It's an interesting question when you consider that in at least two university mathematics department rankings[1][2], the US holds 7 of the top 10 global spots. One could argue that for whatever reason, many of the professors, researchers and postdocs at those schools learned math in other countries, but, if these lists are to be believed, the US does have the richest mathematics knowledge in the world.
So two questions:
1) Why doesn't the preeminence of the US math knowledge appear to seep into the primary and secondary school education?
2) If the primary and secondary education in the ROW produces such a high level of capability relative to the US population, why aren't their universities better represented in the rankings?
We do a great job with the extreme students. The top 5% of private and public schools in the country produce more than enough folks capable in mathematics. It's the rest of the country that struggles.
There are a lot of reasons why the US does well in Universities and poorer up until then relative to the rest of the world:
1 - In much of the world, the school you get into matters more than what you did there. (The lowest University of Tokyo graduate is considered higher than the top grad of any other school - so getting in there is the hard part)
2 - In the US we invest more in higher ed than K-12 relative to the rest of the world on a per-pupil basis - especially at the top schools. (Look at the Harvard or Yale endowment on a per-student basis)
3 - In the US, college professors are at the top of their peer group academically. It's a mixed bad in K-12.
I would say that our problem in K-12 is a self-perpetuating one. The teachers were taught math badly and so never really learned it (and learned to hate it into the bargain). So then they teach it badly.
I'm not sure there's any solution except for tuning the students in to Khan Academy and suchlike programs.
> tuning the students in to Khan Academy and suchlike programs
I think parents have an enormous role to play in the effectiveness of K-12 education. If they are not very much involved (e.g., enforcing, participating, encouraging) then school isn't valued or prioritized, nor will the average student see how fun many subjects can be to learn (assuming the teacher may not be effective at this).
> were taught math badly
> teach it badly
Just an aside, wouldn't this be expressed as "taught poorly"?
But the parents also suffered through incompetent math instruction. Unless they are among the few who loved the subject anyway, they're not going to be able to help.
> Just an aside, wouldn't this be expressed as "taught poorly"?
I don't think there's anything wrong with "badly" here, and at least one dictionary [0] seems to support me; see esp. sense 2. But "poorly" works just as well.
> But the parents also suffered through incompetent math instruction.
Agreed, but that doesn't mean parents are not able to be encouraging. Their support, I believe, is more valuable than their level of absolute education on the common subjects.
I'm a college math professor, and I've talked to people who've taught the required "math for elementary ed majors" class.
From everything I've heard, it's bleakly depressing. The students (i.e., the future teachers) show little aptitude, curiosity, or work ethic. They just want to be shown algorithms that will always lead them to the correct answer.
I haven't taught such a course myself. I hope what I've heard is exaggerated. Liking kids is well and good, but if you're going to be a teacher then you should also like learning. My own elementary school teachers did, and everyone deserves an education as good as the one I got.
I went to a university that used to be a normal school. I am not sure the math ed. majors "showed little aptitude" ( they got better grades than I did ) nor curiosity, but they were quieter people.
I should clarify that these are the elementary ed students. I've personally taught students training to become high school math teachers, and they have been pretty good.
This is likely different state-to-state, but I am aware that in the upper midwest primary education degrees are cross-subject area in focus.
Your elementary teacher may not be a math major because they are expected to teach all subjects. There is also probably some bias against deep content knowledge because "it's elementary school after all".
Secondary Education degrees are subject specific, so secondary education math students would, in fact, be mathematics majors.
In the US, rules are not so strict. In 2007-08, 71.6 percent of secondary math teachers had been math majors. An additional 16.2 percent were certified in math but did not do a math major. 12.2 percent of secondary math teachers had neither a major nor a certification in math [1]. The number of uncertified non-math-major math teachers is much higher in lower grades, as you point out.
In general they won't be, and I wouldn't expect them to be. The subject matter of these classes is usually much simpler than freshman calculus.
It's not math per se that I care about here. If, for example, these future elementary teachers disliked reading and displayed the same attitude towards being asked to write critically about a novel, then I would consider that equally disqualifying.
The limit of learning in a classroom is the teacher. If there's a great teacher, the kids can learn a lot. When the teacher lacks capacity (intelligence, leadership & empathy) then the learning is capped.
Outside of people who explicitly want to teach math, the math skills of most K-8 folks I've seen is abysmal.
our best students to really well. our average students are doing poorly / lagging behind. the US also has an anti-intellectual bent to it. You can be smart, but you can't be too curious or question anything.
Spot on. The US education system is fairly elitist. It supports the top 5% and neglects the lower scoring pupils. That way (and with imported brains), the US can maintain a high level at academia while at the same time affording a fairly bad average education level of the total population in comparison to some other industrialized countries. The university system is also very elitist through student fees, Ivy League schools, academic societies, etc.
That's why it's such a weird contrast for us egalitarian European schmucks when we get to know US colleagues in academia, who are extremely professional and well educated, while watching the daily news makes us think that the majority of US citizens must be mentally retarded and suffers from chronic lead poisoning.
>the majority of US citizens must be mentally retarded and suffers from chronic lead poisoning.
You probably think we only pay attention to news from the US but the Brexit vote is merely the most recent example to show that Europeans can be just as manipulable and unintelligent as Americans.
Brexit Vote is an indication people are not drinking neo-liberal kool-aid. You and I might have benefited from Internationalism, but not every one and they are making their voices heard. They may not be as sophisticated as you, but their concerns are from economic insecurity, either solve or at least empathize rather than reach out to some stupid propaganda play-book and stamp them "bigots".
> The top 5% of private and public schools in the country produce more than enough folks capable in mathematics.
That's because it's a very easy job. Once you've filtered out 95% of students, the rest would thrive if you threw them in a closet with a book and a flashlight.
My guess is that as with most things people need to feel that what they are learning is relevant. Start teaching finance in high school and math will become a lot more useful, quick.
I would hope that conventional K-12 at some point would include instruction on what a balance sheet is, a P+L statement, and a bit on what the jargon is. Should also include the basics of sales, how a business operates, finance (as you said), etc.
I've read (once, somewhere, on the internet) that these types of rankings are also distorted by the fact that in the US the top research institutes are usually teaching and part of a University, while they are not teaching in many other parts of the world.
