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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.

My wife says that the market people here in the Philippines have a lot of trouble calculating fractions of a kilo, etc. They have a set price for a set amount. If you ask for 1/4 they don’t know the price.

Replies:@Anonymous@Foreign Expertin the philippine setting

here's why the chinese are better in math than filipinos: in the private schools where they are enrolled, the schools there teach twice the math an ordinary filipino would get in public and private school--english math in the morning, and chinese math in the afternoon.

Replies: @dixieI Choose C:

So all we need to do to reach these naive mathematicians is to recast problems in calculus and orbital mechanics into the isomorphic problems of making change.

Simple. I can close the achievement gap.

I can haz billions now?

Replies:@AnAnon@The most deplorable one"recast problems in calculus and orbital mechanics into the isomorphic problems of making change." - your billions as soon as you do this.

Speaking of your wife, did she get her new kitchen?

Replies:@Steve Sailer@Sister RaySame old kitchen, but now has, thanks to iSteve commenters, an ADA-compliant (i.e., shorter) dishwasher that fits under the old counters. She says thanks.

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 that would allow the very smartest students to make the theoretical breakthroughs helpful to win the technological competition in the Cold War.I remember my Algebra II/Trig teacher taking the last, free day of class to explain credit card and other real-world math to us because we’d finished all our curricular stuff. I remember wondering why they didn’t always teach math this way. Not that any of it involved Algebra II or Trig.

I like numbers that represent real things.Indeed, that’s the way to get boys interested in math. Girls may be a shade behind boys in terms of math ability, but they are automatically motivated as soon as authority says “you must learn this.” Boys need to have a reason to be interested. Give them real-world examples that appeal to them (artillery strikes, drag races, defusing bombs, demolition, making money, etc., not sharing fruit and the like) and you have a much better chance of hooking their interest.

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).”Boom: “girl who sold lemonade-” zzzzzzzzzzzzzz…

How about selling Ferraris? AR-15s? Speed boats?

teachers tell students to draw butterfliesHow about telling them to draw a sword? A fighter jet?

Lemonade, butterflies, zzzzzzzzzzzzzzzz…

Replies:@Hunsdon@SvigorOn the first day of battle, your platoon suffers 25% casualties. However, on the second day of battle you inflict twice as many casualties as you suffered on the first day. Given a platoon strength of 24 men, how many casualties did you suffer, and how many did you inflict?

Boys will get that one right every time.

Replies: @Steve SailerSame old kitchen, but now has, thanks to iSteve commenters, an ADA-compliant (i.e., shorter) dishwasher that fits under the old counters. She says thanks.

“So all we need to do to reach these naive mathematicians is to recast problems in calculus and orbital mechanics into the isomorphic problems of making change.”

That’s easy, you vote for Obama!

“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.”

A couple of points.

Long time guitar player who is semi-professional musician (and long time reader of you) who comes from a musical family, and can actually read music. By semi-professional, I mean some one who has earned money to play music, but not as a full time career (just on weekends playing for weddings, at local clubs, etc…).

Not surprised that a bunch of great guitar players cannot read music. A while back I was talking to a music professor about some musicians I have played with and some music students I have had. I mentioned that one of the guitar players I played with had “perfect pitch” and after listening to a song once could figure out a song and play it back for the rest of the band even though he did not know any music theory (he barely knew what the names of the notes he was playing) or how to read music (said guitar player ultimately became a biologist IIRC).

He said he was not surprised, he said that people like that are probably in the top 2 to 4 percent of music ability. Apparently, there is a music IQ exam given by music educators you can take to determine your ability. According to him, some professional pop and rock musicians have scored fairly low on the exam according to papers published in music education journals. LOL… not surprised (let’s face a lot of pop and rock music is not very challenging).

He agreed with me that for some musical geniuses, music notes as symbolized on a piece of paper are just not necessary, because they are just an intermediate symbolic representation for some sounds that the musical genius person can directly and viscerally experience and recall without a problem. Written music is a good way to communicate musical information, but for a musical genius it may seem unnecessary.

I would not be surprised if Lennon, Hendrix, Clapton, and Van Halen were in the top 2 to 4 percent or higher like the guitar player I once played with. They certainly were all very inventive and outstanding musicians and probably each was a musical genius in their own way.

Replies:@Steve Sailer@SenecaI took a year of piano lessons at around age 8 and I was quite good at reading music off the page, just as I was at reading words off the page. But it was just a mechanistic skill that didn't do anything to add to my paltry natural musical skills.

Don’t forget Tom Leher’s classic satirical song on the “original” new math:

A couple of points.

Long time guitar player who is semi-professional musician (and long time reader of you) who comes from a musical family, and can actually read music. By semi-professional, I mean some one who has earned money to play music, but not as a full time career (just on weekends playing for weddings, at local clubs, etc...).

Not surprised that a bunch of great guitar players cannot read music. A while back I was talking to a music professor about some musicians I have played with and some music students I have had. I mentioned that one of the guitar players I played with had "perfect pitch" and after listening to a song once could figure out a song and play it back for the rest of the band even though he did not know any music theory (he barely knew what the names of the notes he was playing) or how to read music (said guitar player ultimately became a biologist IIRC).

He said he was not surprised, he said that people like that are probably in the top 2 to 4 percent of music ability. Apparently, there is a music IQ exam given by music educators you can take to determine your ability. According to him, some professional pop and rock musicians have scored fairly low on the exam according to papers published in music education journals. LOL... not surprised (let's face a lot of pop and rock music is not very challenging).

He agreed with me that for some musical geniuses, music notes as symbolized on a piece of paper are just not necessary, because they are just an intermediate symbolic representation for some sounds that the musical genius person can directly and viscerally experience and recall without a problem. Written music is a good way to communicate musical information, but for a musical genius it may seem unnecessary.

I would not be surprised if Lennon, Hendrix, Clapton, and Van Halen were in the top 2 to 4 percent or higher like the guitar player I once played with. They certainly were all very inventive and outstanding musicians and probably each was a musical genius in their own way.

Replies: @Steve SailerI took a year of piano lessons at around age 8 and I was quite good at reading music off the page, just as I was at reading words off the page. But it was just a mechanistic skill that didn’t do anything to add to my paltry natural musical skills.

“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.”

A couple of points.

Long time guitar player who is semi-professional musician (and long time reader of you) who comes from a musical family, and can actually read music. By semi-professional, I mean some one who has earned money to play music, but not as a full time career (just on weekends playing for weddings, at local clubs, etc…).

Not surprised that a bunch of great guitar players cannot read music. A while back I was talking to a music professor about some musicians I have played with and some music students I have had. I mentioned that one of the guitar players I played with had “perfect pitch” and after listening to a song once could figure out a song and play it back for the rest of the band even though he did not know any music theory (he barely knew what the names of the notes he was playing) or how to read music (said guitar player ultimately became a biologist IIRC). I also mentioned that I had a young guitar student who had a similar ability to hear something once and be able to play it without the assistance of sheet music.

He said he was not surprised, he said that people like that are probably in the top 2 to 4 percent of music ability. Apparently, there is a music IQ exam given by music educators you can take to determine your ability. According to him, some professional pop and rock musicians have scored fairly low on the exam according to papers published in music education journals. LOL… not surprised (let’s face a lot of pop and rock music is not very challenging).

He agreed with me that for some musical geniuses, music notes as symbolized on a piece of paper are just not necessary, because they are just an intermediate symbolic representation for some sounds that the musical genius person can directly and viscerally experience and recall without a problem. Written music is a good way to communicate musical information, but for a musical genius it may seem unnecessary.

I would not be surprised if Lennon, Hendrix, Clapton, and Van Halen were in the top 2 to 4 percent or higher like the guitar player I once played with. They certainly were all very inventive and outstanding musicians and probably each was a musical genius in their own way.

As the innumerate say, there are three types of people, those who can count and those who can’t.

I recently came across a neat way to get some boys interested in learning their trigonometric identities.

Superheterodyne radio receivers!

cos(A)cos(B) = 1/2(cos(A+B) + cos(A-B)) … and then they look at what filtering is needed and how it allows you to recover the modulating signal … etc.

I am sure it will work!

Also, it’s better than voting for Obama.

This is actually a misconception. The New Math was nothing like how math professors in general think about math and how they think it should be taught and learned. Mathematicians actually have a much less rigorous view of and approach to math than is popularly assumed. The popular perception is that the rigorous, formal presentation of math is a method itself, rather than just that – the presentation of math, and that mathematicians basically sit at their desks staring at axioms until theorems just magically start pouring out. That is nothing like the actual practice of math in real life.

Solving word problems requires a level of “g” that not everyone has and cannot be taught. Algorithms like the butterfly can be taught and are useful for people who would not be able to figure out how to do a problem by themselves.

I therefore favor algorithmic learning more than most people. You can’t make people smarter, but you can teach them how to do specific types of problems. A supergenius would be able to integrate an expression like x^2 tan(x) in his head. The rest of us mortals need to integrate by parts.

Replies:@jim@BehindTheLinesDo you mean x tan^2(x)?

I also got caught in the New Math craze of the ’60s being an elementary-age student at the time. Definitely agree with Steve that math teachers had difficulty with it. Maybe teachers 20 years from now will be teaching the esoteric underpinnings of the bit coin or the thermodynamics of data server farms.

How arithmetic ultimately works is one of those ultimate questions still hotly debated by philosophers and logicians. It will probably never be answered to complete satisfaction. It’s not something many college students, let alone kids, can really grapple with.

As a kid I was good at math but oh how I hated the conceptual crap. All the dithering over rational and real numbers in the first chapter…meh. It didn’t make any sense until I was well along with manipulation of numbers. I liked the solve all the little problems because they were like puzzles or a game.

Story problems were always a drag. I didn’t even realize I was supposed to translate them into a formula until I was in college.

Same thing with the elaborate metaphors young teachers seem to think are so helpful. They make sense only when a kid already understands something, and usually just muddy the waters.

Replies:@Jim@carolI'm the opposite. I'm not good at computational math but I love the conceptual stuff.

The Chinese are very numerate. Often times the press and even the government will put out economic and business data in the most confusing manner possible. It’s not, as you think, that they’re trying to confuse, because Chinese people can figure it out. It ‘s that they like to play with numbers. When I ask Chinese about it, why the heck don’t you just convey the information in a direct manner, they say Chinese always do that. One of the latest examples is the reliance on land financing. The number is reported as a percentage relative to the percentage of total spending. If a local government gets 50% of its money from taxes and transfers from the central government, and 50% from land sales, then it is 100% reliant on land sales.

As for teaching math, I remember being in junior or senior year of high school and studying pre-calc/trig/calculus. We started getting problems involving the growth of bacteria and widget production at a factory. That was the first time I understood what use this stuff was, and suddenly it became intuitive. Up until that point, it was abstract.

The favela kids are obviously Wittgensteinians:

“It purports to reveal the fundamental basis for arithmetic. However, it is our everyday arithmetical practices such as counting which are fundamental; for if a persistent discrepancy arose between counting and Principia, this would be treated as evidence of an error in Principia (e.g., that Principia did not characterise numbers or addition correctly), not as evidence of an error in everyday counting.”

Replies:@Hacienda@reederIt’s not something many college students, let alone kids, can really grapple with.

---------------

Really? You need a PhD or some specialized training to doubt 1+1 = 2? You're going to surrender

this basic area of intuition because only specialists should hold opinions about it. I hope not.

Personally, I doubt 1 + 1 = 2 because I can think of some many examples where that doesn't seem to

work. Like one apple plus one apple doesn't equal two if one apple happens to get eaten before I get a chance to add them together. I'm good with that. Don't need to go through Russell and Whitehead to try and find flaws, because I don't think they have better experiences of things than I do.

Replies: @JimReplies: @Anonymousin the philippine setting

here’s why the chinese are better in math than filipinos: in the private schools where they are enrolled, the schools there teach twice the math an ordinary filipino would get in public and private school–english math in the morning, and chinese math in the afternoon.

Replies:@dixie@AnonymousThe secret of Chinese math skills :)

http://www.telegraph.co.uk/news/worldnews/1538805/Four-year-olds-study-for-their-MBA.html

"Four year olds study for their MBA"

"There is no room for such traditional Western childhood staples as nursery rhymes. Instead of Baa Baa Black Sheep, the young students are given a computer screen and an animated sheep farm on which they learn to make the business of sheep shearing profitable, building a business structure from the pen to the market place."

