Statistics or Calculus in High School?: Critiquing Matthew Yglesias (& Kareem Carr)
& BRIEFLY NOTED: For 2021-12-13 Mo
First: Statistics or Calculus in High School?
I wish to concur in part with and to dissent in part from Matthew Yglesias here.
(1) Concur: Figuring out what people have learned and retained is absolutely essential to doing education successfully.
(2) Dissent: On the other hand, often scoring badly on standardized tests is used as an excuse to essentially give up.
(3) Dissent: And I really, really disagree that “you can’t really teach statistics properly without calculus”:
Matthew Yglesias: California’s Math Detracking Initiative Seems Pretty Misguided: ‘One big problem… is you can’t really teach statistics properly without calculus, which I think goes to underscore that this is not really a debate about the proper sequencing of math… [but] another manifestation of the dysfunctional tendency in some edu-left circles to stigmatize all efforts at measurement. If you sort kids into different math tracks based on their test scores, that might reveal that Black and Latino kids are doing worse than white and Asian ones. If you refuse to sort, you can pretend you’ve achieved equality. But will you provide useful education?… Lizzy Hull Barnes, who is in charge of San Francisco math education, saying it’s invalid to look at test outcomes to see whether kids are doing better or worse: “In high school they take the test one time in 11th grade…. It’s difficult to use that as a measurement for success in all of mathematics.” I 100% understand that lots of perfectly normal parents and students got test fatigue in the heyday of No Child Left Behind and that there is such a thing as too much assessment. But the authors of the proposed framework are trying to push the state’s schools in a strange new direction with almost no valid research they can cite to indicate that it’s likely to work. It’s particularly egregious because while California schools do exhibit a racial achievement gap, their results are sub-par across the board. They should be trying to imitate other, better-performing states, not inventing a whole new way of doing high school math…
LINK:
Kareem Carr: ‘Controversial opinion: Trying to teach statistics without using calculus actually leads to a lot of bad statistics classes, where the rules seem really arbitrary and the formulas feel like they come out of nowhere…
Here is the fact: the rules of statistics are really arbitrary. The formulas do come out of nowhere.
The formulas come out of the brain of Carl Friedrich Gauss. The formulas for average (rather than median) and for least-squares linear fit make sense if the idea is to minimize the sum-of-squares of the retrodiction errors, which makes sense if the distribution of disturbances to some underlying linear true relationship is:
The Gaussian.
Why should this condition hold? I am told that Carl Friedrich Gauss strongly believed that it must hold because otherwise taking-averages and least-squares would not be the right things to do, and no God could be so cruel as to inflict such non-rightness on his mathematicians.
We, today, however, do not rest our confidence in taking-averages and least-squares procedures on a Benevolent Providence. We rest it on the Central Limit Theorem and the convergence-in-distribution of a sum of independent random variables to a Gaussian distribution if Jari Waldemar Lindeberg’s Condition <https://en.wikipedia.org/wiki/Lindeberg%27s_condition> is satisfied. What is Lindeberg’s condition? That each individual random variable in the sum be sufficiently small compared to the total.
In all this, knowing calculus—in the sense of having it in your intellectual panoply—is very useful for getting from Lindeberg’s Condition to the Central Limit Theorem. Knowing calculus is very useful in getting from the Gaussian distribution to the optimality of taking-averages and least-squares. If calculus is part of your intellectual panoply, you have in your mind a mental structure that makes sense and provides a rationale for standard statistical procedures. If you took calculus once, long ago, you have in your mind… a vague memory… that tells you that the formulas did not come out of nowhere, or, rather, did not come out of nowhere if you jiggered the problem and chose your assumptions carefully so that the formulas do come out of them.
And here’s the crux: in the real world of using statistics, where your sample is finite, where one or a few of the disturbances are large relative to the total, where your sample is non-random, where your observations are not independent, the Central Limit and Gauss-Markov theorems are of little use. Yes, taking averages and least-squares fits are the first things you should do. But then you should not down tools, because calculus! Then you should do other things and see if they agree with taking-averages and least-squares. And if they do not you should think hard about the problem.
Statistics is, IMHO, better taught as if you are teaching engineers rather than mathematicians. And I think that California is probably right in wanting to put engineering-focused statistics before calculus in the sequence.
But somebody should try it. And then we should assess how well it works, through tests and follow-up studies, and see.