I'm not sure whether non-US research institutions are less teaching focused overall than their state-side counterparts but there are plenty of other confounding cultural factors. In Russia, for example, the most elite science/engineering university (PhysTech) isn't as well known globally as MIT or Caltech because it is made up of dozens of research institutes that publish under their own name instead of under the umbrella organization. As a result most academics dont know that all of this research is produced by a single (albeit distributed wrt geography and branding) powerhouse.
We shouldn't mix up the max skill level with the average skill level. They are very different issues, and different educational systems may optimize for different things.
For example, maybe if you are a very good mathematician in the US you can get high-paying jobs for intelligence agencies, data analytics, that sort of thing. But if you are a mathematical genius in another country, maybe they don't have the same job opportunities so you have to go into math teaching.
That would cause the US to have very good mathematicians and at the same time terrible math teachers.
To me the state of math instruction is illustrated by a 3rd grade math book, teacher's edition, with the answer key in it. (3rd grade math answers should be obvious by inspection.)
"Why doesn't the preeminence of the US math knowledge appear to seep into the primary and secondary school education?"
Because preeminence of top tier institutions (which are kind of global centres anyhow) - has absolutely nothing to do with teaching math to the commons.
Here's a hint:
+++ Americans don't suck at Math +++
There's a very un-PC but very large elephant in the room that people won't discuss.
+ European American and Asian American 'testing scores' are actually pretty good - and have been holding steady for a very long time. (Asians do a little better). Nothing has changed.
+ Latino American and African Americans fare poorly, but having been getting better since we've been measuring by standards (i.e. 1950's-1970's).
Here's the trick:
+ European Americans actually do better than Europeans - on average.
+ Asian Americans to better than Asians - on average.
+ Latino Americans do better than Central/South American Latinos
+ African Americans do better than Africans.
The key correlating factor here is 'ethnicity'. 'Ethnicity' is the broad, generalist predictor of educational outcomes. This definitely not 'race' and it's not even 'IQ' (those things are plausible but controversial) - it's a series of behaviours, social norms, examples, attitudes towards work, success, access to services, social networks, mentors, role models, etc. etc. etc..
Educational outcomes (and crime stats, income stats) break down along ethnic lines. In a manner of speaking - America can be thought of as 'four nations' - White, Black, Asian and Latino. Obviously - it's very crude and generalist, and policy based on this would probably be racist - nevertheless - you pretty much have to look at the data given this.
In the end: American test score results have more to do with the changing ethnic composition of the American population than they do anything else. Again: White people and Asians in America have performed consistently he same for decades. Teaching methods haven't changed much, students habits haven't changed much - so the outcome is naturally consistent.
More economic prosperity, access to services and different attitudes + deeper integration have meant Latino A. and African A.'s are doing a little bit better - but because there are so many more Americans of those groups - particularly Latino Americans - it changes the outcome of the 'average american test score'.
Analyzing educational results does not make sense until you break it down along ethnic lines. Once you do - it becomes crystal clear. It's the absolute #1 most important thing about the educational data that turns 'paradox' about educational investment (teaching has remained largely the same) and outcomes into 'perfect sense'.
Unfortunately, it's so sensitive few will want to talk about it - for fear that the general public equates educational outcomes to 'intelligence' and try to strongly correlate ethnicity + race to this, which would be fodder for racist/KKK types, which wouldn't really help the overall social situation in America.
Anyhow - America is actually doing pretty well overall.
> European Americans actually do better than Europeans - on average. + Asian Americans to better than Asians - on average. + Latino Americans do better than Central/South American Latinos + African Americans do better than Africans.
I've always heard this explained as a sort of "selection bias". Since immigration to the US (particularly for university education) is often seen as desirable, the people who manage to pull it off tend to be above the mean. Do you feel that explanation rings false?
That could very well be true, and I'm not stating it as proof of anything, other than to claim that Europeans score 'ballpark the same' on either side of the Atlantic.
I suggest your theory is probably very true for Asian Americans - the one's who came here are 'la creme do la creme'.
But not for Europeans. Europeans that came here were the poorest, the least educated, criminals, fringe religious types etc.. Europe was 100x more civilized than America during early history - why would anyone with any social status leave London in 1800 - to go and live a million miles away, a very, very hard, back-breaking life?
Well - those tenant farmers who could get cheap land and get out from under the thumb of their landlords etc..
> I've always heard this explained as a sort of "selection bias". Since immigration to the US (particularly for university education) is often seen as desirable, the people who manage to pull it off tend to be above the mean.
I think this is an explanation that could only be come up with by the descendants of those who have emigrated.
Thinking here in Scotland, the people who emigrated were not necessarily the most able or genetically superior somehow. Often, they were simply the most desperate. People who were cleared off their farms by landowners, people who had no other options available to them but to roll the dice and go abroad to Canada or Australia or the USA.
Most folk don't want to emigrate, certainly not in the 19th century. It is a last resort that you do if you are out of options. But perhaps the most capable and able have other options to take advantage of?
It's not clear that capability was that much of a factor. As the scion of Scots emigrants from the 19th Century, at least the 20h century version of my family was made up of bright capable people but hardly brilliant.
It's pretty awful to read how people regarded the Highland Clearances, like this from the Scotsman:
"Collective emigration is, therefore, the removal of a diseased and damaged part of our population. It is a relief to the rest of the population to be rid of this part."
We no longer think of people whose ancestors showed up in the 19th century as immigrants, unless we're trying to make a point about indigenous peoples' rights. In 2005, 22% of immigrants to the US came in on an employment-related visa; the only larger category was family reunification. Immigration patterns "back then" are very different than they are now.
I'm talking about contemporary migration mostly; although IIRC some migration waves in the 20th century were more about perceived opportunity as opposed to desperation.
> Unfortunately, it's so sensitive few will want to talk about it - for fear that the general public equates educational outcomes to 'intelligence' and try to strongly correlate ethnicity + race to this, which would be fodder for racist/KKK types, which wouldn't really help the overall social situation in America.
I don't see any statistics in his comment. I actually don't see any supporting info at all. All I see is wild conjecture.
If he did go looking for statistics, he would find that the effect he describes is better explained by the expanding lower class in america, regardless of race.