Replies: @Steve SailerI am out of my gourd pissed at that article, and the hackery that Elizabeth Green has been perpetrating on the public in exchange for a ton of money from the “billionaire philanthropists”.

Green wrote the original PR piece (aka NY Times article) on Doug Lemov, which got her a book deal but also ensure that her site Chalkbeat (where she does reasonable reporting on education politics) got funding not only from Gates, but the guys behind Uncommon Schools and a few other major CMOs. She knows nothing about teaching, did no reporting on it in any of her guises (and unlike Amanda Ripley, she is an education reporter). It’s basic fellation on Doug Lemov. The main reason I can see for her choosing Japan is that Ripley had already snared Korea and Finland.

As I tweeted, it’s not just that the article is crap. It’s REHASHED crap.

“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. ”

I have some bad news. I’ve spent about 20 hours in really fascinating professional development by a highly regarded math professor who has spent a long time fussing about how kids learn math. It is all about the number line, baby. We’re back to that.

I’m not kidding about the PD. I found his insights fascinating and helpful. But he absolutely believes that the reason kids are so bad in math is because of these relatively small insights he’s offering up.

Didn’t Kurt Goedel prove that all this first principles of math stuff like in

Principia Mathematicaled to paradoxes anyway? It’s all over my head but that was the general impression I got from reading Hofstadter.Replies:@AnAnon@Jokah Macphersonhttp://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

"The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency."

Replies: @Jim@Jokah MacphersonIt's definitely over your head.

http://www.celebritynetworth.com/articles/entertainment-articles/how-michael-jackson-bought-the-beatles-catalogue-then-turned-it-into-a-billion-music-empire/

There are only 10 types of people when it comes to math: people who know base 2 and those who don’t.

I attended a Catholic grade school in Philadelphia during the late 50’s and early 60’s .We were separated by sex and their were 50 boys in my class .We were taught by nuns who were not college grads . It was an all white Irish neighborhood . None of the students who entered 1st grade was expelled for any reason and there was no admission test of any kind .It was a lower middle class neighborhood . All the Catholic kids in the neighborhood attended since their wasn’t any tuition . In 8th grade we were required to take a standardized test for high school placement .Eighteen of the fifty boys in my class got a 99 percentile and we had quite a few with 98,97,96,95 etc .The teaching technique was probably in use in 1900 .

“My wife says that the market people here in the Philippines have a lot of trouble calculating fractions of a kilo, etc. They have a set price for a set amount. If you ask for 1/4 they don’t know the price.”

The “stereotype” that Asians are good at math mostly applies to East Asians like the Chinese, Koreans, and the Japanese. Not so much for Southeast Asians like Filipinos, Thai people, and Indonesians for example.

New Math, talk about how to kill interest in math, this was it. I developed a math phobia because of that intellectual garbage. If it was created by math professors, it only proved the math professors were tin eared incompetents who couldn’t teach and didn’t understand young children or teenagers or what they needed.

The really smart kids didn’t need it. They can fend for themselves quite well academically.

As to what kids in Brazil do. Well that’s practical intuitive math and there’s certainly a place for that in school. Even more importantly they can do it in their heads as opposed to most kids today who need a calculator just to do basic arithmetic thanks to the latest developments in teaching. Or should I say the latest regression coming from a bunch of failed academics.

Now one of the big problems in regards to teaching math is that most of it is so abstract, that most kids see it as useless – which most of it is if you’re not bound for a degree in a STEM field. Schools need to find a way of making it relevant or the teachers gets tuned out by the students.

As for useless, Algebra II sort of falls into this category. The problem with it, is that most minority kids are going to blow it no matter what; others who are more interested in repairing cars, plumbing or become a electrician don’t need it and it doesn’t matter for them. These kids are trade school bound and will probably make a lot more money than most math majors sitting in a Herman-Miller cube waiting to be off-shored.

I say make it a elective for those that need it(college bound). Otherwise just focus on teaching kids the practical essentials. That way no trade off.

"It purports to reveal the fundamental basis for arithmetic. However, it is our everyday arithmetical practices such as counting which are fundamental; for if a persistent discrepancy arose between counting and Principia, this would be treated as evidence of an error in Principia (e.g., that Principia did not characterise numbers or addition correctly), not as evidence of an error in everyday counting."

Replies: @HaciendaIt’s not something many college students, let alone kids, can really grapple with.

—————

Really? You need a PhD or some specialized training to doubt 1+1 = 2? You’re going to surrender

this basic area of intuition because only specialists should hold opinions about it. I hope not.

Personally, I doubt 1 + 1 = 2 because I can think of some many examples where that doesn’t seem to

work. Like one apple plus one apple doesn’t equal two if one apple happens to get eaten before I get a chance to add them together. I’m good with that. Don’t need to go through Russell and Whitehead to try and find flaws, because I don’t think they have better experiences of things than I do.

Replies:@Jim@HaciendaNobody, least of all Russell or Whitehead, doubts that 1 + 1 = 2. There's a lot more in the Principia than 1 + 1 =2.

Replies: @HaciendaSteve, you’re a smart guy. You could have enjoyed the “abstract stuff” if you’d given it a second chance. Ya know, abstraction in math is about noticing patterns, and you’re good at that.

Like, there are deep connections between flipping&rotating a mattress and shuffling a deck of cards. And between basic facts about prime numbers, and factoring polynomials.

I hope Derb chimes in on this…

” It’s too abstract for me. I don’t like variables that can stand for different things, I like numbers that represent real things.”

As John D Williams, wrote in ‘The Compleat Strategyst’

“

nis no friend of the working man.”OT but Numbers USA is saying that any bill coming out of the House on the “border surge” can be conferenced with the S.744 amnesty bill. The Cornyn-Cuellar “border surge” bill could be passed by the House by the end of the week. Please everyone, call the Hosue Republican leadership and demand that they pass NO bill on the border surge.

Am I verbally illiterate if I think the title of this post should begin with the word “Are”.

Steve, have you considered looking at the Hungarian history of teaching math? For a long time they were probably considered the leaders in the area of teaching youngsters advanced math. Of course, this really was math-for-students-who-would-become-professors. Donald Knuth is supposed to have said that

“the Hungarian educational system has been the most successful in pure mathematics.”” (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.)”

Well, we know he was able to convert Harold Wilson’s marginal tax rates into a ratio — or did he, seems even math educators are confused, terminology wise on ratio vs. fraction vs. proportion (and I’d add an ‘odds’-like representation of proportions versus ‘percentages’).

Schools spend too much time making math relevant. I can’t even count how many times I fell asleep listening to someone drone on about lemonade stands or percentages of candy I could eat if x did y (I graduated high school in the mid-2000’s). My problem with math is that teachers never went through any trouble to make verbal sense of what math is doing (I’ve been taught to factor maybe a half dozen times in my educational career and I still have no idea what it is or what it does) and that’s probably because it’s impossibly difficult to make verbal sense of math if you have no visceral faculty for doing it.

“Is Numeracy and Mathematical Literacy the Same Thing?”

Numeracy, schnumeracy.

Call me a Grammar Nazi, but shouldn’t that question be posed thusly:

AreNumeracy and Mathematical Literacy the Same Thing?”Now go read the gradations on your sphygmomanometer.

The only torment I’ve known in the course of my sixty-four years on this planet

was the time spent sitting at a desk in school, watching the clock and waiting

for the day to end.

What a waste of precious life.

“I’ve been taught to factor maybe a half dozen times in my educational career and I still have no idea what it is or what it does”

I assume you mean factoring trinomials? Here is a math teacher who isn’t a mathematician answering.

Factoring is the process of converting a sum to a product. There are two major reasons to do this, both of which arguably aren’t as important as they were before graphing calculators.

1) We have no generalized solutions for polynomials past the second degree. We have a few methods for specific cube forms, and there may be some for quartics (even those that don’t fit the quadratic form). It has already been proven that there are no generalized solutions past fifth degree (or is it 4th? Hey, it’s summer.) So finding the solutions of polynomials of degrees greater than 2 often requires some form of guess and check–either for factors or, if it’s prime, just narrowing in. Decartes Rule of Signs, Rational Root Theorem, etc are tools to narrow down the possibilities.

2) Rational expressions may have common factors in the numerator and denominator, which would mean a discontinuity rather than an asymptote.

I have little idea how this is used out in the real world, and I’m sure the explanation I’ve given is missing things. I’m an English major.

Those of you who are arguing for more applied math, less theory—back in the day, we taught advanced math only to the kids who were going to college, because you had to pass calculus in order to get through college. So if you didn’t have skills to get through the math leading up through calculus, then best to know it before you went off to college.

At this point, not only is calculus not required, but you don’t even need a math class to get most degrees. Colleges are still desperately trying to preserve the signaling value of a degree by identifying people who can’t even get through remedial math. However, recent developments are making that difficult because the classes

*after*you get through remedial math are so easy. A few studies proving that students who had remedial level skills were able to pass non-remedial classes have already been published. So now you have people arguing that remediation classes are just crap, just gatekeeping, that we should let everyone through.But remediation has been largely exempt from pressures of racial balance, because it’s a mix of things: AP classes, SAT or ACT scores, placement tests. Let everyone through, and professors and instructors will be under tremendous pressure to have “equitable” pass rates.

College degrees aren’t much of a signal now, but remediation is already under tremendous attack. End requirements for higher math in high school and we’re just plumb out of reliable indicators.

Solution: require calculus for a college diploma, regardless of degree, using a test.

Meanwhile, though, all this talk is technically correct, but boy, given our desire to screw up society already, you all are pulling at the last few dregs that protect college degrees.

Replies:@International Jew@education realist"1) We have no generalized solutions for polynomials past the second degree. We have a few methods for specific cube forms, and there may be some for quartics (even those that don’t fit the quadratic form). It has already been proven that there are no generalized solutions past fifth degree (or is it 4th? Hey, it’s summer)"

Fifth and up: Evariste Galois established that, and it remains one of the greatest accomplishments ever in math. As for 3rd and 4th degree, there are fully general formulas for these. But these formulas can yield results that are very messy. So they're rarely (if ever?) covered in school.

Replies: @Jim@education realistThe basic problem with status competition is that the supply of status is totally fixed. It doesn't respond to demand or technological innovation. So after everybody has a college degree then the possession of such a degree will be useless as it will provide no status whatsoever. Then to get status you will have to get a PhD or whatever.

The nature of status competition in primate societies is that almost everybody will be a loser almost all the time.

I left this comment on your other blog;

“Common Core was funded by Bill Gates not only to ensure a consumer base (the American educational system) for Common Core tech products (created and put out by Microsoft) for decades to come, the Common Core compatible curriculum is designed to NOT make American high school graduates “college ready” in the subject of Math, despite claiming that it would not only make all high school graduates “college ready” but that it would bring American students up to international competitiveness with East and South Asians in the subjects of Math and Science. One of the very creators of the Common Core tests, who was hired by Gates to design the tests, admitted that if teachers stick to teaching Common Core test compatible curriculum alone, their students will NOT be college ready for a typical, mediocre 4 year state college and will have to take a remedial math course.

Is it a plot to make Americans less qualified than H1B visa hopefuls who will work for Gates at much lower pay rates?”

That said, I checked out some videos of a few different ways Common Core plans on teaching math, and I do think it will help kids for whom the abstraction of math is hard to “get”. They are breaking things down visually and while it takes longer, it makes the problem make sense. Its also shown as only one way to solve a problem so I assume that for the kids who don’t like that way, they can use one of the other ways and as long as the reach the correct answer on the test, it will be fine.

Their idea is to go “deep” into math rather than cover a lot of shallow ground quickly in order to include trig and calculus. The problem is, in order to be “college ready” Algebra II, Trig and Pre-Calc IS required, like it or not, in a typical 4 year State college, even for non-STEM majors.

So no matter how “deep” CC goes in Algebra I and Geometry, a remedial math course will be needed for American high school grads who wish to go on to college.

@Foreign Expertin the philippine setting

here's why the chinese are better in math than filipinos: in the private schools where they are enrolled, the schools there teach twice the math an ordinary filipino would get in public and private school--english math in the morning, and chinese math in the afternoon.

