Introduction
Today is perihelion -- the day when our planet is closest to the sun. Earth is, believe it or not, closer to the sun in the winter than in the summer. Of course, in the Southern Hemisphere it really is summer during perihelion (and winter during aphelion -- the opposite of perihelion), but in the Northern Hemisphere the relation between the seasons and the distance to the sun is counterintuitive.
Yule Blog Prompt #15: What's on My To-Be-Read Pile for 2023
It's the new year, and so I choose one of Shelli's Yule Blog topics that refers to 2023. But unfortunately, the books that I read, or plan to read, don't match the ones that Shelli and other MTBoS members read.
For example, I'm still waiting for Eugenia Cheng's newest book, The Joy of Abstraction. Of the two local libraries in my area, one of them has all her older books but has yet to order this one, while the other has just barely ordered -- How to Bake Infinite Pie, her last (children's) book before Joy. So I'm still waiting for a copy of Cheng's latest book to arrive at the library -- and I'm hoping that it will be in 2023, so it does count as being on my to-be-read pile for the year.
But Cheng's fifth adult book, like her first book, is on category theory -- a college-level math topic. Even though her books are in a more recreational format (as opposed to a dry textbook), it's still not the sort of book that MTBoS members blog or tweet about. Instead, their favorite topic usually isn't recreational math, but educational math -- as in better ways to teach math to K-12 students.
For example, one popular book is Necessary Conditions by Geoff Krall, foreword by Fawn Nguyen -- and Nguyen, of course, is one of the leading ladies of the MTBoS. As the title implies, Krall describes the necessary conditions for learning math at the middle or high school level, and he states these conditions right in the subtitle: "academic safety, quality tasks, and effective facilitation." This goes along with the three C's at the heart of the MTBoS -- collaboration, communication, and connection.
Shelli, the leader of the Yule Blog Challenge, responded to this prompt in her eleventh challenge post:
http://statteacher.blogspot.com/2022/12/books-on-my-eduread-list-for-2023.html
Here are the books that she lists in that post:
- The Imperfect and Unfinished Math Teacher by Chase Orton
- Teaching Math in the Visible Learning Classroom by John T. Almarode et al.
- Math Games with Bad Drawings and other books by Ben Orlin
And then maybe, for once, I'll be reading the book as the MTBoS, rather than isolating myself with my reading choices.
The Traditionalist Malcolm Kirkpatrick
While I don't read too many books, I do read lots of blogs. But not all of those blogs are affiliated with the MTBoS -- indeed, some of them are traditionalists who are fully opposed to MTBoS methods. As I've pointed out before, I don't fully agree with the traditionalists, but I do partly agree, and I like to know what they are thinking in any case.
Lately, the blog of choice for many traditionalists is the Joanne Jacobs website. Jacobs is the blog's only official authors, but she links to lots of articles, and many traditionalists post comments. While I've referred to several traditionalists on my blog before, there's one such commenter who's much more controversial than the others -- Malcolm Kirkpatrick.
Recently, Jacobs changed her website, and for some time the commenters' names weren't appearing, so I didn't know which comments were Kirkpatrick's. This week, she finally fixed it, and so the name Malcolm Kirkpatrick now appears in the comment thread at the following website:
https://www.joannejacobs.com/post/if-schools-pay-students-will-they-come
The original article is about declining high school attendance and a possible solution -- pay students to go to school, or at least hire them to work on campus between classes.
One of the commenters there is another well-known traditionalist -- Bruce Smith. He often writes about how juniors should take Calculus, just like their same-aged European peers. Here he mentions another European idea -- having two separate high schools, one academic and one vocational.
But our focus here is on Kirkpatrick. He's a traditionalist when it comes to math instruction, but when it comes to how schools should be operated, he's anything but traditional. But there are several students in my class (and I suspect, most of our classes) who would strongly agree with Kirkpatrick. Yesterday he made the following comment:
Malcolm Kirkpatrick:
US DOE NCES District Directory
Boston school district
2018-19 revenue per pupil: $28,454
Why not: ...