It's widely known that Asian American and European American students outscore Latino and African American students. Nobody would contend that.
For you to indicate that there is a very specific issue i.e. 'poverty' that is the 'driving factor' is wildly speculative and outlandish - a fetish of your ideology - and and the 'impetus' is upon you to prove it one way or another.
My presumption of 'ethnicity' is much better because it would include 'average income' as a natural outcome of an ethnic groups situation regarding education, racism, poverty etc..
And FYI - 'average income' would be a function of ethnic attitudes overall.
Nobody would deny that 'income' has some effect on test scores - but my friend - you have to grasp the obvious reality that cultures who don't value 'education' will have significantly lower incomes, and more likely to be in poverty. 'Poverty' is the result of bad education, as much as the cause.
If you've travelled the world then you know how crazy different people are.
Many of those differences will yield different attitudes to school, to homework, to respect for authority, to criminality, to family etc.. All of those things will affect educational and life outcomes.
By the way - here is PISA scoring across national groups (breakdown for USA students):
Is it 'racist' to say that Sub Saharan Africa is a very, very different place than North Africa / Maghreb? They are like Venus and Mars they are so different. And it's not just 'income' or 'the weather'. They are entirely different cultures.
The world is very diverse. Different values = different behaviours = different outcomes.
Several of your recent posts have crossed the line into incivility. Addressing people as "buddy", telling them about the "fetish of [their] ideology", etc.: all this is patronizing and rude. Please don't do any more of it on Hacker News.
If you're going to engage in difficult topics, you have a responsibility to do so with extra respect, not less.
My comments were reasonable and thoughtful - if borderline controversial - and yet it was directly ad-hominem to call me 'racist' and the comments 'wildly speculative'.
I call you 'Buddy'? Someone (you?) called me 'racist'. Which is more offensive in reasonable discussion?
'Buddy' is a colloquialism, it's not offensive or attacking.
At the same time you offer no insight, rebuttal or points of your own. You're attacking people and chiding them 'not to speak anymore' - while I have no problem with any of the topical discussion.
Your attempt to shut down the discussion while not offering countervailing arguments to is exactly the antithesis of HN, in my opinion.
So if you have some relevant comments about education and potentially relevant factors, however speculative - do so. I'd suggest adding to the discussion instead of trolling and attacking.
This is not a 'safe space' wherein normal ideas are deemed unspeakable because someone, somewhere, might feel uncomfortable.
This is a somewhat worrisome comment. It reaches for a lot of conclusions with zero supporting evidence.
There is no need to bring up race or ethnicity to describe this effect of scores getting worse. Everything you attribute to being Black or Hispanic can more simply and more accurately be described by being poor. SAT score is most directly tied to family income, not race. http://blogs.wsj.com/economics/2014/10/07/sat-scores-and-inc...
Instead of worse scores being the result of changing ethnic composition, they are the result of changing income distribution, and the expansion of the lower class.
" Everything you attribute to being Black or Hispanic can more simply and more accurately be described by being poor. SAT score is most directly tied to family income, not race. "
+ This is not true +
Even when normalized for income - there are still large variations in outcomes.
Again, I'm not really saying it's 'race' or 'genetic' or 'IQ' so be careful with your disdain :).
My friend - do you know the stereotype of the 'Asian who works hard in school because his parents compel him to' - well, it's not just a stereotype, it's true. Some cultures value education more than others. That will show up in the results.
Asian Americans are 600% over-represented in tech (i.e. 5% of population but usually 30% of tech companies). That's not some fluke - and it's not because 'Asians are super rich' - it's obviously a cultural preference. They are choosing STEM and tech for whatever reasons.
It doesn't make some people better or worse than others as human beings, it just means some variations in outcomes whenever you measure something.
This this is what actual 'diversity' really means, though ironically I think most 'pro diversity' advocates really intend to have a hyper-egalitarian situation wherein there is little diversity beyond skin tone :)
People below are saying their poor Scottish ancestors didn't seem that unusually smart. Immigration restrictions have changed who comes to the US. Pre-migration parental socioeconomic status/educational achievement is one of the best predictors of educational success among children of immigrants. Your trick is simply an observation that Asians immigrating legally to the US come on certain visas which bring a non-representative socioeconomic mix to the US, and their children (who are now Asian-American) do very well. The people who stayed in Asia are not the same as the people who could afford to fight for a US visa.
The "immigrant advantage" fades in three generations, at which point achievement by children reflects the US average. This is the influence, if you like, of American "ethnicity", which is often quite anti-intellectual.
"It's an interesting question when you consider that in at least two university mathematics department rankings[1][2], the US holds 7 of the top 10 global spots."
It's a fallacy. US is a large country, so those talented students concentrate in fewer universities. In Europe for instance, due to language and culture barriers, talented students from Czech Republic do not very often go to Cambridge. You need to look at mean or median if you want accurate assessment.
China is a larger country with extremely promising mathematics scores. So why is it not 8/10 China and 2/10 US (in proportion to population) or even more extreme? A large country having access to a large talent poll doesn't get very close to explaining what's going on.
It's hard to tell, but it can be because most maths research is published in Chinese, while these rankings are little anglocentric. Also lot of Chinese probably migrate to the US as soon as they can. Anyway, I was just pointing out that the ranking is not such a simple argument.
I once heard an anecdote that might describe some of what's happening. In the trenches of WWI, when it was time to fight, soldiers would have to climb up a ladder onto a battlefield. The problem was that German snipers could see the tops of the ladders. The Germans would keep their rifles fixed on where they knew the enemy would emerge and simply shoot them down once they saw helmets appear.
The European Allied soldiers were so disciplined that they just kept climbing up the ladders and getting killed one by one, following their orders to their deaths. The Americans saw this and said, "fuck that, I'm not climbing up there."
I think most Americans are pragmatic and they won't do something unless it makes sense. And to be honest, most people don't need to study math. Or at least it's not obvious that they do. I think most of the math professors I've talked to would agree. They view math, as it's taught in core curricula, more as an art than as having vocational value.
> In the trenches of WWI, when it was time to fight, soldiers would have to climb up a ladder onto a battlefield. The problem was that German snipers could see the tops of the ladders. The Germans would keep their rifles fixed on where they knew the enemy would emerge and simply shoot them down once they saw helmets appear.