Replies: @dixieThe secret of Chinese math skills 🙂

http://www.telegraph.co.uk/news/worldnews/1538805/Four-year-olds-study-for-their-MBA.html

“Four year olds study for their MBA”

“There is no room for such traditional Western childhood staples as nursery rhymes. Instead of Baa Baa Black Sheep, the young students are given a computer screen and an animated sheep farm on which they learn to make the business of sheep shearing profitable, building a business structure from the pen to the market place.”

Replies:@Steve Sailer@dixieMy parody alert is going off on this Telegraph article about the Chinese MBA for 3 to 6 year olds, although perhaps Chinese entrepreneurs just come up with most parodic possible offerings for Tiger Mothers.

@AnonymousThe secret of Chinese math skills :)

http://www.telegraph.co.uk/news/worldnews/1538805/Four-year-olds-study-for-their-MBA.html

"Four year olds study for their MBA"

"There is no room for such traditional Western childhood staples as nursery rhymes. Instead of Baa Baa Black Sheep, the young students are given a computer screen and an animated sheep farm on which they learn to make the business of sheep shearing profitable, building a business structure from the pen to the market place."

Replies: @Steve SailerMy parody alert is going off on this Telegraph article about the Chinese MBA for 3 to 6 year olds, although perhaps Chinese entrepreneurs just come up with most parodic possible offerings for Tiger Mothers.

I assume you mean factoring trinomials? Here is a math teacher who isn't a mathematician answering.

Factoring is the process of converting a sum to a product. There are two major reasons to do this, both of which arguably aren't as important as they were before graphing calculators.

1) We have no generalized solutions for polynomials past the second degree. We have a few methods for specific cube forms, and there may be some for quartics (even those that don't fit the quadratic form). It has already been proven that there are no generalized solutions past fifth degree (or is it 4th? Hey, it's summer.) So finding the solutions of polynomials of degrees greater than 2 often requires some form of guess and check--either for factors or, if it's prime, just narrowing in. Decartes Rule of Signs, Rational Root Theorem, etc are tools to narrow down the possibilities.

2) Rational expressions may have common factors in the numerator and denominator, which would mean a discontinuity rather than an asymptote.

I have little idea how this is used out in the real world, and I'm sure the explanation I've given is missing things. I'm an English major.

Those of you who are arguing for more applied math, less theory---back in the day, we taught advanced math only to the kids who were going to college, because you had to pass calculus in order to get through college. So if you didn't have skills to get through the math leading up through calculus, then best to know it before you went off to college.

At this point, not only is calculus not required, but you don't even need a math class to get most degrees. Colleges are still desperately trying to preserve the signaling value of a degree by identifying people who can't even get through remedial math. However, recent developments are making that difficult because the classes *after* you get through remedial math are so easy. A few studies proving that students who had remedial level skills were able to pass non-remedial classes have already been published. So now you have people arguing that remediation classes are just crap, just gatekeeping, that we should let everyone through.

But remediation has been largely exempt from pressures of racial balance, because it's a mix of things: AP classes, SAT or ACT scores, placement tests. Let everyone through, and professors and instructors will be under tremendous pressure to have "equitable" pass rates.

College degrees aren't much of a signal now, but remediation is already under tremendous attack. End requirements for higher math in high school and we're just plumb out of reliable indicators.

Solution: require calculus for a college diploma, regardless of degree, using a test.

Meanwhile, though, all this talk is technically correct, but boy, given our desire to screw up society already, you all are pulling at the last few dregs that protect college degrees.

Replies: @International Jew, @Jim“1) We have no generalized solutions for polynomials past the second degree. We have a few methods for specific cube forms, and there may be some for quartics (even those that don’t fit the quadratic form). It has already been proven that there are no generalized solutions past fifth degree (or is it 4th? Hey, it’s summer)”

Fifth and up: Evariste Galois established that, and it remains one of the greatest accomplishments ever in math. As for 3rd and 4th degree, there are fully general formulas for these. But these formulas can yield results that are very messy. So they’re rarely (if ever?) covered in school.

Replies:@Jim@International JewBefore Galois, Abel showed that the general quintic could not be solved by radicals. The general quintic can be solved by modular functions.

Reading this made me think of an amusing episode from my high school days.

I got awful grades on the standardized tests they used to place kids at different math instruction levels, so I always got placed in the absolute lowest math classes. Even so, I usually was one of the better students in those classes. (Also, at the same time I was taking the lowest level math classes, I was taking AP level courses in every other subject — for what it’s worth…)

Anyhoo, the teachers in these low-level math courses were always careful to explain everything in very simple terms, which was why I did so well. “Regular” math teachers always spoke in a manner I found incomprehensible. I remember one week, though, when our regular math teacher (a very straightforward, no-nonsense black man who was a former Marine) was out sick, and for the class period I was in, instead of bringing in a substitute, the school brought in the advanced math teacher from across the hall. (Evidently she was free during that period, so the school was looking to save some money.)

The advanced math teacher had some damn fancy degree from somewhere — I remember she was actually a “doctor,” though she didn’t make us call her that. She might as well have been speaking Attic Greek, though, as far as we were concerned. She kept on referencing these high level concepts that none of us dumb kids had ever heard of. I actually started mocking her by using her own high-level language to ask complicated nonsense questions, just to see her response. (I believe I was the only one who got the joke…)

I suppose my point is that I understood then that there’s a big difference between teaching math to people with a deep understanding of math and teaching math to people who just need to know the basics. Steve’s citation of “Principia Mathematica” is actually pretty sharp — more than once, when looking at the Common Core curriculum my teacher fiancée has to push, I’ve found myself asking the same question: “Hell, why don’t they just pull Russell and Whitehead off the shelf and try to teach the kids that? If they’re gonna dump all this stuff on little kids, why not go directly to the source?”

Not exactly kids from the favela, but I have always been impressed with how fast blackjack dealers can count.

A few statements below have been mentioned around here. They can’t all be true, nor can they all be untrue. It is helpful to look at data and statistics rather than going with gut feelings and anecdotes.

1) In the 1980s, American-born university students who had been through the New Math era in their childhoods, as they would have been grade school children in the ’60s to ’70s, attained science and engineering degrees in far higher proportions than before or since. Thousands more American females got science and engineering degrees than any time since the 1980s. Then the male patriarchy drove female students out of studying things like computer science as journalists say.

2) The Department of Education puts out somewhat fraudulent or falsified statistics on university degrees including degrees from for-profit diploma mills and such. While false information is purposefully or unknowningly propagated by some for propaganda reasons, some investigative journalist could do a FOIA.

3) “New Math was a failure.”

“Well, we know he was able to convert Harold Wilson’s marginal tax rates into a ratio …”That was a George Harrison song.

Steve, your schooldays from 1970 to 1972 seem idyllic – living the hippy dream at the world epicenter of hippidom and right at the peak of the so-called ‘love and peace’ era with its misplaced optimism and faith in human nature. Listening to the ‘Hair’ soundtrack in class whilst an at the peak of energy aand confidence in the future just sounds so cool it’s just a crying shame thag the world turned out to be the vicious as the old gang said it was.

@ Svigor:

” Boys need to have a reason to be interested. Give them real-world examples that appeal to them (artillery strikes, drag races, defusing bombs, demolition, making money, etc., not sharing fruit and the like) and you have a much better chance of hooking their interest.”

Actually, they used to do this. They turned all the word problems into girls selling lemonade thanks to feminists who thought girls were being left behind. And, yeah, girls did better, but boys did worse. As Sailer says, things don’t have to be zero-sum, but they often are.

@Charles

I went to a country school with a lot of older teachers in the 70’s and 80’s. They all looked at the New Math crud in the textbooks, told everyone to ignore it, and then proceeded to teach math the old way. So, even though New Math was around, I think a lot of older teachers just ignored it. Back then there wasn’t the CE requirements that force teachers to constantly go back to college to learn how to improve teaching 2nd graders. Instead state standards were simply you had to pass so many classes and schools got to dictate that however they wanted. My physics teacher, who was tough but great, graduated at 19 from college with a two year teaching degree. Never would happen today.

In the 80’s, engineering/computer science was cool. It stayed that way until the new teaching methods worked their way everywhere (use those old textbooks until they fall apart and the old teachers retire) and wages for those degrees stagnated.

Not getting point #2.

New Math is a failure. It being a failure does not contradict item #1 because the looser state standards meant a slower adoption rate.

I like the point that there is a tradeoff between objectives of teaching math. MBA students are terrified of calculus because they remember their college math classes. For my economics course, though, all they need to know are about three cookbook formulas (one, really), which means all they have to do is to overcome the terror they have from their science-major-designed undergrad math.

Most people don’t need calculus or trigonometry at all, even for college. They need statistics and the Idea of a Proof (i.e., Euclidean geometry). And algebra– maybe. But if you

*do*want to be a science or engineering major, calculus is best, and if you want to be a math major or do PhD work in another technical field, rigorous calculus is best.USSR humor. Two Russians are talking:

Ivan: Is it true that Karl Marx invented socialism?

Vladimir: Yes, Karl Marx was the inventor of the socialist economic system.

Ivan: Is is true that Karl Marx was a scientist?

Vladimir: Yes, indeed! He was the greatest scientist in the history of the world!

Ivan: If Karl Marx was such a scientist, why didn’t he test it on rats first?

I was two years behind you, Steve, but my Catholic grade/junior high school had the old school nuns as 80% of the staff. Punishment for major infractions went beyond a bit of knuckle rapping. Which is how a 5′ tall, 100 lb. fifty-something woman could keep control of a room with 35 11-year olds, half of whom were boys on the cusp of deciding that maybe girls were something worthy of more attention than they had previously paid them. The lay staff toed the party line or found other employment; the feeling was that if you want to “rap about the movement”, go to a public school where you belong.

IJ–thanks. I always point out Galois during my precalc class (never remember his name until I look it up). I can never remember that there are fully generalized forms because the book doesn’t show them. We get an example of the formula for a “depressed cube” and compare the formula to guess and check–the latter wins.

For those people thinking good lord, a math teacher who doesn’t know this stuff! Well, I know it to teach it. I only started teaching pre-calc a year ago, so my deeper understanding of math analysis is relatively shaky.

But for the average kid sitting in a precalc class knowing they’re unlikely to use this again, here’s what I point out:

In your math education thus far, you’ve spent 2-3 years on first degree equations, 2 years on second degree equations, and a year or so on exponential equations. You probably thought that in more advanced math, you’d learn about third degree, and then fourth degree, and then by the time you get to calculus, by golly you’re on fifth and sixth degree! Because you’re really out there, baby.

But it turns out that polynomials are more like polygons. You learn about triangles, circles, and quadrilaterals in great details. Then it turns out that, once you get past four sides, everything is nothing more than an increasing number of triangles—which is why we spend so much time beating triangle facts into your head. By the time you get out of geometry, you’ve vaguely figured out that triangles and circles are 90% of the ballgame, with parallelograms having some use, too.

Likewise, you spend so much time on linear and quadratic terms because they are the building blocks for the higher degree polynomials.

That, I would argue, is what the educated non-mathie needs to take away from math analysis, along with an understanding that the significant math theorems after this point involve tools to break down the polynomials more quickly.

However, the question remains does this study have any relevance in a world where we can graph everything and see it so quickly? I remember in the Laura Ingalls Wilder book reading the kids quickly find square roots in their head. Holy crap, we can’t do that. But we don’t need to, because we have calculators and before that slide rules.

Today, the kids’ smartphone calculators are starting to include an option for log base. So you can type in log(3,15), which is the log of 15 in base 3. I mean, holy crap. The whole reason we teach the change of base formula, so far as I can figure, is so that we can get the equation into base 10, because our tables (and now, our calculators) needed that. And it’s already been a stretch to pretend we need log properties, or the sine and cosine angle addition formulas, which were also designed for a time before calculators.

So just as we stopped teaching kids how to find square roots, are we going to stop teaching kids the log properties and the trig properties? And is math analysis something else we don’t need?

Replies:@Jim@Education RealistEdRealist - You are profoundly ignorant of the mathematics of polynomials.

On the word problems: the misogyny is getting moronic.