Mandate that all Boston District schools must hire parents of enrolled children, on personal service contracts, to provide for their children's education if (1.1) the parents apply for the contract and (1.2) the child is at or above age-level expectations on standardized tests of Reading (any language) and Math and
Mandate that all Boston District schools must administer an exit exam (the GED or SAT will do) that students may take at any age and subsidize private-sector employment or post-secondary tuition to age 18 from the $28,000 that the district would have spent.
So as we see, Malcolm would completely change how schools are funded. His "personal service contracts" are sort of like vouchers, except that the money is paid directly to parents. He expects some parents to spend the money on private school tuition, but he also expects many just to homeschool.
Before we continue, some readers might expect Malcolm to be yet another anti-teacher commenter (like Bruce Smith, possibly) who opposes anything that we teachers do. And so I must point out one important thing: Malcolm Kirkpatrick is a teacher -- indeed, a high school math teacher. For that reason alone, his opinion carries more weight than most non-teachers who comment there.
In previous comments (that he wrote before Jacobs changed her website), Malcolm explains a bit more on how his funding will work. He writes about a hypothetical fraction a/b < 1 of funding that normally would go to the school -- instead, that amount is paid to the parent.
In this post, he mentions that one particular district (Boston, since the original article is about that district) spends $28K per student. If we assume that his a/b < 1 fraction is 3/4, then that works out to be $21K paid to each parent. Malcolm's theory is all about incentives -- districts have an incentive to do this because they save $7K per student, while parents have an incentive to sign up since any amount above $0 is more than what they normally receive.
Notice that another commenter mentions paying $5K per student, but this refers to Texas. There are numerous political reasons (that I don't care to discuss here) why Bostonian schools cost five times as much per student as Texan schools. That guest proposes paying the student the entire $5K and then having the student pay the school if he "agrees that the school has helped him prepare." I suspect that Malcolm's idea is more like paying private school tuition -- the tuition is paid to the school before any classes are taught, out of the amount that the family received.
All of the money is contingent on passing some sort of standardized tests along the way. Malcolm mentions a capstone test in his comment -- in previous comments it's usually the GED, although in this comment he also mentions the SAT. Once the capstone test is passed, the student is considered to be fully educated. The same amount of money (the a/b < 1 fraction above) continues to be paid to the student, except it's now in the form of a wage subsidy or college tuition -- that is, the student must show proof of employment or enrollment in order to get the money. The money ends at age 18 -- that is, when the student would have graduated from a traditional public high school.
Notice that while the lower tests are based on "age-level expectations," the capstone test can be taken at any age. If a one-year-old (however unlikely) can pass the GED, then that child is considered fully educated and thus qualifies for the wage subsidy.
Several other commenters in the past besides Malcolm have proposed not requiring the last few years of high school. In response, I've blogged about a "Dickens age" -- below a certain age, I believe that sending students to work instead of school would be torture, similar to a Charles Dickens novel.
In parts of Europe, students are allowed to leave school after the equivalent of sophomore year. (Once again, my knowledge of the British educational system comes from Harry Potter -- the O.W.L.s that wizards in that book are known as "O-levels" or "GCSEs" for Muggles. After they complete them, students are allowed to leave Hogwarts -- and in fact, the Weasley twins do exactly this in Book Five, well after they completed their O.W.L.s in Book Three.) This implies a Dickens age of 16. In my posts, I allowed for dropping the Dickens age one year below the British standard, or 15. (The vocational schools mentioned by Bruce Smith often start at age 15 as well.)
But Malcolm disagrees with the idea of a Dickens age altogether. To him, it's having to go to school, not having to go to work, that is Dickensian. He believes that the students who hate going to school would enjoy working more, even if the work is hard.