> The European Allied soldiers were so disciplined that they just kept climbing up the ladders and getting killed one by one, following their orders to their deaths. The Americans saw this and said, "fuck that, I'm not climbing up there."
Wow, that anecdote explains American supremacy better than anything to date. /s
Without a citation I'm going to have to call bull-shit on that one, I'm just trying to imagine their CO standing there with an ever mounting heap of corpses at the bottom of the ladder and not once thinking 'this doesn't seem to work'.
Some googling does not turn up any evidence for your story either.
I don't know about any americans, but isn't the Souain affair pretty much a CO ordering his soldiers to keep going despite an ever mounting heap of corpses?
Sure, but that's not structural and there are definitely parallels in the American civil war.
war is an exercise in stupidity to begin with, it shouldn't be surprising there are pockets of even worse. But to claim that structurally Americans refused to get out of the trenches in a certain way in order not to get killed whereas docile Europeans were led like lambs to the slaughter is not something I've found in any history of World War I (or II for that matter).
Both wars had extremely heavy casualties on both sides, and in both wars there were quite a few instances of CO's treating their men like disposables. The Christmas Truce is a beautiful story about such behavior. Even so, both sides were desperately trying to win the war and the rule would have been to not take action hastening the demise of the men on one's own side.
I've yet to come across any substantiation of the anecdote related above, if it was structurally true you'd think it would be more than a mere anecdote.
It's a stupid anecdote to point out the fact that differences in cultural attitudes can explain and somewhat justify test scores. I'm not even American, and I never claimed that the anecdote is true.
> It's a stupid anecdote to point out the fact that differences in cultural attitudes can explain and somewhat justify test scores.
It would if it were true.
> I'm not even American
That's immaterial.
> and I never claimed that the anecdote is true.
Well, you didn't claim that it wasn't true either, but the whole thing hinges on whether or not the anecdote is true so if you bring it up I'm going to assume that you at least believe it to be true and that the conclusion is supported by the anecdote.
If we're all just going to make stuff up to prove some point then it becomes very hard to reach conclusions.
OTOH, the terrible truth is that it was necessary to order thousands of soldiers to their almost-certain deaths in order to win the war, and without a disciplined army this would not be possible. This says something about the value of discipline (in war or maths!) even when it's not obviously in your personal interest.
Is that true? I mean: is it true that the actions where it was necessary to "order thousands of soldiers to their almost-certain deaths" were significant causes of the final outcome?
WWI was the first 'industrialized' war, trench warfare implied the certain death of a huge number of men if the lines were ever to move, it's basically a never ending meat grinder until one of the parties runs out of warm bodies, supplies or ammo.
The final assault on the remains of the entrenched opposition were without exception extremely bloody and the side that would take the others trench never did so without significant losses.
Most math (and physics) programs, especially at higher tier universities, have not only a disproportionate number of foreign professors and researchers but also students. So it's largely people who had superior primary and secondary school math education teaching others who had superior primary and secondary school education.
You can see this especially strongly in graduate programs, where US citizens can often get a sort of "affirmative action" because they are the only ones eligible for NSF student fellowships (incentivizing universities to admit them). There's just very little domestic interest in math and physics, despite both feeding into rather favorable job markets (as long as you aren't dead set on being in academia long-term).
As for your second question, it's an extension of US universities' preeminent standing in most fields (on average, there are of course exceptions). I don't think there are really any math specific effects going on there.
Both quantitative finance and more general data science draw heavily from physics PhDs (you can see this if you look at job listings for these jobs; physics is almost always listed next to math in the list of desired PhD degrees). Depending on area there can also be a significant number of industry pure-research positions available; quantum information is a good example.
As long as you are willing to go into industry, you're looking at a 90-100k starting salary[1]. That's pretty damn good considering that in the US nobody pays for a physics PhD; the average TA stipend is around $30k with tuition and medical insurance covered. You also have a decent chance of getting a research assistanship to provide the stipend once you start working seriously with an advisor.
The high number of physics PhDs getting postdocs off the bat is driven by people who want to go into academia/research, not by a lack of industry options. Admitidely if you're dead set on going into research (especially right off the bat) you better be ready to play the same low-paid tenure lottery as everybody else, but if you're just looking for a good career a physics PhD is a pretty good bet.
Why would you get a PhD in physics if you would go into data science? Just get a data science job. Why would you think that someone that spent six years of their life thinking about light, energy, matter and the universe would be satisfied thinking about graphics to show the results of A/B testing an advertisement?
If all you want is a career in data science, sure. But with a Physics PhD you (a) get those six years to study physics, which many people enjoy and (b) get to take a shot at getting a research gig, and fall back on a solid foundation in a good career track. Not to mention a PhD makes your chances of not having to do something as boring as A/B testing advertisements markedly higher.
If getting an art degree meant you could try and become an artist (with the same astronomically low chances of success you see today) and then immediately fall back on a solid career if you fail, don't you think it would represent a pretty good option? If you enjoy physics or math, you already live in a world with that possibility.
US society/culture in general is the main cause of this problem as most people here do not value math. Math requires deep, creative and original thinking and thinking is rather difficult to perform act. It seems that the US culture derides analytical thinking in general, with many popular public figures flaunting their "inability to do math" as some sort of great achievement and sending out a loud and clear messages, like, "don't analyze just enjoy your life and avoid math as math means analysis". This paper titled "Student Apathy: The Downfall of Education" [1] is a good read in this respect too.
The bad effects of such societal level bias against math can be seen in public spending in schools also. Enormous amounts of money are spent on football coaches/teams[2] in schools whereas not enough money for math teachers/students.
The society seems to equate success only with one's ability to earn money. Then you can see that many such popular public figures "getting successful" without math as they can be seen to earn large amounts of money.
This is sad as such people even if are successful on money front, cannot understand any of the complex aspects of modern societal/business/political structures (be it, things like privacy issues, or many public policy issues like taxes) and of course cannot understand any of the modern technological/scientific discussions (be it discussions regarding global warming or genetically modified food or statistical significance of some tests).