First, I’d say that word problems remain the area least taught by high school teachers. I have colleagues routinely tell me that their kids just flatly refuse to do word problems and would fail if they insisted. They fail far more kids than I do, anyway.

So the idea that we’re turning our word problems into girly girl situations that bore boys is not just wrong, but stupid wrong. In high school, many math teachers aren’t doing word problems nearly as often as they should–one of the more legitimate criticisms of high school math. Kids know the formulas of higher math, but have limited understanding.

In my classes, I start each section with word problems. (Quadratics are the most difficult to simply model). My weakest ability kids are stunned to realize they find this helpful, and often end my class realizing they are better at word problems than anything, because it gives them a frame of reference—-I am speaking particularly of linear modeling. Here’s how I teach it: https://educationrealist.wordpress.com/2013/02/16/modeling-linear-equations-part-3/

As a result, my courses are considered easy to pass, but difficult to get an A in. I find this incomprehensible. My colleagues give these five or six page tests, filled with problems more difficult than I ever dreamed of making. My tests are two pages, occasionally three. Kids say mine are brutal, except the lowest ability kids, who find them manageable. I am puzzled, but have tentatively concluded that the more difficult tests cover material the kids never see again, so have the incentive to memorize just long enough for the test. The lower ability kids can’t manage this, so do poorly. My tests, on the other hand, run through all the material of the course, so if you’re the sort that just memorizes, you’re screwed.

The point of this long screed is that a) many high school teachers never do word problems, or do so minimally. I wouldn’t dispute that mine tend to be straightforward–I’m no Dan Meyer. b) the textbooks provide

*tons*of word problems, and while I occasionally dislike something about them, it’s not the fact that they are too girly girl. Texts have gotten good at coming up with word problems in a huge variety of situations.So you all talking about lemonade stands are wrong that this is a general approach. There may be teachers that emphasize approaches that girls may prefer, but seriously, if you’re a guy who can’t cope with lemonade stands, grow a pair.

Examples of modeling quadratics and exponential equations:

https://educationrealist.wordpress.com/2013/12/16/the-negative-16-problems-and-educational-romanticism/

and

https://educationrealist.wordpress.com/2013/05/05/modeling-exponential-growthdecay-interspersed-with-a-reform-rant/

One last thing: the careful reader will note that I am not in ANY WAY a formula based teacher. Yet I think Green’s article is utter, profound crap that is funded by reformers and business boys that want a piece of the education market. For what it’s worth.

Replies:@Shuddh Bharatiyaan@Education RealistGreat Comment!

I am all in favor of presenting math in various ways so that kids of varying abilities and learning styles have a frame of reference that works for them.

One of the critiques of common core is that it gives to many alternative methods to work out a problem and arrive at the answer. I say the more the merrier because what makes sense to one kid may not, and often does not, make sense to another.

Common Core does have serious problems, but that is not one of them.

“Carefully taught, the assignments can help make math more concrete.”

Hmmm…. Start out with making math more concrete. Let them play cards for money. That’s concrete. Or handicap horse racing. Craps is nice also.

As far as the real world…. everyone uses simulation. In the real world of things, anyway. I have no idea what they do in the real world of particle physics.

Teach them simulation.

If you think the previous fads sucked, try ‘everyday math’ . Sucked. I suppose it was worst for parents who couldn’t teach their kid long devision because of the bizarre algorithm used.

Kids that are going to be engineers don’t need specialized math teaching because they can learn it using any method. Regular kids … teach them to measure twice, cut once.

Out top 5 percentile is as good as any other countries top 5 percentile. Other than that, why does anyone care.

Personally, I would like people to know the GDP. And the difference between a billion, 10 billion, 100 billion and a trillion. And a hundred million. Amounts that are theoretically possible to win in a lottery tend to be understood in practical terms. Sort a like, people should get that as far as the federal government is concerned, a trillion is a big number. A billion? Not so much.

The typical reaction to a number with a lot of zeros … once again, if it is the amount of the super lotto, it is understandable. A couple of private jets. Houses, Cars, relatives, hanger on’s and it will disappear in a few years.

Larger numbers? Just call em a zillion.

Does anyone notice that the header says 50 comments, but there are actually 53 (now 54) comments? Entering this to see if the counter updates.

Fifth and up: Evariste Galois established thatActually Abel in 1824, when Galois was 13.

Peter Pesic wrote a good accessible book about it, which I recommend.

Replies:@dearieme@John DerbyshireBell's Men of Mathematics says "Abel" too, if memory serves - which it doesn't always.

Anyway: calculus. We learnt it in the context of the-best-way-to-build-a-railway. Complex numbers: we learnt those in an abstract way, and then, poof!, e^(i*pi) = -1. Bloody hell!!! What it is to be sixteen.

@John DerbyshireThanks for setting the record straight. But Galois does deserve more credit than that ;-)

Replies: @John Derbyshire, @Jim, @Jim@John DerbyshireThe quintic solution in radicals (for special cases) or elliptic curves is a beautiful sight to behold

They turned all the word problems into girls selling lemonade thanks to feminists who thought girls were being left behind.One female college instructor in Australia presented a word problem about a female destroyer captain and a female submarine commander where the submarine had to dive or something.

The crux of the problem was about the submarine diving to a depth of 1 kilometer.

Stupid women. (If you know anything about submarines then you will know why I say that. You don’t even have to know about the limitations of the Collins class submarines Australia fields to understand.)

Mish gets it wrong:

http://globaleconomicanalysis.blogspot.com/2014/07/bad-day-for-bad-teachers-good-day-for.html

I think that minority behavior is so bad that good teachers can, and do, flee to the better areas.

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 that would allow the very smartest students to make the theoretical breakthroughs helpful to win the technological competition in the Cold War.I remember my Algebra II/Trig teacher taking the last, free day of class to explain credit card and other real-world math to us because we'd finished all our curricular stuff. I remember wondering why they didn't always teach math this way. Not that any of it involved Algebra II or Trig.

I like numbers that represent real things.Indeed, that's the way to get boys interested in math. Girls may be a shade behind boys in terms of math ability, but they are automatically motivated as soon as authority says "you must learn this." Boys need to have a reason to be interested. Give them real-world examples that appeal to them (artillery strikes, drag races, defusing bombs, demolition, making money, etc., not sharing fruit and the like) and you have a much better chance of hooking their interest.

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).”Boom: "girl who sold lemonade-" zzzzzzzzzzzzzz...

How about selling Ferraris? AR-15s? Speed boats?

teachers tell students to draw butterfliesHow about telling them to draw a sword? A fighter jet?

Lemonade, butterflies, zzzzzzzzzzzzzzzz...

Replies: @HunsdonOn the first day of battle, your platoon suffers 25% casualties. However, on the second day of battle you inflict twice as many casualties as you suffered on the first day. Given a platoon strength of 24 men, how many casualties did you suffer, and how many did you inflict?

Boys will get that one right every time.

Simple. I can close the achievement gap.

I can haz billions now?

Replies: @AnAnon“recast problems in calculus and orbital mechanics into the isomorphic problems of making change.” – your billions as soon as you do this.

Principia Mathematicaled to paradoxes anyway? It's all over my head but that was the general impression I got from reading Hofstadter.Replies: @AnAnon, @Jimhttp://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

“The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an “effective procedure” (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.”

Replies:@Jim@AnAnonThe first incompleteness statement is the same as saying that the true statements of elementary artithmetic are not recursively enumerable. This is the same as saying that there is no finite state automaton whose output could consist exactly of the true statements of elementary arithmetic.

The second incompleteness theorem I don't understand. I learned mathematical logic from Kleene and he refers to Hilbert & Ackerman for a complete treatment of the second incompleteness theorem. I never read Hilbert & Ackerman.

Goedel also proved Tarski's Theorem on the Undefinability of Truth (he never got around to publishing it before Tarski did). The truth predicate for any significant mathematical theory cannot be formalized within the theory. So for example the set of true statements of elementary arithmetic cannot be given an arithmetical definition. One can define arithmetical truth in say Zermelo-Fraenkel set theory but then the truth predicate for Zermelo-Fraenkel set theory is not formalizable in Zermelo-Fraenkel set theory.

It seems that mathematics makes sense but understanding it is beyond human cognitive abilities.

Fifth and up: Evariste Galois established thatActually Abel in 1824, when Galois was 13.

Peter Pesic wrote a good accessible book about it, which I recommend.

Replies: @dearieme, @International Jew, @grey enlightenmentBell’s Men of Mathematics says “Abel” too, if memory serves – which it doesn’t always.

Anyway: calculus. We learnt it in the context of the-best-way-to-build-a-railway. Complex numbers: we learnt those in an abstract way, and then, poof!, e^(i*pi) = -1. Bloody hell!!! What it is to be sixteen.

Btw, there is a link between psychosis and math. Paranoia-schizophrenia seems to be the case with: John Nash, Ted Kaczynski, Kurt Gödel, Isaac Newton, et al.

Look, you don’t want to innumerates giving you change on a hundred-dollar bill, but at least they won’t imbed a hammer in your skull if they get a math problem wrong in a textbook:

Colleagues later said Petryshyn, who taught advanced mathematics at Rutgers for 29 years, had been in a mental tailspin since realizing he had made an error in his second published math textbook, entitled “Generalized Topological Degree and Semilinear Equations”. Associates also said he worried the mistake would make him the subject of ridicule.Replies:@Anonymous@VincentTYeah, I hate being wrong too.

Your preference for ‘actual’ numbers over ‘pretend’ numbers recalls to mind a young Jung’s struggles with algebra, as recounted in

Dreams, Memories, Reflections. From aIt’s a lengthy passage, but amusing and worth reading.

Given how well Chinese-orgin and Indian students in the US test on math, it would be interesting to see the immigrant results with those two groups stripped out.

Not that I can remember needing to solve a polynomial with n >5.

Newton’s method is pretty efficient. There are probably a lot of brute force techniques that work but are less efficient. Like just guessing. You can just grind through a hell of a lot of guesses, and when signs change, you can work on it till it gets as precise as you need. I am sure there are a lot of counter examples where this is bad. Using brute force or an algorithm like Newtons seems a lot safer than doing a lot of algebra.

The ‘Wizard of Odds’ uses simulation extensively, even though (it seems to me) that most of them could be solved analytically.

http://wizardofodds.com/site/about/

When money is involved, you will find simulation.

Replies:@Jim@FWIWNewton's Method is second-order. Ostrowski's "Solutions of Equaitons" gives a third-order method that works for any polynomial with distinct roots.

@FWIWOstrowski also has a pretty through error analysis of Newton's Method.

OT and mild Sailerbait:

http://www.newgeography.com/content/004440-to-fight-inequality-blue-states-need-to-shift-focus-to-blue-collar-jobs

Replies:@Shuddh Bharatiyaan@countenanceBlue collar jobs went the way of the do do bird when manufacturing was outsourced. Bring back Big Plant with Big Paycheck, and we will see a rise in the Middle Class again.

We need another Henry Ford, without the wackiness and corruption.

Canadian (Ontario) public schools all the way here, from JK at the age of just short of 4 in September 1974. High school class of ’88. [Go Phoenix!] Two degrees since then, most emphatically not STEM. Though I did remember for a while a bunch of astronomy calculations and formulae from first year undergrad.

Damn. Still getting used to the fact that as of next year, the distant 30 year future cited at the end of Back to the Future will be here, and nary a sign of antigrav skateboards, Mr Fusion auto engines, etc.

Now that’s out of the way…

We experienced what, in retrospect, must have been the absorption of American new math at some point in the 1970s, but it seemed very sporadically implemented and dependent on principals’ and teachers’ tastes during that time, and must have faded away or been partially retained at best. There was a lot of references to sets at various points. I don’t know let alone remember enough to say for sure what was new or old.

I also concede I was never among the best at math. Worst subject and there WERE times I was having trouble getting it at a conceptual level. With all that in mind, I find it interesting to reflect on what I can and can’t do now, and what it seems to mean for math in everyday life.

Calculus, trig, geometry and algebra are basically gone now from complete atrophy. I miss them on the level of, “wow that would be nice to remember and I’d feel a little more comfortable in certain situations”, but it is not a worry for my work or daily life.