One Video:
Ancient Athens 3D: A Tour of Classical Athens (5th Century BC) <https://www.youtube.com/watch?v=ulAxMLJ7O7M&t=1s> <
Very Briefly Noted:
Ben Thompson: Intel vs. TSMC, How Samsung and TSMC Won, MAD Chips: ‘Globalization generally, and chips specifically, are a sort of weapon of mutually assured economic destruction… more comforting than you might expect when considered systematically… <https://stratechery.com/2021/intel-vs-tsmc-how-samsung-and-tsmc-won-mad-chips/>
Barry Ritholtz: Business Not As Usual: Foundations of ESG: ‘Robert Schwartz and Robert Zevin were Wall Street curiosities. At the height of an era defined by Ivan Boesky’s mantra of greed, the two leftists pioneered what became known as sustainable investing… <https://ritholtz.com/2021/12/business-not-as-usual-the-foundations-of-esg/>
Philip Ball: The Chase for Fusion Energy: ‘An industry of nuclear-fusion firms promises to have commercial reactors ready in the next decade… <https://www.nature.com/immersive/d41586-021-03401-w/index.html>
Jay Daigle: What is the Axiom of Choice?: ‘One of the easiest ways to start a (friendly) fight in a group of mathematicians is to bring up the axiom of choice… <https://jaydaigle.net/blog/what-is-the-axiom-of-choice/>
Jim Baggott: ‘Shut Up & Calculate’ Does a Disservice to Quantum Mechanics: ‘The cliché has it that the Copenhagen interpretation demands adherence without deep enquiry. That does physics a disservice… <https://aeon.co/essays/shut-up-and-calculate-does-a-disservice-to-quantum-mechanics>
William Quinn & John D. Turner: Riding the Bubble or Taken for a Ride? Investors in the British Bicycle Mania__: ‘A hand-collected sample of 12,000 investors during an asset price reversal in the shares of British bicycle companies between 1895 and 1900…. Informed investors reduced their holdings substantially during the crash, suggesting that they were riding the bubble. Those who performed worst were… gentlemen living near a stock exchange, who had the most time, money, and opportunity to engage in speculation… <https://www.econstor.eu/bitstream/10419/246500/1/1776320050.pdf>
Paragraphs:
Cosma Shalizi: Review of Ashworth, Berry, & Bruno de Mesquita, Theory & Credibility: ‘The big point that the authors want to drive home is that rigorous formal modeling and careful analysis of empirical data ought to be seen as complements…. It is by articulating formal models that we see what our ideas imply, and what assumptions are needed to reach various conclusions…. The “credibility revolution”, the rather self-congratulatory name given to the wide adoption of more careful methods for non-experimental causal inference in the social sciences since the 1980s. (I wish I could remember who joked that it’s really been a revolution of in-credulity about everyone else’s identification assumptions.) Good courses in econometrics and data analysis for the social sciences will now give these methods a lot of attention…. Causal-inference techniques. Rather, they’re relevant here because they often give us way to estimate the quantities we’re theorizing about, and not other quantities which resemble our favorites only superficially…
LINK: <http://bactra.org/reviews/theory-and-credibility.html>
Rachel Donadio: Lampedusa’s ‘The Leopard,’ Fifty Years on: ‘One novel is the key to Sicily: “The Leopard,” Giuseppe Tomasi di Lampedusa’s masterpiece… the decline and fall of the house of Salina, a family of Sicilian aristocrats…. Don Fabrizio, the world-weary, cleareyed Prince of Salina, scion of an old feudal family and lover of astronomy. It opens in 1860 with the landing in Sicily of forces intent on unifying Italy and ends in 1910, when a priest comes to assess the reliquaries of the prince’s now aged spinster daughters. In between, it recounts the fortunes of the prince’s favorite nephew, Tancredi, who supports the unification efforts of Giuseppe Garibaldi more out of opportunism than idealism and eventually becomes a diplomat. Tancredi’s career is made possible only by his marrying money—which inevitably means marrying down. To the horror of his aunt, the devastation of a cousin who loves him and the wry comprehension of his uncle, Tancredi falls in love with Angelica, the beautiful daughter of an upwardly mobile landed peasant father and an illiterate mother…. Tancredi… speaks the novel’s most famous line: “If we want things to stay as they are,” he tells his uncle, “things will have to change.” Tancredi’s declaration lies at the heart of “The Leopard,” at once a loving portrait of a vanished society and a critique of its provincialism…
LINK: <https://www.nytimes.com/2008/07/29/arts/29iht-booktue.1.14826755.html>
Ellen Kushner: Hallelujah, Hallelujah …& Jewish Space Lasers: ‘I finally got my Introduction for the French centennial edition of THE WORM OUROBOROS done!… 3,900 words, which is longer than some papers I wrote in college. And possibly better-researched. God bless the internet!… Here’s the most requested bit: "It took my friend Caroline Stevermer to show me the way in to E.R.Eddison’s world. We had met our first day at University, quickly discovered our love of the same books (which always makes a friendship!), and gradually admitted to each other that we both wanted to be writers of fantasy.1 Caroline is much better-read than I. And happens to be one of the world’s great readers-aloud. She picked up my copy of The Worm, leafed through it a little, and then began to read to me, her voice pitched low with amusement: ‘Then fared Juss to the guest-chamber, where Lord Brandoch Daha lay a-sleeping, and waked him and told him all. Brandoch Daha snuggled him under the bedclothes and said, “Let me be and let me sleep yet two hours. Then will I rise and bathe and array myself and eat my morning meal, and thereafter will I take rede with thee and tell thee somewhat for thine advantage. I have not slept in a goose-feather bed and sheets of lawn these many weeks. If thou plague me now, by God, I will incontinently take horse over the Stile to Krothering, and let thee and thine affairs go to the devil.”’ And there it was at last! A hero I could recognize: the sharp-tongued, witty one who also just happened to be the best swordsman in the world…. In his company, I was willing to venture along with his companions…