China comes to mind. They seem more technocratic than we (USA) do (legalist hierarchy?). Can you imagine Deng Xiaoping saying, "Ha, I'm just a policy guy" or something alike
>> It seems that the US culture derides analytical thinking in general, with many popular public figures flaunting their "inability to do math" as some sort of great achievement and sending out a loud and clear messages, like, "don't analyze just enjoy your life and avoid math as math means analysis"
I am generally not a fan of speaking in terms of "privilege" and "identity", but this is precisely an example of "developed Western country privilege". The attitude goes, "See, I can still be successful/well-off without having to hunch down and study math like those third-world FOB immigrants do." It comes down to status signalling: not having to do math becomes a status signal. It also appears it is mostly this attitude and not the widely blamed "bro culture" that is the major reason why women (in the West, because this didn't happen elsewhere) abandoned CS sometime in the late 80s/90s.
>Math requires deep, creative and original thinking and thinking
That is absolutely not true of anything the average non-math-major has ever been exposed to under the heading "math."
It's rote symbol manipulation requiring diligence, practice, and attention to detail. American K12 math education asks for fast and reliable algorithm execution, not insight.
Only math majors and attendees of a few exceptional private high schools will ever seriously engage with proofs.
You're probably right. those rankings have a lot to do with the research papers published by those schools from non Americans.
Relative to the US population, the primary and secondary education do not produce a high level of capability.
Looking at the fields medals per capita[1] you can see that the US doesn't have as much as the UK, Russia, or France. You can also see that the university with Fields medals recipients [2] for more details, and indeed you can see that of the mathematicians associated with Princeton are not Americans.
As a specific example, in [1] you can see that France is generating more than 4 times more fields medalists per capita than the U.S. Why is that?
It probably has to do with the more rigorous Math education.
Look for instance at this translated Math final[3] exam for French high school students. This is for the "Literary" students, those who focus more with the worst level of Math.
You can see examples of the "Scientific" math test here [4]. It's in French, but it's Math, just by looking at the symbols it's possible to understand. It has some differential equations solving, probabilities, geometry with vectors/planes in 3D spaces, Series analysis, etc. Integrals and derivatives are also part of the program. In Physics these concepts are applied to calculate velocity/speed/radioactive decay etc.
The single biggest factor on student performance is not any in-school factor, but rather the socioeconomic status and educational level of the parents.
The best American student mathematicians are the equal of anywhere in the world. But when people talk about poor math performance in the US, they're talking about the average—average math performance is terrible. This is because the U.S., more than any other advanced country, tolerates a high level of poverty and economic inequality. This inequality is reproduced within the educational system, and brings down our averages.
The average is lower not because the U.S. has lower performance across the board, but because so many more students lack the most basic numeracy. Too many students are going to school hungry, come from families torn apart by joblessness, abuse, and addiction, or where the single breadwinner has to work 80 hours/week in order to achieve a higher level of poverty, and simply doesn't have time to help the kids with their school work or ensure they are disciplined students, much less ferry them to all sorts of enrichment activities.
Funny fact: a couple of my former classmates won gold/silver medals on the international mathematical olympiad. One minute googleing sais that at least 4 of them are teaching on US universities you mentioned. I am from eastern Europe.
The US does have the richest mathematics knowledge in the world because the US can buy it.
Math is the lingua franca of sciences. It should be taught like a language as well. In our elementary schools we don't have teachers dedicated to teaching math. Instead we have teachers who are afraid of math themselves.
I've wondered about that. The imperial system is positively awkward. What about other countries that also use the imperial system instead of metric? Are they also behind?
Not really. You still go to the shops and buy things in litres and kilograms, and their entire construction industry is in metric. The only commercial product that isn't in metric, funnily enough, is beer and cider. I guess getting pissed people to change proved harder than everything else.
...and milk is always pints. An awful lot of products come in imperial sizes but metric labelling - 454g jars of jam etc.
Timber sometimes gets called 2x4 even though it's 50x100, but that seems less common now.
Oddly just about everyone still uses feet and inches for their height. I hear kilos more and more often for weight, I don't think I've ever heard cm for height in the UK.
Returned milk is (you know, in dem bottles). But milk you buy in the store is 1L, 2L, etc.
> Oddly just about everyone still uses feet and inches for their height.
True. Probably because of the recognition of the 6' centering measurement. People know that if you're 6', that's kind of the cut-off for what is referred to as 'tall enough'. Over that, you're generally 'tall'. Under that, and you're average height.
Not in any supermarket I've been in, 4 and 6 pints are the commonest. Maybe us northerners aren't trusted with metric milk yet!
Only time I see litres is some of the niche brands (like Cravendale and branded organic) in bigger supermarkets, or 500ml in corner shops and some garages. So they can charge more. Presumably people compare 2l price with 2.3l / 4 pint instead of £/l.
It's a very small group: apart from the US, the only other non-metric countries are Burma and Liberia. It would be very difficult, I'd imagine, to do a meaningful comparison between these countries.
I found math class stultifying and boring until the focus switched from learning algorithms for computation to doing proofs of relations and concepts. Also I began to discover the actual applications for advanced math, instead of learning it for its own sake.
I wish in hindsight that instead of taking endless years of calculus there was a high school version of real analysis, and exercise problems rooted in real-life to motivate the learning.
Reading this comment as a programmer, I had the horrific thought that we treat our children the same way we treat our computers: programming them with algorithms, debugging them with failing grades, executing them for test scores. It's barbaric.
Leave the program execution to the computers, give the children critical thinking skills.
I teach calculus on the side sometimes for beer money (ie. adjunct), and has made me stronger in my long-running feeling that we teach way too much calculus in the STEM fields.
Most people "in the trenches" writing tools that would make use of the fundamentals in their field (either in the physical sciences or engineering disciplines) tend to agree that linear algebra has disproportionately little representation compared to calculus. To be fair, that is somewhat of a downstream problem from mathematics departments, and speaks more about the possibility of stagnation in the undergraduate science and engineering curricula.
fair caveat though: my PhD is in numerical linear algebra...
I didn't really get into math until calculus, personally. There was this moment of realization that calc starts to describe the real world, opening up the ability to do all kinds of useful and interesting things. I ate up calc 1-3 and differential equations, even though I have no real use for them today. OTOH, my brain apparently just is not equipped to deal with proofs, either in mathematics or comp sci theory. They were always an enormous struggle for me.
