I wanted to come back to the popular NYT Magazine article “Why Do Americans Stink at Math?” about how they teach math better in Japan, as you can tell because Japanese students average a higher PISA score than American students. According to the article, the Common Core now offers us another opportunity to teach math better. But, American teachers have consistently failed to exploit the opportunities offered them by educational theorists:
It wasn’t the first time that Americans had dreamed up a better way to teach math and then failed to implement it. The same pattern played out in the 1960s, when schools gripped by a post-Sputnik inferiority complex unveiled an ambitious “new math,” only to find, a few years later, that nothing actually changed. In fact, efforts to introduce a better way of teaching math stretch back to the 1800s. The story is the same every time: a big, excited push, followed by mass confusion and then a return to conventional practices.
You see, it’s not that the math fads of the past failed, it’s that they were never really tried.
In reality, the New Math mostly failed because it was an attempt by math professors to design a curriculum that makes sense to math professors wanting to create new math professors. To students, however, it was repetitious (every September from 1965-1970 I had to study the Number Line in the first chapter of each math textbook), boring, and pointless. The Number Line didn’t do anything to help me think more interesting thoughts about baseball statistics.
The trouble always starts when teachers are told to put innovative ideas into practice without much guidance on how to do it. In the hands of unprepared teachers, the reforms turn to nonsense, perplexing students more than helping them.
The trouble starts earlier when the Powers that Be adopt some smooth-talking salesman’s pitch for a whole new way to teach math without making him test it first on real students. The reason we have the Common Core is not because it aced its Phase I, II, and III experiments involving real students. It was never tested before roll-out.
No, we have the Common Core because David Coleman impressed Bill Gates as significantly less stupid than the typical education theorist, so Gates bribed the educational establishment to get behind Coleman’s baby and make it a fait accompli before anyone had a chance to ask: “Shouldn’t we test this first?” (And keep in mind that I’m relatively positive toward the Common Core versus most of the other junk out there. If our country is going to let one guy control education according to his whims, Bill Gates would be among the less bad choices for that guy.)
Carefully taught, the assignments can help make math more concrete. Students don’t just memorize their times tables and addition facts but also understand how arithmetic works and how to apply it to real-life situations. But in practice, most teachers are unprepared and children are baffled, leaving parents furious.
This paragraph reflects today’s education establishment worldview about the past up until about last week. Until yesterday, children were forced to sit up perfectly straight in their desks and chant the time tables and get rapped on the knuckles with a ruler when they made a mistake. That’s why students “just memorize their times tables and addition facts” instead of developing Critical Thinking Skills and Concern about Social Justice.
In reality, of course, large fractions of students these days fail to memorize their times tables and addition facts.
In other words, liberals are completely amnesiac about how they’ve been running education for a long, long time.
For instance, I went to a Catholic parochial school with nuns, and there was a little knuckle-rapping still going on in the mid-1960s. But by the time I got to St. Francis de Sales’ 7th grade in 1970, the younger teachers had staged a coup and organized a junior high school teaching collective that was more relevant. Most of my schooling in 1970-72, as far as I can remember, consisted of listening in class to album sides from Abbey Road, Deja Vu, Hair, and Jesus Christ Superstar for examples of symbols and metaphors, and sitting in a circle and rapping about how the deaths of Hendrix, Joplin, and Morison bummed us out.
And this was at a prim parochial school. I went to public Millikan Junior High for summer school those years and it looked like Dazed and Confused. Granted, St. Francis de Sales is just over Coldwater Canyon from the Sunset Strip, so we were probably a year or two out in the lead of the rest of the country, but your junior high school probably went through the same changes within a half decade.
Let me repeat this NYT explanation of how things will be better if the educational theorists ever get their full funding:
Students don’t just memorize their times tables and addition facts but also understand how arithmetic works and how to apply it to real-life situations.
Look, forcing students to memorize their times tables and addition facts (e.g., 6+7=13) is not something the current liberal-run system is all that great at. It’s boring for teachers. But you sure can’t apply arithmetic to real-life situations without being instantly aware and really confident that 6+7=13.
As for “understand how arithmetic works,” well, that’s a rabbit hole that more than a few of the greatest minds of the later 19th and early 20th Centuries went down:
“From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2.”
That’s on p. 379 of Volume I of Principia Mathematica by Bertrand Russell and Alfred North Whitehead in 1910. (I haven’t actually read the previous 378 pages.)
There’s a difference between how to work with math and how math works. But the article on why Americans stink at math seems oblivious to that:
The new math of the ‘60s, the new new math of the ‘80s and today’s Common Core math all stem from the idea that the traditional way of teaching math simply does not work. As a nation, we suffer from an ailment that John Allen Paulos, a Temple University math professor and an author, calls innumeracy — the mathematical equivalent of not being able to read. On national tests, nearly two-thirds of fourth graders and eighth graders are not proficient in math. More than half of fourth graders taking the 2013 National Assessment of Educational Progress could not accurately read the temperature on a neatly drawn thermometer. (They did not understand that each hash mark represented two degrees rather than one, leading many students to mistake 46 degrees for 43 degrees.)
May I suggest that numeracy and mathematics are not necessarily the same thing. The New Math of the 1960s, for example, was definitely not intended to emphasize the kind of practical numeracy that say, a carpenter needs. It was intended to make students better at the higher, more abstract forms of mathematics that would form the underpinnings of their college and postgrad math courses that would allow the very smartest students to make the theoretical breakthroughs necessary to win the technological competition in the Cold War and/or create better grad students for math professors.
