation has n solutions, but it doesn't tell us how to find any of these n solutions -- as opposed to something like the Quadratic Formula, which does tell us the solutions of the equation.

    A physicist and a mathematician are sitting in a faculty lounge. Suddenly, the coffee machine catches on fire. The physicist grabs a bucket and leap towards the sink, filled the bucket with water and puts out the fire. Second day, the same two sit in the same lounge. Again, the coffee machine catches on fire. This time, the mathematician stands up, got a bucket, hands the bucket to the physicist, thus reducing the problem to a previously solved one.
      Another version:
    A mathematician and an engineer are on desert island. They find two palm trees with one coconut each. The engineer climbs up one tree, gets the coconut, eats. The mathematician climbs up the other tree, gets the coconut, climbs the other tree and puts it there. "Now we've reduced it to a problem we know how to solve." 

Mathematicians are also fond of reducing new problems to previously solved old problems. Chapter 11 on Coordinate Proofs reduces many Geometry problems to Algebra I problems. And right here in Chapter 10, we reduced the problem of a cone's volume to that of a pyramid, which in turn is reduced to that of a prism, which in turn is reduced to that of a box.

A mathematician, a physicist, and an engineer are all given identical rubber balls and told to find the volume. They are given anything they want to measure it, and have all the time they need. The mathematician pulls out a measuring tape and records the circumference. He then divides by two times pi to get the radius, cubes that, multiplies by pi again, and then multiplies by four-thirds and thereby calculates the volume. The physicist gets a bucket of water, places 1.00000 gallons of water in the bucket, drops in the ball, and measures the displacement to six significant figures. And the engineer? He writes down the serial number of the ball, and looks it up.

I wish to end this post with one more article that's related to the content of this post. Last week, the researcher Tom Loveless wrote an article called "Tracking and Advanced Placement":

As we know, tracking is controversial -- which is why once again, I'm burying this at the bottom of a long post in the middle of a vacation period. Indeed, Loveless himself writes:

"But tracking is controversial. By definition, it involves differentiating students in terms of their skills and knowledge. Black, Hispanic, and socioeconomically disadvantaged students are historically underrepresented in accelerated tracks."

Loveless concludes that overall, the net results of tracking are positive. It's ironic that, given the racial controversy involved in tracking, it was an experiment in Kenya that led to his conclusion:

"In 2005, an experiment in Kenya could be conducted because schools were granted extra funds to hire first grade teachers ... Both high- and low-achievers in the tracked schools gained more on achievement tests compared to students in the untracked schools."

The Brookings link above doesn't have a comment page. But here's a link to a blog which discusses the Brookings article and allows for comments:

The most prolific commenter in that thread goes by the username "Momof4." I've mentioned her once before on the blog -- I'd definitely count her as a traditionalist. One of her posts goes as follows:

Somehow, this mindset [that tracking is bad -- dw] is never in operation in either athletics or the fine/performing arts, where kids are openly sorted by ability and motivation, even across age groups.

I've already mentioned the traditionalist complaints that progressives apply drill and practice to sports and music, but not academics. Here Momof4 makes a similar complaint, except about tracking.

Here's my response -- it's okay to track in sports and art because being stuck on the lower tracks in those disciplines does not ruin a person's prospects over their entire lifetime. Sure, a person on the lower sports or art track will never make millions as a professional athlete or artist -- but it's possible to have a comfortable lifestyle without being a pro athlete or artist. On the other hand, being stuck on the lowest academic track means that a person is doomed to work in the lowest-paying jobs for his or her entire life!

Momof4 is aware, of course, of the racial implications of tracking, as we see in this next comment:

During the period that all SF [San Francisco -- dw] kids took 8th-grade algebra, I can't help but wonder if there were easily identifiable demographic differences between those who failed and those who passed.

And I don't need to quote the rest of the comment, since we know exactly what those "identifiable demographic differences" are.

Once again, let me emphasize that I favor a pure meritocracy for subjects like sports, art, and entertainment, since so few people of any demographic group are professionals. On the other hand, for subjects that affect whether one can get a job to put food on the table and a roof over one's head, we must make sure that all people -- including all racial groups -- have the opportunity to be successful.

In another comment Momof4 discusses some of the non-racial benefits of tracking as well as of traditional pedagogy. She lauds Singapore Math -- a curriculum favored by traditionalists -- and is happy to see an entire class of sixth graders taking Algebra I.

Another commenter in the thread, BT, also mentions the racial angle:

And of course, removing tracking backfires on the social justice measure. Parents with the means (disproportionately White and Asian) will send their child to private school, home school, use private tutors, test prep, study groups, or just Khan Academy to ensure that the education the schools are denying them.

We've seen some traditionalists (most notably SteveH) mention the idea that students are receiving outside tutoring to bypass Common Core and get into AP Calculus. Here BT says the same thing, but emphasizes race.

Here's my opinion of this -- if members of certain races get more of education because of outside tutoring, then the district gets to wash its hands of any racial discrimination. On the other hand, if the district were simply to provide the accelerated opportunities that BT prefers they do, then it's easy to blame the district of using racially discriminatory tracking.

Recall my story of the monkey and the fish -- the monkey is placed on the tree-climbing track, and the fish is placed on the swimming track. Both animals are happy because they are placed on a track where they understand the material and can be successful. But the fish will become unhappy when it learns that the monkey will be paid ten times as much as the fish is, because tree-climbing jobs pay ten times as much as swimming jobs.

And so I have no problem with having different proportions of races on different tracks provided that all tracks lead to equal-paying jobs. But if there are differences in the pay scales of different tracks -- especially if the pay of one track is so low that one has trouble making ends meet -- then care must be taken to keep the tracks racially balanced.

The only form of tracking that I've ever advocated here on the blog is a mild form of tracking, which I call the Path Plan. Students progress along the paths based on achievement, but along the way we make sure that not too many of any demographic group end up on the lower paths.

Also, notice that to implement Momof4's ideas fully, young students are tracked based on subject -- so a first-grader who's above grade-level in math, below grade-level in reading, and on grade-level in science may wind up with three, four, or five teachers in one day. (This is one non-racial problem I have with Momof4's ideas.) My Path Plan only gradually increases the number of teachers a child sees in one day, as opposed to forcing high school schedules on six-year-olds.

This is the end of today's post. Spring break is over, and regular Geometry posts will resume on Monday, April 4th.

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