Put 100 mathematicians, math educators, and policy makers into a room and ask them to come up with a good mathematics curriculum. They'll come out with 101 proposals. That's not a problem, really. Mathematics is not merely about mathematical substance, but also mathematical process (the two are intimately related), and not just process, but the process of discovering processes. Curricula, standards, directives, are all substance. Policy makers want to put their name on a thing, a substance and give it to everyone else and hope that process develops... somehow.
I think you have to contextualize mathematical in the broader problem with american public schools: they're awful, awful places for many students including me. I ended up studying higher mathematics in college and still study on my own for my own pleasure (and I get to apply some really high powered ideas to programming once in a while which grants a satisfaction that lingers). I think my math education could have been advanced 5 or even 10 years if public school weren't such a soul-crushing stultifying nightmare of procedure and compliance.
Lately I've been experimenting with comonads to structure stateful code. The idea is that a comonad describes the interface for a state machine, and you can convert any comonad in Haskell into a monad transformer which acts as a restricted form of the state monad. The state monad describes expressions that can arbitrarily manipulate a state of some type. The monad you derive from a comonad does not have direct access to its state, the comonad describes all the allowed manipulations and queries.
data CoT w m a = CoT { runCoT :: forall r. w (a -> m r) -> m r }
Let's unpack this. The following type represents a state machine whose nodes are labeled with a continuation demanding an `a` and executing an effect:
w (a -> m r)
The forall quantification in CoT means that it does not care what the result value is. It will take a state machine, manipulate it for a bit, produce an `a`, pick a continuation from the state machine and execute it.
That's a lot going on! I don't have time or space to explain how this is actually useful but here's a handy approximation:
- `w` is an interface
- `w s` is a model labeled with a value of `s` for each state it can occupy. These labels are a sort of view.
- `CoT w m a` is a controller that can manipulate a model and compute in some effectful context.
High powered MVC. The upshot is that because we're leaning on Functors, Monads, and Comonads there are extremely well behaved and natural composition operations. For example, `fmap` allows us to change the view of a model by applying a pure function. The fact that `CoT` is a monad means that we can run controllers in serial, as well as combine our controller languages in sensible ways. The fact that `w` is a comonad means we can reason about the behavior of models equationally. This lets us transform, compose, and compare models with mathematical precision.
This is a project I only seldom work on in my spare time. I have already applied it to writing GUI's and game logic to good effect but I'm still exploring the design space regarding reactivity and temporal behavior. This has led me to reading the mathematical literature pertaining to products of comonads/sums of monads (they're dual).
Everyone believes they are bad at maths, so they are. It applies just as much to the UK. My youngest believes this strongly, yet when we were quizzing, or helping with homework, she didn't seem to find it too hard, just believed it so. She certainly didn't get those beliefs from home!
Looking at my kids maths lessons, especially in late Junior, so much effort was spent to actually hide the maths that I wonder they learnt anything!
OK, not everyone, but it seems to be the subject that people are most likely to claim to be terrible at. Schools don't help by sucking all the fun out of it.
Was always my favourite subject, but that was despite school rather than because of.
If they would teach kids to play with math (or any form of knowledge, for that matter), and not just run through 10,000 rote problems, maybe we'd rank better.
Especially word problems -- most of them felt like they were written for students wearing intellectual blinders... if you had any modicum of relevant knowledge outside of the lesson oftentimes word problems were impossible to solve
> If they would teach kids to play with math (or any form of knowledge, for that matter), and not just run through 10,000 rote problems, maybe we'd rank better.
Don't the countries that rank better use the rote method even more?
>Though lesson study is pervasive in elementary and middle school, it is less so in high school, where the emphasis is on cramming for college entrance exams
I think the implication is that kids are supposed to be taught to play with math in addition to the rote memorization, not that rote memorization is evil.
Many of the more "fun" math techniques rely on knowledge acquired through rote memorization, just like dynamic programming uses the results of inefficient calculations to speed up subsequent calculations.
I think rote learning is a really really important thing, perhaps the most important thing, in math. Math is like any sport... you get good at it by practice practice practice. Check check check check. You have to train your brain to be good at it, and it can be hard, just like anything you train at.
> Many of the more "fun" math techniques rely on knowledge acquired through rote memorization
I wasn't aware of this. I have a nine-year-old. Are there any examples of this?
Ignore the stupid rectangles, and ignore the part where it says "They actually use the distributive property, but we do not need to explain that to 4th grade students." (seriously, wtf)
A child who practices lots of sums, such that she knows how to add 420 and 56 without counting with their fingers or writing things down, will be able to learn this kind of multiplication with ease. Once they learn that and practice a lot, they will be able to generalize the method to multiply any two two-digit numbers.
Memorizing things is important because if you have to count with your fingers to calculate 3x3, you will never be able to calculate, for example, 23 squared in your head. But if you know the times tables your thought process might look like this:
20x20 = 400
23x20 = 400 + 60
23x23 = 460 + 23x3 = 460 + 69 = 529
(23 times 3 is "the hard part" where most kids who know all the theory (distributive property) but are out of practice (not enough rote memorization) will lose track)
A more advanced example is what engineers used to do a lot before they had calculators: They memorized log tables, so when they wanted to multiply big numbers they just added their logs together, because log(a x b) = log(a) + log(b).
An unrelated example is converting miles to kilometers. The official relationship is that a mile equals 1.609 kilometers, an ugly number that doesn't work well with mental arithmetic. But that number is kinda close to the golden ratio, don't you think? So if you are the kind of weirdo who memorizes the Fibonacci sequence, you can quickly calculate that 13 miles should be about 21 kilometers (and 21 miles is about 34km, and so on), because the ratio between a Fibonacci number and the next approaches phi.
But the whole point of play is to run through all those rote learning steps while simultaneously learning about the system and environment they live in. Sure you can cram, and then you'll test well on standardized entrance exams, but then how many of these people are able to effectively apply that knowledge to real-world problems? If you play, or tinker, or experiment, as part of your learning, then you already have this experience.
It's not about "fun", it's about experimentation. Free-form intellectual play.