Curiously, I have dim memories as a really small child of having trouble with subtraction, not the concept of course but the mechanics of doing it on paper as opposed to the brute force in the head method, still faster. I had it down soon enough but once you stop doing paper math in class it seems to fade as well. I still sometimes have to make myself remember that procedure. Never really comes up, but that’s a bit more embarrassing. Funny to me that it is harder than addition.Again, just the paper and pencil mechanics, not the concept or the accurate result.

I also remember something with division. Somehow or other I had learned short division first and it struck me as easy and I had trouble learning the mechanics for long division. Once I had learned the latter, I struggled with the former. Which was new and which old, or whether both were “old”, I don’t know. Now I can do long division effortlessly, so it must have really stuck. Still can’t remember the short division procedure. As with subtraction, do most daily divisions in the noggin anyway. As you can see, not an engineer.

So I think of math as one of those subjects we are taught to see which of us really get it and will make our careers in a closely related field, which of us need to know it as part of knowing the breadth and scope of human knowledge and how what we know and retain and use fits into that, and which of us need to be aware of it so we go through our own endeavours without being wholly ignorant of human achievements and our own limitations within them. But we all need some kind of basic numeracy.

I generally assume that I have managed to achieve items 2-4 with flying colours and I haven’t needed more, though I wish it had been more within my capacity.

With all that in mind, I am still broadly willing to accept that most people do not need math in everyday life beyond a certain point, and that point generally does not include calculus, trig, geometry or algebra in any serious way. Although I would not wish to understate the number of fields of professional endeavour in which they are needed, either. Neither would I wish to suggest that it would not be better for individuals and society to retain more of these skills if they could be taught more effectively, presuming there is such a way. But on the whole, ability to count, familiarity with the scale of numbers, units of measure, fractions/percentages, and the four basic arithmetical functions have served me well. In my foolishness, I have assumed these skills were omnipresent in society.

I have resisted drawing the conclusion that they are not, despite the numbers of people unfamiliar with the distinctions among thousand, million, billion and trillion, despite people who cannot make change or tell the difference between positive and negative [I don’t mean using them in arithmetic, either, just that they exist], to the detriment of their understanding of personal and public finance.

And then I read that example about the thermometer and had the terrifying thought that there may well be North American adults of whom that is true. Along the lines of those who cannot read an analogue clock for the time. With kids, I can at least understand. They might never have seen the analogue forms of any kind of gauge.

Sorry to have banged on to get to that rather simple concern. In retrospect, it seems rather futile…

That lemonade example seems like one of those situations in which familiarity with test requirements would be a plus.

Given a set of choices, I would easily recognize 15+(2×15) as correct [I forgot to add that order of operations is something I retained in core memory from algebra; go figure]. Simply told to write it in arithmetical form I might well put 15+30 if not feeling snotty enough to write 45.

Just being a tad facetious.

Still, I’d be satisfied if a numerate citizenry could arrive at 45, with an awareness that Sunday was clearly the better sales day by a factor of 2. I concede that the basic algebraic symbols seem simple enough that the formula they want should come easily to most.

Skip Foreplay #25-

I didn’t expect to laugh out loud on this post. Many thanks.

I was pretty good in math up until I got to Algebra II/Trig(scoring at National Merit level on the PSAT) yet I am better at manipulating numbers than my wife who has a Math degree. I also went through Catholic school where we focused on drilling through addition, subtraction, multiplication, and division first through eighth grade. Most of my classmates could easily do triple digit problems in their head. In schools now the focus is on the higher level concepts and the students struggle with single digit multiplication. I fail to see the benefit of going into Algebra or Geometry without mastering the basics. In fact my oldest daughter and her friends struggled in math due to the focus on word problems early on. Why give second and third graders a few word problems to do instead of hundreds of simple problems to master the basics? Once the basics are mastered the word problems become easy.

http://www.newgeography.com/content/004440-to-fight-inequality-blue-states-need-to-shift-focus-to-blue-collar-jobs

Replies: @Shuddh BharatiyaanBlue collar jobs went the way of the do do bird when manufacturing was outsourced. Bring back Big Plant with Big Paycheck, and we will see a rise in the Middle Class again.

We need another Henry Ford, without the wackiness and corruption.

How about ordering Jewish oligarchs to pay 5 trillion dollars for looting Russia in the 1990s? How about ordering Zionists to pay 10 trillion to the Palestinians for stealing Palestine?

Look, you don't want to innumerates giving you change on a hundred-dollar bill, but at least they won't imbed a hammer in your skull if they get a math problem wrong in a textbook:

Colleagues later said Petryshyn, who taught advanced mathematics at Rutgers for 29 years, had been in a mental tailspin since realizing he had made an error in his second published math textbook, entitled "Generalized Topological Degree and Semilinear Equations". Associates also said he worried the mistake would make him the subject of ridicule.Replies: @AnonymousYeah, I hate being wrong too.

Fifth and up: Evariste Galois established thatActually Abel in 1824, when Galois was 13.

Peter Pesic wrote a good accessible book about it, which I recommend.

Replies: @dearieme, @International Jew, @grey enlightenmentThanks for setting the record straight. But Galois does deserve more credit than that 😉

Replies:@John Derbyshire@International JewYes, and I misread: Abel proved for n=5, Galois proved all n>=5.

Galois has been so over-romanticized one just instinctively wants to rebut any praise anyone gives him. But yes, he deserves a lot.

Men of Mathematicsis great fun to read & the world would be poorer without it; but factwise, it has a lot of soft brown spots.@International JewGalois showed that solvability is equivalent to the solvability of the Galois group of the equation. Abel's result that the general quintic is not solvable is a corollary of this since the general quintic has the symmetric group on n letters as Galois group and this group is not solvable when n is greater than 4.

Solvability for groups means that the group can be constructed from abelian groups. So a polynomial equation is solvable ie it's solution is reducible to equations of the form x^n = a if and only if it's Galois group can be built from abelian groups. A very famous theorem of Kronecker asserts that the solution of a polynomial (with rational coefficents) can be reduced to equations of the form x^n = 1 that is the extraction of roots of unity if and only if it's Galois group is abelian.

Since the Galois group of a quadratic is always abelian any quadratic with rational coefficents can be solved by roots of unity. This is equivalent to the fact that the square roots of integers can be expressed as rational linear combinations of roots of unity. The famous formulas for doing this were discovered by Gauss.

The story is told that Gauss struggled for four years to prove these formulas, spending many hours attempting to find the proof. Then one day when he was not thinking about the problem the proof suddenly flashed through his mind. He wrote in a letter to Bessel "Wie der Blitzen einschragen, das Raechtsel is geloest" or something like that - "As the lighting strikes the riddle is solved."

Replies: @International Jew@International JewAlso Lagrange made a very penetrating analysis of the existence of solutions by radicals before either Abel or Galois. I'm not sure to what extent Abel or Galois were influenced by Lagrange, however a lot of the ideas in Galois theory can be traced back to Lagrange.

Of course all of these three - Lagrange, Abel and Galois - are very great mathematicians.

@John DerbyshireThanks for setting the record straight. But Galois does deserve more credit than that ;-)

Replies: @John Derbyshire, @Jim, @JimYes, and I misread: Abel proved for n=5, Galois proved all n>=5.

Galois has been so over-romanticized one just instinctively wants to rebut any praise anyone gives him. But yes, he deserves a lot.

Men of Mathematicsis great fun to read & the world would be poorer without it; but factwise, it has a lot of soft brown spots.I may catch some flack from the STEM people here, but my take is that verbal is simply harder, and possibly a better predictor of IQ than math. If Americans suck at math, rest assured they are even

worseat writing, grammar, and reading. But I think we’re simply expecting too much from the double digit IQ masses. Studies show that the average IQ of a HS graduate is around 100, so many should be expected to fail. Harder standards will mean more failures and or more coddling to make slow kids catch up. The former is bad for the morale of the student and the later will drive up education costs even more.Replies:@Shuddh Bharatiyaan@grey enlightenment"I may catch some flack from the STEM people here, but my take is that verbal is simply harder, and possibly a better predictor of IQ than math. If Americans suck at math, rest assured they are even worse at writing, grammar, and reading."

The Gates-hired CC writer/creator mentioned that CC has verbal issues as well. This program was implemented in schools hot off the stove without any testing for its efficacy. The students are the test tubes and the results will be seen a decade from now.

Fifth and up: Evariste Galois established thatActually Abel in 1824, when Galois was 13.

Peter Pesic wrote a good accessible book about it, which I recommend.

Replies: @dearieme, @International Jew, @grey enlightenmentThe quintic solution in radicals (for special cases) or elliptic curves is a beautiful sight to behold

The “New Math” of the 1960’s included a lot of computer math, as it was thought computers might become important in society.

The subjects included binary arithmetic, matrix algebra, boolean algebra, symbolic logic, predicate calculus, and I think some statistics.

O, the foolishness to think that computers would become a more important part of society!

“How about ordering Jewish oligarchs to pay 5 trillion dollars for looting Russia in the 1990s? How about ordering Zionists to pay 10 trillion to the Palestinians for stealing Palestine?”

Honestly: stupid idea. We’ll all have to pay reparations to black people if that goes through. They have at least as good a case.

worseat writing, grammar, and reading. But I think we're simply expecting too much from the double digit IQ masses. Studies show that the average IQ of a HS graduate is around 100, so many should be expected to fail. Harder standards will mean more failures and or more coddling to make slow kids catch up. The former is bad for the morale of the student and the later will drive up education costs even more.Replies: @Shuddh Bharatiyaan“I may catch some flack from the STEM people here, but my take is that verbal is simply harder, and possibly a better predictor of IQ than math. If Americans suck at math, rest assured they are even worse at writing, grammar, and reading.”

The Gates-hired CC writer/creator mentioned that CC has verbal issues as well. This program was implemented in schools hot off the stove without any testing for its efficacy. The students are the test tubes and the results will be seen a decade from now.

You’re hardly going to get any argument from me about verbal being harder. I wrote a whole piece about it two and a half years ago: https://educationrealist.wordpress.com/2012/01/28/the-gap-in-the-gre/ Steve linked it in–it was my first big link, in fact.

I remember in some PD class I saw a graph that I wished I could get a copy of and a cite for, although it made intuitive sense: our high school English difficulty was significantly

*less*than was needed at the top echelons, whereas our high school level math was significantly*more*than was needed.That’s because if we’re going to say we teach advanced math, we need right answers. If we’re going to say we teach advanced English, we can fake it.

On Galois–I know my precalc book says Galois, so it must be for N>5.

Replies:@Carl G.@Education Realist@educationrealist

I only have a BS in math albeit from a top program. But I feel confident with enough time and effort, I could parse and understand the most difficult literary texts in a meaningful sense. That is not the case with some higher level maths where though I am able to follow the logic of a proof, no amount of study will allow me a deep and intuitive understanding. BTW, my pre-95 verbal GRE was 750. My math was 800. This is a reflection of the low skill ceiling of the math portion rather than an indication of the ceiling of math. I don't think the average person has any conception how abstruse pure math can be. Differential and multivariate calculus on the level taught to college freshman and sophomores are with little if any exaggeration considered to be on the level of the alphabet in the math world in terms of difficulty.

I have read your blog. Based on your relatively meager level of mathematical education, how comfortable are you in making an informed assessment on the difficulty of math? You seem to have a propensity to make bold claims based on a paucity of information.

Replies: @SenecaI remember in some PD class I saw a graph that I wished I could get a copy of and a cite for, although it made intuitive sense: our high school English difficulty was significantly *less* than was needed at the top echelons, whereas our high school level math was significantly *more* than was needed.

That's because if we're going to say we teach advanced math, we need right answers. If we're going to say we teach advanced English, we can fake it.

On Galois--I know my precalc book says Galois, so it must be for N>5.

Replies: @Carl G.@educationrealist

I only have a BS in math albeit from a top program. But I feel confident with enough time and effort, I could parse and understand the most difficult literary texts in a meaningful sense. That is not the case with some higher level maths where though I am able to follow the logic of a proof, no amount of study will allow me a deep and intuitive understanding. BTW, my pre-95 verbal GRE was 750. My math was 800. This is a reflection of the low skill ceiling of the math portion rather than an indication of the ceiling of math. I don’t think the average person has any conception how abstruse pure math can be. Differential and multivariate calculus on the level taught to college freshman and sophomores are with little if any exaggeration considered to be on the level of the alphabet in the math world in terms of difficulty.