So Jacobs asks the question, "if the schools pay students, will they come?" -- and the answer that Malcolm provides is a resounding yes. According to him, schools should definitely pay students -- or at least their parents -- money via the "personal services contract" to learn, and the students will come if they know that money is on the line.
I often wonder what my education would have looked like if I'd grown up under Malcolm's vision. I point out that on the original timeline, I took the SAT in January of my seventh grade year as part of a special program, Johns Hopkins. My math score was 710, but my verbal score was 400. My verbal score wasn't high enough to qualify for Johns Hopkins.
Malcolm never states what his cutoff score for the SAT capstone test would be. He likely wouldn't be as strict as Johns Hopkins -- maybe my 400 would have been good enough for him. If it's good enough, then that means I would have qualified for the wage subsidy midway through my 7th grade year (just after my thirteenth birthday).
If 400 isn't good enough, I might have taken the test in October of my freshman year -- the last month before I moved to another city (and thus a convenient time to try to qualify). I believe that I would have scored high enough to qualify at that time -- if not on the SAT, then at least the GED. This would have been just before my fifteenth birthday (that is, close to my original Dickens age).
OK, so what would I have done with all that free time after qualifying for the subsidy? Since I would ultimately attend UCLA on the original timeline, I'd likely try to attend some college. So I might try to spend some money on a college class or two. If I were to go for employment instead -- hmm, I'm not sure whether I'm up for any sort of physical labor (which is what made Dickens Dickensian). For example, on CBS tonight is the season premiere of Tough as Nails, where players compete to finish tasks in factories and on construction sites. I'm not "tough as nails," so I don't think I can do anything I see on that show. On the original timeline, I ultimately work at the UCLA library, so maybe if I could get hired at a library, then I'd take the employment.
It's a good exercise to see what Malcolm's vision would look like for some of my students -- especially those who are struggling in my actual class.
My Soccer Player in Malcolm World
Let's start with the guy in my sixth period Math I -- the guy who's argued the most about not enjoying math and believes that he shouldn't have to take it. So what would Malcolm World look like for him?
In December, he was asking me to round his grade up to a D- so that he could be academically eligible to play soccer -- at the time, his grade was in the high 50's. But he ended up never taking the final exam, and his grade dropped to the low 50's. He blamed this struggles and absences on the soccer practice and games that took place during sixth grade.
Now let's take him to Malcolm World. First of all, in Malcolm World he wouldn't be asking me to round up his grade -- instead, he'd be asking me to help him pass the GED so he can get the wage subsidy.
Some teachers lament the increase in standardized tests required in school. Many often see them as encroaching on teacher autonomy -- they think teachers can't be trusted to grade students, so those in power have to find ways to evaluate students themselves.
But for Malcolm (who, I remind you, is a teacher himself), standardized tests are good because they use incentives to change the direction of discussion. In the real world, my student asked me to make the class easier so that he can pass with less work. In Malcolm World, he knows that I lack the power to make the GED any easier. So he'd instead ask me to give him more work so he can pass the test, instead of less work.
I've glanced at the GED test. It appears that the math required to pass it is on the level of Math I -- there are many linear equations to solve and functions to graph. The Geometry section is mostly on measurement formulas. So the math I'm teaching my student now will appear on the GED.
Once he passes the GED, my student will be out of my class and out of our school forever. But how would he be able to play soccer? Well, soccer clubs exist, but they cost money. (Of course, some clubs, like the champion LA Football Club, pay its players to play soccer. But my student isn't as good as the professionals, so he'd have to pay the club to let him play.) So he'd have to earn enough money from the wage subsidy to pay the club -- meaning he'd have to have a job, since there's no subsidy without a job.
In our world, he practices during first period, and matches are played in the afternoon. But the coach will know that in Malcolm World, most of his players would have jobs. So instead, practice would be held the same time of the day as competitions, which is more convenient for employers. My guy might get, say, a fast food job during breakfast and lunch hours, so he can play soccer after work.