Could not agree with you more. Another thing I've noticed anecdotally is that many of the math teachers in the secondary system in North America do not have a broad understanding of math themselves. Math is given the most rote treatment of all the subjects, and I suppose this is understandable given the abstract nature of math.
But students are never told why they should care about abstractions in the first place, which is unfortunate. Many high schools simply refuse to speak the students' language.
I suspect this will change in this century. IMO, if a high school really wanted to be progressive, they would totally reform their math curriculum to include more exposure to applied math and computer sciences. Young folks should be using math to build their own Instagram or Minecraft clones that they can deploy to their devices that VERY DAY - using the concepts they've been introduced to in mathematics.
Ok so taken literally, that was a bad example. But instead of giving me a textbook of problems where I solve y = mx + b, can I use mathematical concepts in a creative capacity? To do things I care about as a high schooler? Can I see those concepts on my device today?
> Without the right training, most teachers do not understand math well enough to teach it the way Lampert does.
My high school math teacher majored in math and then got a certificate in teaching. She wasn't a teacher who was told to teach math among other classes. She knew all the advanced stuff (beyond what a high school curriculum required), she was excited about it, she could explain things in various ways, give analogies, was available after class to ask questions and so on. And that is post-collapse Soviet Union full of corruption, poverty and other crap like that. Surely if we can spend trillions of dollars on F-35 we can get us some good math teachers...?
In this country I see a large disconnect between words "Oh kids are so very very important, they are our future, they can't play outside too far because they will be abducted and we care so much for them" and deeds: I see large classrooms, not enough teachers, teacher are underpaid, not interested in math. Funding comes from local property taxes so rich neighborhood get more money, poor ones get less.
Another thing is I remember teachers were respected. Imagine how we react to someone saying "Oh they are doctor. And then everyone nods, right, they are very successful. Or lawyer, or works for Google and so on". Why isn't teaching like that? My mom tells me someone back home thought she was a teacher, because she at her age spoke some English. And she took it as a great complement. In this country you tell someone you thought they were a high-school teacher and they might get offended. Something is very wrong here...
As for making it more exciting -- initially I studies math by repetition and it worked pretty well for me. I think works because kids are amazing at memorizing stuff. Why not first take advantage of that? Starting them out with set theory sounds all cool on paper that is not how humans learn. Multiplication table, basic patterns, even operations are fine to memorize first. Later on it makes most sense to introduce proofs, word problems (I remember doing lots of word problems, our teachers were crazy for them) and so on.
I don't know of any country where teachers are considered as elites. The reason your mother took that as compliment is not because teacher is a high social class occupation, but because being good at english is considered a great skill in many countries. If you really had to compare, in most cases people would feel more flattered to be mistaken as a doctor (or an engineer who works at Google than as an elementary school teacher.
I know people will throw stones at me for this, but the reason teachers are not the top 1% of the society is because functionally their job is a commodity. (We're talking about elementary/middle/high school teachers, not professors here).
It is not hard to find someone who knows high school math, for example. I'm not discounting the fact that there are sometimes really outstanding teachers, but the thing is, it's hard to objectively measure their performance since their teaching talent is not directly related to how well their students do. On the other hand, the "teachers" at universities are well respected since their talent is not only limited to how well they teach but the quality of their own research--the value is much easier to quantify.
Nope, at least not in Norway or Sweden. Or possibly only in the sense that non of those professions are held in particularly high regard. And they certainly aren't paid like doctors of lawyers.
Sure if you ask people, they respect teachers in the abstract sense that they're people doing a very tough job for very little money, but it's hardly a career people aspire to, and certainly most teachers I know will admit it was their second or third choice that they kind of fell into.
This seems unlikely. Perhaps in Finland. But the Danish teachers that I know have complained about the collapse of respect for the job as well as the system for as long as I've known them.
This is true, for some odd reason, teaching in America is not socially respected institution as much as it is elsewhere.
Teaching in Canada is along the lines of nursing or something: it's professional, requires a degree of competence and certification, there is social value. It's not exactly 'doctor' but it's respected for what it is, as it is in most nations.
I think it's because people assume American teachers are not paid much, but the one's I know are paid reasonably.
> She knew all the advanced stuff [...] was excited about it, could explain things in various ways, give analogies
I envy you this experience. I still remember telling a math teacher in 8th or 9th grade that I wanted more information on why a formula worked, how it was originally derived, or any different way of looking at it to better understand it. She essentially said, "you don't need to worry about all that, just memorize it." It was intensely frustrating for me, and contributed to me losing interest in academic pursuits. I don't know how you can train teachers to be more open to this kind of thing, but it sure would help.
> Funding comes from local property taxes so rich neighborhood get more money, poor ones get less.
I used to think so too, but I don't think that is exactly the case. School districts usually span many neighborhoods, rich and poor. Within school districts, there are school zones (school boundaries), or mappings of residential addresses to assigned schools. While it's true that richer neighborhoods tend to have better schools, the property tax money is collected centrally at school district level, and then distributed across schools, both rich and poor. I haven't seen evidence that districts distribute more money to schools in richer neighborhoods.
If anyone is more knowledgeable on this, I would love to learn more.
Because there are a thousand other things you can do that are worthwhile and there is no anxiety to be perceived as "smart" in the way that math requires. Just party on and network and land yourself in a start up. You can pay someone else to do the math because the real money is in MAKING DECISIONS.
Math is hard work. I was bad at math. I went to college a couple years after I graduated high school where I made little effort to challenge myself. But after figuring things out, at 20 I decided I wanted to do computer science as I enjoyed some of the programming skills I picked up (C++ of all things!).
I was very unprepared for the math part of it and it was hard. I tested into a remedial math level and when I looked at the CS requirement I knew that wouldn't be a good way to start. So I bought a used math book and spent a few weeks studying hard so I could get into a decent math placement so I could be where I needed for CS. This was just the start.
I had a long road with a lot of frustration but I made it. I made it because I never worked so hard at something up to that point in my life. There were times I thought it was hopeless but I just continued to do rote math problems, over and over again. And slowly the concepts started to sink in more and more. But there was always the frustration of missing the little details and forgetting a concept or not having enough experience honing a certain skill. But if I kept at it, over and over again, I would learn it and it suddenly became easy.