I have read your blog. Based on your relatively meager level of mathematical education, how comfortable are you in making an informed assessment on the difficulty of math? You seem to have a propensity to make bold claims based on a paucity of information.

Replies:@Seneca@Carl G.I agree with your comment. I scored a 780 on the Verbal GRE and a 720 on the Math GRE (both Pre 1995), and recall that I ran into some road blocks on some upper level math courses when I was in college. The math GRE score, while not extremely high, is still very good.

As an undergraduate I could follow the logic of most of the proofs, such as for Cantor's theorem for example, but an intuitive understanding and grasp of these higher level mathematical theorems eluded me as an undergraduate. My grades in these upper level math courses were good because of great effort, but I knew I was not gifted and that I was beginning to hit a wall in my level of understanding.

When you finally hit a wall in your ability to intuitively grasp and understand something, it is a humbling epiphany. As Clint Eastwood as Dirty Harry once said ... "A man has got to know his limitations."

High level math is not very intuitive, unless you are very gifted IMO.

Replies: @JimI have read your blog. Based on your relatively meager level of mathematical education, how comfortable are you in making an informed assessment on the difficulty of math? You seem to have a propensity to make bold claims based on a paucity of information.

I’m pretty sure their have been top mathematicians and physicists who were just as critical of the “new math” movement in the 1960’s as Steve as been. Both Morris Kline and Richard Feynman I recall were critical as well.

Replies:@Carl G.@unladen swallowMy post was in response to educationrealist's claim that verbal is harder than math. I have no opinion regarding New Math of which I know little. Ambitious students with any natural mathematical ability more often than not will find their way to rigorous STEM fields usually by college. Whether they choose to make it a career is another matter and has little to do with teaching methods in grade school. Perhaps that's a naive view, but I simply do not have strong feelings on the matter, especially as one who rarely paid in attention in school, but could figure out things on his own.

I'm pretty sure their have been top mathematicians and physicists who were just as critical of the "new math" movement in the 1960's as Steve as been. Both Morris Kline and Richard Feynman I recall were critical as well.

Replies: @Carl G.My post was in response to educationrealist’s claim that verbal is harder than math. I have no opinion regarding New Math of which I know little. Ambitious students with any natural mathematical ability more often than not will find their way to rigorous STEM fields usually by college. Whether they choose to make it a career is another matter and has little to do with teaching methods in grade school. Perhaps that’s a naive view, but I simply do not have strong feelings on the matter, especially as one who rarely paid in attention in school, but could figure out things on his own.

@Education Realist@educationrealist

I have read your blog. Based on your relatively meager level of mathematical education, how comfortable are you in making an informed assessment on the difficulty of math? You seem to have a propensity to make bold claims based on a paucity of information.

I only have a BS in math albeit from a top program. But I feel confident with enough time and effort, I could parse and understand the most difficult literary texts in a meaningful sense. That is not the case with some higher level maths where though I am able to follow the logic of a proof, no amount of study will allow me a deep and intuitive understanding. BTW, my pre-95 verbal GRE was 750. My math was 800. This is a reflection of the low skill ceiling of the math portion rather than an indication of the ceiling of math. I don't think the average person has any conception how abstruse pure math can be. Differential and multivariate calculus on the level taught to college freshman and sophomores are with little if any exaggeration considered to be on the level of the alphabet in the math world in terms of difficulty.

Replies: @SenecaI agree with your comment. I scored a 780 on the Verbal GRE and a 720 on the Math GRE (both Pre 1995), and recall that I ran into some road blocks on some upper level math courses when I was in college. The math GRE score, while not extremely high, is still very good.

As an undergraduate I could follow the logic of most of the proofs, such as for Cantor’s theorem for example, but an intuitive understanding and grasp of these higher level mathematical theorems eluded me as an undergraduate. My grades in these upper level math courses were good because of great effort, but I knew I was not gifted and that I was beginning to hit a wall in my level of understanding.

When you finally hit a wall in your ability to intuitively grasp and understand something, it is a humbling epiphany. As Clint Eastwood as Dirty Harry once said … “A man has got to know his limitations.”

High level math is not very intuitive, unless you are very gifted IMO.

Replies:@Jim@SenecaYes you are right that intuition can only take you so far in mathematics. At some point very strange things begin to happen. I remember being astonished when I learned that there are orientable manifolds in 3 dimensions and higher where the two orientation classes are different

from each other. Right is different from left! Among these manifolds are CPn for n>=2. When the cup product in cohomology is available a simple algebraic argument shows that CPn for n>=2 admits no oreintation reversing homeomorphisms. Although the algebraic argument is simple there seems no intuitive reason to expect this. The same phenomenon occurs for some of the lens spaces in dimensions 3. When you glue such manifolds together you get different results depending on how you orient them before gluing. Nothing like this happens in two dimensions. When you join two tori to make a pretzel it makes no difference whether you glue from the inside or the outside. But the intuition gained from the two dimensional case is misleading in higher dimensions.

Principia Mathematicaled to paradoxes anyway? It's all over my head but that was the general impression I got from reading Hofstadter.Replies: @AnAnon, @JimIt’s definitely over your head.

@Jokah Macphersonhttp://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

"The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency."

Replies: @JimThe first incompleteness statement is the same as saying that the true statements of elementary artithmetic are not recursively enumerable. This is the same as saying that there is no finite state automaton whose output could consist exactly of the true statements of elementary arithmetic.

The second incompleteness theorem I don’t understand. I learned mathematical logic from Kleene and he refers to Hilbert & Ackerman for a complete treatment of the second incompleteness theorem. I never read Hilbert & Ackerman.

Goedel also proved Tarski’s Theorem on the Undefinability of Truth (he never got around to publishing it before Tarski did). The truth predicate for any significant mathematical theory cannot be formalized within the theory. So for example the set of true statements of elementary arithmetic cannot be given an arithmetical definition. One can define arithmetical truth in say Zermelo-Fraenkel set theory but then the truth predicate for Zermelo-Fraenkel set theory is not formalizable in Zermelo-Fraenkel set theory.

It seems that mathematics makes sense but understanding it is beyond human cognitive abilities.

I therefore favor algorithmic learning more than most people. You can't make people smarter, but you can teach them how to do specific types of problems. A supergenius would be able to integrate an expression like x^2 tan(x) in his head. The rest of us mortals need to integrate by parts.

Replies: @jimDo you mean x tan^2(x)?

@reederIt’s not something many college students, let alone kids, can really grapple with.

---------------

Really? You need a PhD or some specialized training to doubt 1+1 = 2? You're going to surrender

this basic area of intuition because only specialists should hold opinions about it. I hope not.

Personally, I doubt 1 + 1 = 2 because I can think of some many examples where that doesn't seem to

work. Like one apple plus one apple doesn't equal two if one apple happens to get eaten before I get a chance to add them together. I'm good with that. Don't need to go through Russell and Whitehead to try and find flaws, because I don't think they have better experiences of things than I do.

Replies: @JimNobody, least of all Russell or Whitehead, doubts that 1 + 1 = 2. There’s a lot more in the Principia than 1 + 1 =2.

Replies:@Hacienda@JimDavid Hume would doubt 1 + 1 = 2. Or that 1 + 1 = 2 works in all situations. It actually doesn't apply in any real world situation perfectly.

1 + 1 = 2 is true if you say it is true, i.e., a tautology- the numbers representing nothing, just symbols.

Replies: @JimI assume you mean factoring trinomials? Here is a math teacher who isn't a mathematician answering.

Factoring is the process of converting a sum to a product. There are two major reasons to do this, both of which arguably aren't as important as they were before graphing calculators.

1) We have no generalized solutions for polynomials past the second degree. We have a few methods for specific cube forms, and there may be some for quartics (even those that don't fit the quadratic form). It has already been proven that there are no generalized solutions past fifth degree (or is it 4th? Hey, it's summer.) So finding the solutions of polynomials of degrees greater than 2 often requires some form of guess and check--either for factors or, if it's prime, just narrowing in. Decartes Rule of Signs, Rational Root Theorem, etc are tools to narrow down the possibilities.

2) Rational expressions may have common factors in the numerator and denominator, which would mean a discontinuity rather than an asymptote.

I have little idea how this is used out in the real world, and I'm sure the explanation I've given is missing things. I'm an English major.

Those of you who are arguing for more applied math, less theory---back in the day, we taught advanced math only to the kids who were going to college, because you had to pass calculus in order to get through college. So if you didn't have skills to get through the math leading up through calculus, then best to know it before you went off to college.

At this point, not only is calculus not required, but you don't even need a math class to get most degrees. Colleges are still desperately trying to preserve the signaling value of a degree by identifying people who can't even get through remedial math. However, recent developments are making that difficult because the classes *after* you get through remedial math are so easy. A few studies proving that students who had remedial level skills were able to pass non-remedial classes have already been published. So now you have people arguing that remediation classes are just crap, just gatekeeping, that we should let everyone through.

But remediation has been largely exempt from pressures of racial balance, because it's a mix of things: AP classes, SAT or ACT scores, placement tests. Let everyone through, and professors and instructors will be under tremendous pressure to have "equitable" pass rates.

College degrees aren't much of a signal now, but remediation is already under tremendous attack. End requirements for higher math in high school and we're just plumb out of reliable indicators.

Solution: require calculus for a college diploma, regardless of degree, using a test.

Meanwhile, though, all this talk is technically correct, but boy, given our desire to screw up society already, you all are pulling at the last few dregs that protect college degrees.

Replies: @International Jew, @JimThe basic problem with status competition is that the supply of status is totally fixed. It doesn’t respond to demand or technological innovation. So after everybody has a college degree then the possession of such a degree will be useless as it will provide no status whatsoever. Then to get status you will have to get a PhD or whatever.

The nature of status competition in primate societies is that almost everybody will be a loser almost all the time.

@education realist"1) We have no generalized solutions for polynomials past the second degree. We have a few methods for specific cube forms, and there may be some for quartics (even those that don’t fit the quadratic form). It has already been proven that there are no generalized solutions past fifth degree (or is it 4th? Hey, it’s summer)"

Fifth and up: Evariste Galois established that, and it remains one of the greatest accomplishments ever in math. As for 3rd and 4th degree, there are fully general formulas for these. But these formulas can yield results that are very messy. So they're rarely (if ever?) covered in school.

Replies: @JimBefore Galois, Abel showed that the general quintic could not be solved by radicals. The general quintic can be solved by modular functions.

For those people thinking good lord, a math teacher who doesn't know this stuff! Well, I know it to teach it. I only started teaching pre-calc a year ago, so my deeper understanding of math analysis is relatively shaky.

But for the average kid sitting in a precalc class knowing they're unlikely to use this again, here's what I point out:

In your math education thus far, you've spent 2-3 years on first degree equations, 2 years on second degree equations, and a year or so on exponential equations. You probably thought that in more advanced math, you'd learn about third degree, and then fourth degree, and then by the time you get to calculus, by golly you're on fifth and sixth degree! Because you're really out there, baby.

But it turns out that polynomials are more like polygons. You learn about triangles, circles, and quadrilaterals in great details. Then it turns out that, once you get past four sides, everything is nothing more than an increasing number of triangles---which is why we spend so much time beating triangle facts into your head. By the time you get out of geometry, you've vaguely figured out that triangles and circles are 90% of the ballgame, with parallelograms having some use, too.

Likewise, you spend so much time on linear and quadratic terms because they are the building blocks for the higher degree polynomials.

That, I would argue, is what the educated non-mathie needs to take away from math analysis, along with an understanding that the significant math theorems after this point involve tools to break down the polynomials more quickly.

However, the question remains does this study have any relevance in a world where we can graph everything and see it so quickly? I remember in the Laura Ingalls Wilder book reading the kids quickly find square roots in their head. Holy crap, we can't do that. But we don't need to, because we have calculators and before that slide rules.

Today, the kids' smartphone calculators are starting to include an option for log base. So you can type in log(3,15), which is the log of 15 in base 3. I mean, holy crap. The whole reason we teach the change of base formula, so far as I can figure, is so that we can get the equation into base 10, because our tables (and now, our calculators) needed that. And it's already been a stretch to pretend we need log properties, or the sine and cosine angle addition formulas, which were also designed for a time before calculators.