LINK:
Steven Johnson: Seven Types of Serendipity: ‘So much of the creative process is about being open to happy accidents. But how do you make them more likely to happen?… Exaptation…. Gutenberg borrowing the screw press that he’d observed winemakers using and applying it to his nascent printing technology…
LINK:
David French: Deconstructing White Evangelical Politics: ‘My response to the question of whether theology and doctrine were of primary importance to the movement was alway… Republican Christian conservatism is mainly driven by deeply rooted, theologically coherent faith convictions and not by the perhaps more deeply rooted “folkways” or customs of a disproportionately white, disproportionately rural, and disproportionately Southern American subculture. I no longer believe this to be true…. The theological convictions of Christian conservatism were put to a profound stress test, and the convictions failed. Partisanship prevailed. Populism prevailed. In some ways, the South prevailed…. I’m old enough to remember the words and expressed beliefs of even some of the most enthusiastic Trumpist Evangelicals before they supported Trump, and this much I know: If I’d told them in December 2014 that white Evangelicals would shortly vote in overwhelming numbers for a thrice-married man who bragged about grabbing women by their genitals, appeared in a Playboy movie, paid hush money to cover up an affair with a porn star, and was facing multiple corroborated claims of sexual harassment and sexual assault, they’d say that only Democrats were that hypocritical. Then, if I followed that up by saying that a disproportionate number of that same Evangelical community would shun pre-vaccine mask-wearing and social distancing in the midst of a deadly pandemic that would claim hundreds of thousands of American lives and then disproportionately reject life-saving vaccines, they’d think I was an anti-Christian bigot. If I capped off my prophecy by noting that white Evangelicals would be far more likely than virtually any other American community to embrace wild election conspiracy theories and then a subset of that community would literally storm the Capitol with prayers on their lips, then their assessment of me would be clinched. They’d think I’d simply lost my mind. But that’s where we are. And don’t think for a minute that those views are the only Evangelical outliers…
LINK:
The problem with quantum physics is that it is hard to find a good human scale analogy for understanding it. Every analogy one might choose has a comprehensible and visible mechanism underlying it. Try trains running every 15 minutes with a standard deviation of three minutes, and you can get a wave function for trains, but there are still physical trains out on the tracks with observable positions and velocities. In quantum mechanics, trains can only be observed when they arrive at the station, and no one is exactly sure what it means to observe or arrive or exactly what a station is.
"Shut up and compute" is useful advice if you want to use quantum physics, but it is philosophically unsatisfying. People have devised more satisfying philosophical approaches, but a lot depends on what one finds satisfying. As with most of philosophy, it's more about the human mind, and less about the world in which that mind exists. The Book of Job covers some of this. Some people feel it is important to understand the "essence" of things. Others don't. I'll stop here and quote Joan Didion, “What makes Iago evil? Some people ask. I never ask.”
(I'm not very religious, but now and then I pray to god or the gods that no one finds it necessary to do a version of Othello with a tedious backstory explaining Iago's motivation and, possibly worse, has Othello and Desdemona meeting cute.)
So, algebra's essential for all math. But so's formal logic, which is missing from high school curricula.
I think inductive reasoning, standards of evidence, and critical thinking should be a mandatory part of the curriculum. Then logic. Also rhetoric, so people can defend themselves against fallacies. All are missing from the K-12 curriculum, or not taken seriously.
Then we can get into arithmetic and algebra. *Everyone* needs algebra. From there, the next step is probability.
And after that, statistics. Which is hard. And it's not really mathematics. It's a different, though closely related, field. Calculus doesn't tell you shit about statistics, though it can be one of many tools for trying to analyze statistical distributions. Half the statistics course needs to be on data collection, uncertainty, survey design, data presentation, all that stuff, none of which is actually math. I believe at this point that Bayesian approaches are better than frequentist, but both should be taught.
These need to be mandatory courses, and if the K-12s won't do it, they should be required college courses.
Geometry, trigonometry, and analytic geometry are important for those continuing in the sciences, though frankly not nearly as important as probability and statistics for everyday life. And analytic geometry isn't in the high school curriculum. Calculus... should come after analytic geometry, which it doesn't; when it comes after analytic geometry, it's easy.