Malcolm knows that many businesses will be reluctant to hire teens. And so he provides another incentive to hire them -- eliminate the minimum wage entirely for teens (which is why he's giving them a wage subsidy), and have their wages be determined by the free market only. Since the California minimum wage is $15.50, let's just drop the 1, and say that the going wage for teens at fast food joints would be $5.50 per hour in the absence of a minimum wage. That isn't very much money, but if we add this to the wage, this might be enough for him to afford the soccer club.
By the way, I was considering telling this class about Malcolm World before winter break. But in the end, I didn't -- it would only lead to an argument. I reckon that this guy would have told me that he wished he were living in Malcolm World -- and then that's it. It's not as if I have any power to make Malcolm World a reality.
My Calculus Student in Malcolm World
Another case study is that girl who was in my AP Calculus class last year. During the pandemic, she took an after-school job, and she continued to work there once the schools reopened. Once she turned 18 in early January (yes -- this week is probably her nineteenth birthday), she began working full-time.
Oh, and here's the kicker -- her job was at a factory. And she didn't work just 2-4 hours after school -- it was more like 10-12 hours. I'm reminded of the Square One TV song "Time Keeper," where Tempestt Bledsoe sang about how she had to work ten hours, from 4 to 2 (since 4 + 10 = 2 on the clock). Well, for my student, this wasn't just a song -- her actual work hours were 4 PM to 2 AM.
You may ask, how did she manage to work long hours, go to sleep, and show up to school on time? The answer is that she didn't. She regularly showed up to our third period Calculus class very late -- or, more often, not at all.
She's a bright math student, or she wouldn't have been placed in AP Calculus. The few times I saw her, I was able to get her to catch up quickly. But as the AP exam approached, she attended class less often -- and she ended up not taking the AP. She did return for finals -- I allowed her to make up the final project, and I gave her enough points to get a D- and pass the class. But I still feel guilty for not making enough effort to get her to show up for the AP exam. Perhaps I could have sent her or her an email message each day in April, reminding her that the AP was coming soon.
OK, but all of this is in the real world. How would things be different for her in Malcolm World? Well, a student in an AP class is likely to have passed the capstone GED or SAT before entering the AP class, and so she would already have the wage subsidy.
I often wonder whether AP classes and exams would even exist in Malcolm World. Well, Malcolm often writes that credit by exam will apply to the entire K-PhD system. so there must be some way for my student to take a Calculus exam, even if it's not specifically called AP.
But she might take the class directly at some sort of college. In Malcolm World, colleges might allow her to take just one class in a semester -- the Calculus class. If the class is at, say, 9 AM, she might then be able to work at the factory from noon to 10 PM. Then she can get home at a reasonable hour, study for math, and wake up in time for her class. It's only because high schools, not colleges, require a full (or nearly full) schedule that she's having so much trouble. Perhaps if she works at the factory for a year, her wages plus the subsidy for that year may be enough to pay for a full load at a college.
Why I Can't Fully Support Malcolm World
I often point out that work is Dickensian -- especially long hours of factory work. To me, going to school protects students from having to work long hours. But we saw above that school didn't save my Calculus student from long hours at the factory -- she had to go to school and work there.
But still, there are a few reasons why I can't agree with Malcolm Kirkpatrick. For one thing, I wonder what math classes will look like in Malcolm World.
We already know that many students don't enjoy math. If math were an elective rather than required, they'd drop math the instant they can. Of course, STEM will require more math than the GED, but those students will just say that they don't want a STEM job.
I can easily see the following -- the math haters will drop math as soon as possible. Some stronger math students might consider continuing math, but peer pressure is strong at this age -- they see their friends dropping math, so they decide not to take math as well. In the end, no one takes math past GED-level.
Then what would happen to the STEM jobs? Well, STEM jobs will still exist, but no teenagers will be studying math. Instead, more mature people will realize that it's more important to learn what's needed for jobs than what their friends are learning -- but they might be well into their 30's.