In general, numeracy and abstract higher math skills correlate, just as the ability to harmonize and the ability to read music correlate. But lots of star musicians are bad at reading music. For example, here’s a list of 15 guitarists who couldn’t read sheet music, including John Lennon, Jimi Hendrix, Eric Clapton, and Eddie Van Halen. Similary, from Wikipedia on the Beatles’ song “Golden Slumbers” on Abbey Road:
“Golden Slumbers” is based on the poem “Cradle Song“, a lullaby by the dramatist Thomas Dekker. The poem appears in Dekker’s 1603 comedy Patient Grissel. McCartney saw sheet music for Dekker’s lullaby at his father’s home in Liverpool, left on a piano by his stepsister Ruth. Unable to read music, he created his own music.
My impression is that while McCartney lacks musical literacy, he’s quite good at numeracy and could probably tell you off the top of his head his annual after-tax royalties on “Golden Slumbers” and how much that bitch Yoko made off his song before Paul wrestled the rights back. (I don’t know specifically about “Golden Slumbers,” but there was a period of years in which 100% of the royalties from Paul’s “Yesterday” went to Yoko, and that sum is no doubt carved in Paul’s soul.)
By the lowly standards of pundits, and even by the higher standards of MBAs, I’m pretty numerate. I can do arithmetical stunts like calculating a weighted average in my head. But I let my wife help my sons with their high school math because all that stuff is over my head. It’s too abstract for me. I don’t like variables that can stand for different things, I like numbers that represent real things. If I didn’t like working with actual numbers so much, I might care more about working with pretend numbers.
Unlike most people, however, I don’t advise children to Be Like Me. But, I think people who theorize in the New York Times about education should try at least to be aware of these tradeoffs.
On the same multiple-choice test, three-quarters of fourth graders could not translate a simple word problem about a girl who sold 15 cups of lemonade on Saturday and twice as many on Sunday into the expression “15 + (2×15).” Even in Massachusetts, one of the country’s highest-performing states, math students are more than two years behind their counterparts in Shanghai.
Adulthood does not alleviate our quantitative deficiency. A 2012 study comparing 16-to-65-year-olds in 20 countries found that Americans rank in the bottom five in numeracy. On a scale of 1 to 5, 29 percent of them scored at Level 1 or below, meaning they could do basic arithmetic but not computations requiring two or more steps.
This PIAAC test of adults from the PISA people showed that immigrants and blacks were pulling the U.S. scores way down versus other rich countries in Europe and Northeast Asia. From the New York Times last year :
The new study shows that foreign-born adults in the United States have much poorer-than-average skills, but even the native-born scored a bit below the international norms. White Americans fared better than the multicountry average in literacy, but were about average in the math and technology tests.
The NYT Magazine article assumes that numeracy is the same as understanding how math works. For example, in reactionary America in contrast to progressive Japan, according to the article,
Students learn not math but, in the words of one math educator, answer-getting. Instead of trying to convey, say, the essence of what it means to subtract fractions teachers tell students to draw butterflies and multiply along the diagonal wings, add the antennas and finally reduce and simplify as needed. The answer-getting strategies may serve them well for a class period of practice problems, but after a week, they forget. And students often can’t figure out how to apply the strategy for a particular problem to new problems.
In contrast, street children in Brazil are numerate and understand the essences:
But our innumeracy isn’t inevitable. In the 1970s and the 1980s, cognitive scientists studied a population known as the unschooled, people with little or no formal education. Observing workers at a Baltimore dairy factory in the ‘80s, the psychologist Sylvia Scribner noted that even basic tasks required an extensive amount of math. For instance, many of the workers charged with loading quarts and gallons of milk into crates had no more than a sixth-grade education. But they were able to do math, in order to assemble their loads efficiently, that was “equivalent to shifting between different base systems of numbers.” Throughout these mental calculations, errors were “virtually nonexistent.” And yet when these workers were out sick and the dairy’s better-educated office workers filled in for them, productivity declined.
The unschooled may have been more capable of complex math than people who were specifically taught it, but in the context of school, they were stymied by math they already knew. Studies of children in Brazil, who helped support their families by roaming the streets selling roasted peanuts and coconuts, showed that the children routinely solved complex problems in their heads to calculate a bill or make change. When cognitive scientists presented the children with the very same problem, however, this time with pen and paper, they stumbled. A 12-year-old boy who accurately computed the price of four coconuts at 35 cruzeiros each was later given the problem on paper. Incorrectly using the multiplication method he was taught in school, he came up with the wrong answer. Similarly, when Scribner gave her dairy workers tests using the language of math class, their scores averaged around 64 percent. The cognitive-science research suggested a startling cause of Americans’ innumeracy: school.
But of course the favela kids making change don’t understand the “essence” of arithmetic, not in the sense that say Bertrand Russell understood its essence. They have rules of thumb they follow that work fine for their tasks. Their techniques aren’t necessarily generalizable, however. Their change-making techniques aren’t going to be much use in getting them through Algebra II, which is now required to graduate high school in some regions in America.
So, in the real world, inculcating the numeracy to make change and getting all students through Algebra II turn out to be somewhat contradictory goals for the bottom half or so of the population. I don’t know what’s the best way to deal with this partial trade-off. But certainly the first step is to be able to publicly admit there is a tradeoff.