So just as we stopped teaching kids how to find square roots, are we going to stop teaching kids the log properties and the trig properties? And is math analysis something else we don't need?

Replies: @JimEdRealist – You are profoundly ignorant of the mathematics of polynomials.

Not that I can remember needing to solve a polynomial with n >5.

Newton's method is pretty efficient. There are probably a lot of brute force techniques that work but are less efficient. Like just guessing. You can just grind through a hell of a lot of guesses, and when signs change, you can work on it till it gets as precise as you need. I am sure there are a lot of counter examples where this is bad. Using brute force or an algorithm like Newtons seems a lot safer than doing a lot of algebra.

The 'Wizard of Odds' uses simulation extensively, even though (it seems to me) that most of them could be solved analytically.

http://wizardofodds.com/site/about/

When money is involved, you will find simulation.

Replies: @Jim, @JimNewton’s Method is second-order. Ostrowski’s “Solutions of Equaitons” gives a third-order method that works for any polynomial with distinct roots.

Not that I can remember needing to solve a polynomial with n >5.

Newton's method is pretty efficient. There are probably a lot of brute force techniques that work but are less efficient. Like just guessing. You can just grind through a hell of a lot of guesses, and when signs change, you can work on it till it gets as precise as you need. I am sure there are a lot of counter examples where this is bad. Using brute force or an algorithm like Newtons seems a lot safer than doing a lot of algebra.

The 'Wizard of Odds' uses simulation extensively, even though (it seems to me) that most of them could be solved analytically.

http://wizardofodds.com/site/about/

When money is involved, you will find simulation.

Replies: @Jim, @JimOstrowski also has a pretty through error analysis of Newton’s Method.

@John DerbyshireThanks for setting the record straight. But Galois does deserve more credit than that ;-)

Replies: @John Derbyshire, @Jim, @JimGalois showed that solvability is equivalent to the solvability of the Galois group of the equation. Abel’s result that the general quintic is not solvable is a corollary of this since the general quintic has the symmetric group on n letters as Galois group and this group is not solvable when n is greater than 4.

Solvability for groups means that the group can be constructed from abelian groups. So a polynomial equation is solvable ie it’s solution is reducible to equations of the form x^n = a if and only if it’s Galois group can be built from abelian groups. A very famous theorem of Kronecker asserts that the solution of a polynomial (with rational coefficents) can be reduced to equations of the form x^n = 1 that is the extraction of roots of unity if and only if it’s Galois group is abelian.

Since the Galois group of a quadratic is always abelian any quadratic with rational coefficents can be solved by roots of unity. This is equivalent to the fact that the square roots of integers can be expressed as rational linear combinations of roots of unity. The famous formulas for doing this were discovered by Gauss.

The story is told that Gauss struggled for four years to prove these formulas, spending many hours attempting to find the proof. Then one day when he was not thinking about the problem the proof suddenly flashed through his mind. He wrote in a letter to Bessel “Wie der Blitzen einschragen, das Raechtsel is geloest” or something like that – “As the lighting strikes the riddle is solved.”

Replies:@International Jew@JimYou could have just linked to Wikipedia but as it happens your confusion about its and it's provides us with the more valuable contribution to what, was a more prominent theme of this thread.

Replies: @Jim@International JewGalois showed that solvability is equivalent to the solvability of the Galois group of the equation. Abel's result that the general quintic is not solvable is a corollary of this since the general quintic has the symmetric group on n letters as Galois group and this group is not solvable when n is greater than 4.

Solvability for groups means that the group can be constructed from abelian groups. So a polynomial equation is solvable ie it's solution is reducible to equations of the form x^n = a if and only if it's Galois group can be built from abelian groups. A very famous theorem of Kronecker asserts that the solution of a polynomial (with rational coefficents) can be reduced to equations of the form x^n = 1 that is the extraction of roots of unity if and only if it's Galois group is abelian.

Since the Galois group of a quadratic is always abelian any quadratic with rational coefficents can be solved by roots of unity. This is equivalent to the fact that the square roots of integers can be expressed as rational linear combinations of roots of unity. The famous formulas for doing this were discovered by Gauss.

The story is told that Gauss struggled for four years to prove these formulas, spending many hours attempting to find the proof. Then one day when he was not thinking about the problem the proof suddenly flashed through his mind. He wrote in a letter to Bessel "Wie der Blitzen einschragen, das Raechtsel is geloest" or something like that - "As the lighting strikes the riddle is solved."

Replies: @International JewYou could have just linked to Wikipedia but as it happens your confusion about its and it’s provides us with the more valuable contribution to what, was a more prominent theme of this thread.

Replies:@Jim@International JewI apologize for writing "it's" when it should have been "its". I have no idea what Wikipedia says about any of this.

@HaciendaNobody, least of all Russell or Whitehead, doubts that 1 + 1 = 2. There's a lot more in the Principia than 1 + 1 =2.

Replies: @HaciendaDavid Hume would doubt 1 + 1 = 2. Or that 1 + 1 = 2 works in all situations. It actually doesn’t apply in any real world situation perfectly.

1 + 1 = 2 is true if you say it is true, i.e., a tautology- the numbers representing nothing, just symbols.

Replies:@Jim@HaciendaWhere does Hume express doubt that 1 + 1 = 2? Numbers don't represent anything. The symbol "1" represents the number 1.

Replies: @HaciendaTexas-style word problem, elementary school level:

Teacher: You have ten chocolate cupcakes, and someone asks you for two of them. How many do you have left?

Student: Ten.

Teacher: You have ten chocolate cupcakes, and someone takes away two by force. How many do you have left?

Student: Ten and a dead body.

How about selling Ferraris? AR-15s? Speed boats?You are a soldier and shoot several enemy soldiers. If an AR-15 kills 80% of targets and the remainder are wounded. Of the wounded, 30% are still well enough to play dead until you walk by and they can shoot you. You have a 35% chance of walking by them after you have shot them. How many enemy must you kill, before on average, your time is up?

What they should really do is pick out 100 schools, norm by race, SES, IQ, and trial 100 different likely education methods. See how well the smart kids, the medium kids and the slow kids do. Then after the better methods relative to the underlying other factors make themselves known, do some more testing on more schools with those superior methods and see how they go. Whichever one wins is what gets replicated at all the other schools. Perhaps with a different methodology for different streams. It will take a few years but the results will be better for everyone but the book publishers.

“You could have just linked to Wikipedia but as it happens your confusion about its and it’s provides us with the more valuable contributionto what, was a moreprominent theme of this thread.”Dude, like, what’s that comma doing in the middle of your sentence? They call that some sort of run-on or a dangling sump’in? Next thing ya know you’ll be writing enginerish like a real engineer!

Replies:@International Jew@anonymousYeah, sorry, it was my flaky phone keyboard interacting badly with Javascript.

But to borrow from Winston Churchill, this morning I've rebooted my phone, while Jim likely still doesn't know its from it's.

First, I'd say that word problems remain the area least taught by high school teachers. I have colleagues routinely tell me that their kids just flatly refuse to do word problems and would fail if they insisted. They fail far more kids than I do, anyway.

So the idea that we're turning our word problems into girly girl situations that bore boys is not just wrong, but stupid wrong. In high school, many math teachers aren't doing word problems nearly as often as they should--one of the more legitimate criticisms of high school math. Kids know the formulas of higher math, but have limited understanding.

In my classes, I start each section with word problems. (Quadratics are the most difficult to simply model). My weakest ability kids are stunned to realize they find this helpful, and often end my class realizing they are better at word problems than anything, because it gives them a frame of reference----I am speaking particularly of linear modeling. Here's how I teach it: https://educationrealist.wordpress.com/2013/02/16/modeling-linear-equations-part-3/

As a result, my courses are considered easy to pass, but difficult to get an A in. I find this incomprehensible. My colleagues give these five or six page tests, filled with problems more difficult than I ever dreamed of making. My tests are two pages, occasionally three. Kids say mine are brutal, except the lowest ability kids, who find them manageable. I am puzzled, but have tentatively concluded that the more difficult tests cover material the kids never see again, so have the incentive to memorize just long enough for the test. The lower ability kids can't manage this, so do poorly. My tests, on the other hand, run through all the material of the course, so if you're the sort that just memorizes, you're screwed.

The point of this long screed is that a) many high school teachers never do word problems, or do so minimally. I wouldn't dispute that mine tend to be straightforward--I'm no Dan Meyer. b) the textbooks provide *tons* of word problems, and while I occasionally dislike something about them, it's not the fact that they are too girly girl. Texts have gotten good at coming up with word problems in a huge variety of situations.

So you all talking about lemonade stands are wrong that this is a general approach. There may be teachers that emphasize approaches that girls may prefer, but seriously, if you're a guy who can't cope with lemonade stands, grow a pair.

Examples of modeling quadratics and exponential equations:

https://educationrealist.wordpress.com/2013/12/16/the-negative-16-problems-and-educational-romanticism/

and

https://educationrealist.wordpress.com/2013/05/05/modeling-exponential-growthdecay-interspersed-with-a-reform-rant/

One last thing: the careful reader will note that I am not in ANY WAY a formula based teacher. Yet I think Green's article is utter, profound crap that is funded by reformers and business boys that want a piece of the education market. For what it's worth.

Replies: @Shuddh BharatiyaanGreat Comment!

I am all in favor of presenting math in various ways so that kids of varying abilities and learning styles have a frame of reference that works for them.

One of the critiques of common core is that it gives to many alternative methods to work out a problem and arrive at the answer. I say the more the merrier because what makes sense to one kid may not, and often does not, make sense to another.

Common Core does have serious problems, but that is not one of them.

“Based on your relatively meager level of mathematical education, how comfortable are you in making an informed assessment on the difficulty of math?”

1) My level of mathematical education isn’t relatively meager, but non-existent.

2) I wasn’t making an informed assessment. If you read the link, which you say you did, you see that I pointed out that verbal high scores were far less frequent than math high scores. Generally, the less frequently something occurs, the more likely we are to call that achievement “harder”. I readily acknowledge the difficulty of abstruse math.

” I could parse and understand the most difficult literary texts in a meaningful sense. ”

Probably. So? As I said, verbal skills aren’t “reading ability”. Please note that you are the one doing what you accused me of–making an “informed assessment” of verbal ability without really understanding what it is.

And, for the record, I’m really, really, REALLY far from being an “average person”, thankyouverymuch.

“You are profoundly ignorant of the mathematics of polynomials.”

I’m pretty sure this isn’t true. What is true is that I’m ignorant of polynomial math as used by mathematicians. Not profoundly. But still. However, my description was pretty accurate, apparently.

I will say this: I am not good at describing math in a way that makes mathematicians happy. More than once, both at school and in PD, I’ve said something and a math teacher or mathematician has said confidently “No, that’s completely wrong.” and I furrow my brow and say X isn’t true? Y isn’t true? and they look again and say “Oh. Yeah. Okay, you’re right.” I suspect this is why I don’t do well in advanced math. I understand math very differently than mathematicians, which is why I had to teach myself what I do know.

A number of my students go on to higher math, and they always say I was helpful. I’m happy with that.

And it’s “equations”.

Replies:@Carl G.@Education RealistYour understanding of statistics and how to interpret them properly is lacking to put it gently. I encourage people to visit EdRealist's site and take a gander at what I am talking about.Why don't we start from scratch. What does 'verbal is harder than math' mean in precise terms? Do you mean to say it's a better predictor of IQ? What evidence do you have of this? Surely, you do not intend to offer your GRE post as corroboration? Why bring it up in the first place? More anecdotes then?

I did not comment on your ignorance regarding polynomials. That was another poster.

1) My level of mathematical education isn't relatively meager, but non-existent.

2) I wasn't making an informed assessment. If you read the link, which you say you did, you see that I pointed out that verbal high scores were far less frequent than math high scores. Generally, the less frequently something occurs, the more likely we are to call that achievement "harder". I readily acknowledge the difficulty of abstruse math.

" I could parse and understand the most difficult literary texts in a meaningful sense. "

Probably. So? As I said, verbal skills aren't "reading ability". Please note that you are the one doing what you accused me of--making an "informed assessment" of verbal ability without really understanding what it is.

And, for the record, I'm really, really, REALLY far from being an "average person", thankyouverymuch.