The Fields Medals are given to the top mathematicians in the world who are under 40 years old. But by the time anyone in Malcolm World studies enough math to do research, they're already over 40. So no one will win Fields Medals at all in Malcolm World.
Of course, the type of person who would earn a Fields Medal is someone who's always enjoyed math from an early age. But then again, the word "nerd" exists as well. Requiring everyone to take math removes some of the stigma of being a strong math student -- not requiring it discourages even strong students from continuing in math as peer pressure takes over.
In Malcolm World, incentives matter. A STEM student might apply to an easier non-STEM job, which means that there are fewer jobs for non-STEM workers. The only way to convince a STEM worker to take the STEM job is for the STEM job to pay more money.
Now there's another thing to consider -- gender and racial gaps. As we already know, there are more males than females in STEM, and more whites and Asians in STEM jobs. Thus a world where STEM jobs pay more money is one where whites, Asians, and males make more. This is already true to some extent, but it's even stronger in Malcolm World. But in a world where everything is determined by the capstone and other gatekeeper tests, many will accuse the tests of being biased in favor of males, whites, and Asians.
Malcolm himself has addressed this in some of his comments. His generic term for "gender or race" is "variety of human." But all he says is that different varieties of human have different interests or even abilities -- eliminating the gaps is not a problem worth solving.
This takes us right back to my old "magic red button" analogy. If these exists a red button that, when pressed, increases gaps between groups yet improves outcomes for all, should we press the button,
Malcolm would probably press that magic button -- and indeed, he might even consider his world (where personal service contracts with districts exist) to be the world of the magic button.
But as I said before, I'm not sure I can press that button. As long as gaps exist, people will be skeptical that they are really better off. I want to see my students -- most of whom are Hispanic, nearly half of whom are female -- be successful at STEM and earn those high paying jobs. In Malcolm World, they might not even choose to study math. In my world, I want them to study math and go into STEM.
Rapoport Question of the Day
Today on her Mathematics Calendar 2023, Rebecca Rapoport writes:
-x^4 + 16x^3 - 96x^2 + 256x - 256 = 0
Our first question on the new Daily Epsilon of Math calendar is a quartic equation. We can solve this quartic by using trial and error. We first try +1, -1, +2, and -2, and skip +3 and -3 because 3 doesn't divide 256 evenly.
Let's try 4. (Here I've already factored out -1 from the original quartic.)
-(x - 4)(x^3 - 12x^2 + 48x - 64) = 0
So 4 is a solution -- and of course, today's date is the fourth, so we're already done.
In reality, of course we can't just find one root of a polynomial and claim that we're done. We should continue with the cubic x^3 - 12x^2 + 48x - 64 to find the other three roots.
At this point, we might suspect that the other roots are negative or imaginary -- but Rapoport doesn't say "find the positive real solution." In fact, now we might suspect that 4 is a root of multiplicity 4 -- that is, the original factors as -(x - 4)^4. And that turns out to be the case.
Polynomial equations like this one will appear in Chapter 8 in my Math III classes, which is why I decided to highlight today's question.
Conclusion
I have no power to implement Malcolm's World -- but then again, neither does Malcolm. He must deal with the fact that his students need his class to graduate, and there is no capstone test or wage subsidy that they can take to avoid it.
He writes that there is one thing that he can give his classes right now -- free time. His Exit Passes are real Exit Passes -- if they pass, he writes them a pass to the library. He says that his students are then motivated to work harder because they know they can get out of class faster.
In fact, he often writes that "a hour of class" is a a complete nonsense term, similar to an inch of love or a pound of friendship. Class should end when the lesson is learned, not when a bell rings. (His GED at any age idea is an extension of the idea that learning shouldn't be mentioned by time.)
Of course, as a new teacher, if I wrote too many library passes it would look suspicious, so I won't try to implement this idea. Still, it gives me something to think about as our planet starts to move farther away from the sun on its annual journey.
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