"You are profoundly ignorant of the mathematics of polynomials."

I'm pretty sure this isn't true. What is true is that I'm ignorant of polynomial math as used by mathematicians. Not profoundly. But still. However, my description was pretty accurate, apparently.

I will say this: I am not good at describing math in a way that makes mathematicians happy. More than once, both at school and in PD, I've said something and a math teacher or mathematician has said confidently "No, that's completely wrong." and I furrow my brow and say X isn't true? Y isn't true? and they look again and say "Oh. Yeah. Okay, you're right." I suspect this is why I don't do well in advanced math. I understand math very differently than mathematicians, which is why I had to teach myself what I do know.

A number of my students go on to higher math, and they always say I was helpful. I'm happy with that.

And it's "equations".

Replies: @Carl G.Your understanding of statistics and how to interpret them properly is lacking to put it gently. I encourage people to visit EdRealist’s site and take a gander at what I am talking about.

Why don’t we start from scratch. What does ‘verbal is harder than math’ mean in precise terms? Do you mean to say it’s a better predictor of IQ? What evidence do you have of this? Surely, you do not intend to offer your GRE post as corroboration? Why bring it up in the first place? More anecdotes then?

I did not comment on your ignorance regarding polynomials. That was another poster.

@JimYou could have just linked to Wikipedia but as it happens your confusion about its and it's provides us with the more valuable contribution to what, was a more prominent theme of this thread.

Replies: @JimI apologize for writing “it’s” when it should have been “its”. I have no idea what Wikipedia says about any of this.

@JimDavid Hume would doubt 1 + 1 = 2. Or that 1 + 1 = 2 works in all situations. It actually doesn't apply in any real world situation perfectly.

1 + 1 = 2 is true if you say it is true, i.e., a tautology- the numbers representing nothing, just symbols.

Replies: @JimWhere does Hume express doubt that 1 + 1 = 2? Numbers don’t represent anything. The symbol “1” represents the number 1.

Replies:@Hacienda@Jim"There is no Algebraist nor Mathematician so expert in his science, as to place entire confidence in any truth immediately upon his discovery of it, or regard it as any thing, but a mere probability. "

Hume Treatise, 1.4.1.2

1 + 1 = 2 , as humble as it appears, can surely qualify as a mathematical truth.

Replies: @gdpbull@Carl G.I agree with your comment. I scored a 780 on the Verbal GRE and a 720 on the Math GRE (both Pre 1995), and recall that I ran into some road blocks on some upper level math courses when I was in college. The math GRE score, while not extremely high, is still very good.

As an undergraduate I could follow the logic of most of the proofs, such as for Cantor's theorem for example, but an intuitive understanding and grasp of these higher level mathematical theorems eluded me as an undergraduate. My grades in these upper level math courses were good because of great effort, but I knew I was not gifted and that I was beginning to hit a wall in my level of understanding.

When you finally hit a wall in your ability to intuitively grasp and understand something, it is a humbling epiphany. As Clint Eastwood as Dirty Harry once said ... "A man has got to know his limitations."

High level math is not very intuitive, unless you are very gifted IMO.

Replies: @JimYes you are right that intuition can only take you so far in mathematics. At some point very strange things begin to happen. I remember being astonished when I learned that there are orientable manifolds in 3 dimensions and higher where the two orientation classes are different

from each other. Right is different from left! Among these manifolds are CPn for n>=2. When the cup product in cohomology is available a simple algebraic argument shows that CPn for n>=2 admits no oreintation reversing homeomorphisms. Although the algebraic argument is simple there seems no intuitive reason to expect this. The same phenomenon occurs for some of the lens spaces in dimensions 3. When you glue such manifolds together you get different results depending on how you orient them before gluing. Nothing like this happens in two dimensions. When you join two tori to make a pretzel it makes no difference whether you glue from the inside or the outside. But the intuition gained from the two dimensional case is misleading in higher dimensions.

Story problems were always a drag. I didn't even realize I was supposed to translate them into a formula until I was in college.

Same thing with the elaborate metaphors young teachers seem to think are so helpful. They make sense only when a kid already understands something, and usually just muddy the waters.

Replies: @JimI’m the opposite. I’m not good at computational math but I love the conceptual stuff.

“What is true is that I’m ignorant of polynomial math as used by mathematicians.”

Best just to stop digging, at this point.

@John DerbyshireThanks for setting the record straight. But Galois does deserve more credit than that ;-)

Replies: @John Derbyshire, @Jim, @JimAlso Lagrange made a very penetrating analysis of the existence of solutions by radicals before either Abel or Galois. I’m not sure to what extent Abel or Galois were influenced by Lagrange, however a lot of the ideas in Galois theory can be traced back to Lagrange.

Of course all of these three – Lagrange, Abel and Galois – are very great mathematicians.

"You could have just linked to Wikipedia but as it happens your confusion about its and it’s provides us with the more valuable contributionto what, was a moreprominent theme of this thread."Dude, like, what's that comma doing in the middle of your sentence? They call that some sort of run-on or a dangling sump'in? Next thing ya know you'll be writing enginerish like a real engineer!

Replies: @International JewYeah, sorry, it was my flaky phone keyboard interacting badly with Javascript.

But to borrow from Winston Churchill, this morning I’ve rebooted my phone, while Jim likely still doesn’t know its from it’s.

The advent of the so called new math starting in the 1960s had an interesting and as yet unmentioned result with far reaching consequences. It made it much more difficult for college educated parents to help their children with their math homework. At the same time it made it almost impossible for parents with lesser educational levels to help their kids.

In effect the new math made the math teachers the only ones with the “secret code” to decipher math exercises. This made these teachers much more powerful arbitrators of children’s math education since most parents, especially those from poorer homes, were no longer involved.

I taught public elementary school for thirteen years before departing for “greener” climes. I found that assigning any sort of homework to children from poorer families was a waste of time since the homes they came from were so beset by noise and general confusion that it made concentrating on homework impossible. It would be difficult for even child with a high IQ to do homework where TVs are perpetually on full volume and both adults and children are constantly trying to heard by shouting over the general noise level. It takes just one visit by an outsider to determine that expecting homework to be completed in such an atmosphere to an exercise in futility. Unlike their more well to do schoolmates, none of these poor children have individual bedroom they could retreat to in order to finish their homework in comparative silence. Instead this work is most likely done in front of the blaring TV with other noisy kids present in what passes for a living room. It is sort of like trying to learn in the Bedlam Asylum. The day after these kids would simply copy their homework from the smartest kid in the classroom before handing it it. This copying would be allowed because the “less motivated” kids would threaten or bribe the brighter kids.

The cheapest cure for all of the educational malaise that fogs our education system is to simply extend the school day to five or six o’clock.Then the poorer children could complete their homework in the comparative silence of study halls overseen by teacher aides who could help them with their assignments.

This extended school day would also lessen other social problems. One of these problems arises directly out of the fact that that when both Hispanic parents work, they don’t supply any sort of child care or supervision for their children after about age six or eight. There are no costly after school ballet, piano or karate classes of the type that more well to do parents assign their kids to to make sure they are supervised. Poor Hispanic and black kids are expected to take care of themselves at a very young age. when parent work. They end up roaming around after school creating all sorts of social problems.

” “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.”

McCartney’s father, by the way, was a jazz musician himself and had his own little local band in the 20s and early 30s, so Paul came by musical talent naturally.

@HaciendaWhere does Hume express doubt that 1 + 1 = 2? Numbers don't represent anything. The symbol "1" represents the number 1.

Replies: @Hacienda“There is no Algebraist nor Mathematician so expert in his science, as to place entire confidence in any truth immediately upon his discovery of it, or regard it as any thing, but a mere probability. ”

Hume Treatise, 1.4.1.2

1 + 1 = 2 , as humble as it appears, can surely qualify as a mathematical truth.

Replies:@gdpbull@HaciendaThere are 10 types of people in the world. Those that understand binary and those that don't

Many very competent mathematicians cannot do arithmetic well and fast in their heads. Mathematics is much more than that. One can even be a slow, but thorough thinker and be a very good mathematician. Deep insight does not require a quick mind. One of the first realizations a good mathematician has is that non-linear relationships can defy shallow insight and they quickly learn and understand the power of theorem proving logic.

@Jim"There is no Algebraist nor Mathematician so expert in his science, as to place entire confidence in any truth immediately upon his discovery of it, or regard it as any thing, but a mere probability. "

Hume Treatise, 1.4.1.2

1 + 1 = 2 , as humble as it appears, can surely qualify as a mathematical truth.

Replies: @gdpbullThere are 10 types of people in the world. Those that understand binary and those that don’t

Cryptic de-basement humour. 1100100% LOL.

“What does ‘verbal is harder than math’ mean in precise terms? Do you mean to say it’s a better predictor of IQ? What evidence do you have of this? Surely, you do not intend to offer your GRE post as corroboration? Why bring it up in the first place? More anecdotes then? ”

1) That high verbal ability is less frequent than high math ability, in the way we measure it on tests that are taken as proxies for IQ. This is demonstrably true; the GRE and the old SAT were both accepted as proxies for IQ, and both had far more high math scores than high verbal. I don’t think I said “harder” in the post. I just riffed off the comment. 2) No. 3) None, since I’m not asserting it. 4) No. 5) Because I thought people would be interested in the different frequency level. 6) I linked in the post because it had data about the relative difficulty, as expressed in score distribution, of an IQ proxy.

I didn’t address “you” at all. I merely took the quotes I wanted to respond to. Two of them happened to be yours.

Replies:@Carl G.@Education RealistThe distribution on these tests is wider in verbal than math. So what? I explained in a previous post why your conclusions are not necessarily true; however I will try one more time. GRE and SATs are good proxies for IQ, but there are limitations to the conclusions we can draw. This is a simple case where a low ceiling does not accurately reflect the far right tail of the curve for math students. A 800 math score on the GRE tells an admission committee the student is competent in basic math and not much more than that. I would consider a 780 to be a requisite for a good math, physics or engineering graduate program. ETS could create another section designed to distinguish between the top 0.1% and 2% of test takers in mathematical ability but that would be impractical for a number of reasons, the least of which being that 95% of test takers would find most of the questions incomprehensible.What if ETS made the verbal section easier and the math section harder? The respective distributions would shrink and expand, but it would be shortsighted to assume the new results revealed anything about the underlying inherent difficulty of verbal and math skills in general.

This is very simple to understand. If you don't get it, then you don't get it.

If you’re really good at math, you get to work for the secret government and build really cool weapons.

1) That high verbal ability is less frequent than high math ability, in the way we measure it on tests that are taken as proxies for IQ. This is demonstrably true; the GRE and the old SAT were both accepted as proxies for IQ, and both had far more high math scores than high verbal. I don't think I said "harder" in the post. I just riffed off the comment. 2) No. 3) None, since I'm not asserting it. 4) No. 5) Because I thought people would be interested in the different frequency level. 6) I linked in the post because it had data about the relative difficulty, as expressed in score distribution, of an IQ proxy.

I didn't address "you" at all. I merely took the quotes I wanted to respond to. Two of them happened to be yours.

Replies: @Carl G.The distribution on these tests is wider in verbal than math. So what? I explained in a previous post why your conclusions are not necessarily true; however I will try one more time. GRE and SATs are good proxies for IQ, but there are limitations to the conclusions we can draw. This is a simple case where a low ceiling does not accurately reflect the far right tail of the curve for math students. A 800 math score on the GRE tells an admission committee the student is competent in basic math and not much more than that. I would consider a 780 to be a requisite for a good math, physics or engineering graduate program. ETS could create another section designed to distinguish between the top 0.1% and 2% of test takers in mathematical ability but that would be impractical for a number of reasons, the least of which being that 95% of test takers would find most of the questions incomprehensible.

What if ETS made the verbal section easier and the math section harder? The respective distributions would shrink and expand, but it would be shortsighted to assume the new results revealed anything about the underlying inherent difficulty of verbal and math skills in general.

This is very simple to understand. If you don’t get it, then you don’t get it.

Hurray for New Math,

New-oo-oo Math,

It won’t do you a bit of good to review math. . .

If GRE and SATs use math problems from international mathematical olympiad, you will have harder math than verbal sections.

To those who drew conclusion from simple observation at superficial level, you know your mental limitation.