I was "bad at math". It would have been so easy to quit and try something else. But I knew I wanted to not only pass my tests but really understand the calculus and differentials, etc I had to do. And it was the best thing I ever did for myself.
Not only does working so hard at something prove you can work hard and achieve something but it shapes the way you see the world.
Math is hard. Some of us just have to work harder. People should realize that failing is normal and if you keep at it you will eventually get it. For some people it takes longer than others. I could never create math. But with enough time I believe I could eventually truly learn anything because of this experience.
I mean that's really the problem. For some reason we think things will just make sense once we are taught the lesson and we try an exercise. But that's so often not the case. Accept you'll never be the genius who creates new things - maybe you will but put that aside. Focus on trying hard and committing yourself. It doesn't matter if it's math or cooking food, the more you try it and think of it and accept failure the better you'll be. Accept your limits as a creator but never accept that you can't understand a concept if you spend the time you personally need at it.
It isn't that they cannot learn X, they just cannot learn it in the same time frame
This is exactly the argument I use when seeing people criticizing or laugh at others over their profession or education. I also think that the time frame one needs to "learn X" greatly depends on what they already know, i.e everyone is born a genius; training is what makes the difference.
[...] and they shouldn't be expected to!. Hats off for this too. Probably the biggest flaw in most of the world's educational systems.
I'm no fan of the American education system, having suffered through it a full 12 years, but I have to believe it's not the primary cause here. Math is hard, and near impossible if you're stressed. I excelled at math, despite relatively boring math curricula. Why? Because I wasn't stressed as a kid, my family was stable and did not suffer from any serious physical and mental illness, and one parent always made enough money so that the other could stay at home throughout my entire childhood. I had an enormous advantage, and most all of the kids I knew through advanced math classes and math competitions had a similarly charmed existence.
By failing at reading comprehension. There's a vast difference between public school students and school-aged children. In Chicago, only 10% of public school students are white[1], with a 45% white population[2] (for example.) Conflating the two things seems almost willfully deceptive.
What's wilfully deceptive is pointing to one city and claiming it is relevant to national statistics. The WP Article assets a majority of public school students IN THE US are in poverty, that is objectively false. So the reading comprehension problem isn't with the article I cited.
The WP article isn't really central to my point -- that outside factors have more to do with poor math performance than anything happening inside the schools. I worked in a public school as a tutor for kids slightly behind their grade level (not with the kids who were really struggling) and I have plenty of anecdata from that experience. Kids with drug-addicted parents, kids with parents in prison, kids not having enough food at night, medicated kids, obese kids, a kid who had to move mid school year because his house was shot up in a driveby, etc.
One of my friends graduated with an engineering degree from MIT. I once asked him if he could have traded that to be better at Football, would he? His response was that if there was even a remote chance he was good enough to play in the NFL he would have traded education for that chance.
Humans innately crave fame, and the lack of scarcity that is perceived to come with it. The USA has sadly, and unknowingly groomed that romanticism of fame, towards consumables (this might be the long term effects of Capitalism). Rather than grooming that romanticism of fame towards research/intelligence/creation (the renaissance).
Even local to the film industry, almost 100% of 10 year olds want to grow up to be Brad Pitt or Salma Hayek, not George Lucas or Stephen Spielberg[0]. Even fewer want to be the writer.
Even local to football, everyone knows the names of the player that caught that hail marry, got that big hit, recovered that fumble. Few know the names of the people who create those plays.
Only in recent years have the masses truly started to recognize creators/intelligence with the title famous (Gates, Jobs, Zuck). But again for the wrong reasons, i.e. for their money. Even artists Picasso, Beethoven (adored for generations) have only truly been appreciated by the aristocracy, i.e. the rich and "mathematically" acute.
Which way do you suspect the causal link lies? Are we first intelligent, therefore we appreciate the intelligent? Do we first appreciate the intelligent, therefore we become intelligent?
[0] I mean no disrespect to Brad Pitt or Salma Hayek, or actors in general. Good Actors need to be extremely intelligent, but this isn't why they are adored. I'd argue this part of them is even sadly shunned by the media and populace.
It's not just Mars or the moon, or decades ago, the US also is where most operating systems and software and CPUS are conceived of and designed. As well as countless other things that people who are so ignorant would not reasonably be expected to be able to do.
I agree USA has an impressive track record for space exploration and is at the forefront of computing, but that seems like more of a reflection of our economy and top talent than overall math literacy. Ideally we would have the best of both worlds: good high school education, and a good economy, but for some reason the former lags behind.
Some of it is cultural: Americans don't want their kids to have too much homework, in many cases they don't want to help with homework or are unable to; they want a silver bullet. Some of it is lack of qualified teachers, and part of that is the way public schools are funded from the local tax base--do it cheaply as possible. Some of it is political pressure to dumb things down so that students can be graduated from HS before they reach the legal drinking age. Some of it is the rise of the education ``experts'' who always have some new…silver bullet (Feynman had a lot to say about such experts after reviewing math textbooks for California in the early '60s). The new math came about as an almost hysterical reaction to Sputnik, when, in fact, the U.S. had plenty of highly qualified scientists and engineers who just happened to get beaten to the punch by the URSS. But no, the math curriculum had to change into some kind of Bourbaki for tots, meanwhile in fact the Soviets were teaching math the old-fashioned way--nice cognitive dissonance there. There are a lot of factors, and I don't see any magical way of improving curricula, getting better teachers, requiring more homework, etc.
In 4th grade, I got a C in math. During my teacher student grade review meeting, my male teacher told me "it's okay, girls don't do as well at math as boys."
Yeah, so there's this as a reason. I definitely wasn't a good student until high school. I definitely had a bad teacher who gave me an excuse for YEARS to accept bad results in math. My mother still bitches about that guy to this day. I suspect I have a slight learning disorder that affected my ability to process numbers. I mix 6/9s. 7/9s when transcribing numbers. Made homework difficult. Once I got to the point where letters started replacing numbers in homework, I went from a C student to an A student in math and was competing for the top grade in my math classes. Once I learned that I could do the work, I did the work and excelled.
I suspect that this type of bias affected more girls than just me, and these discouraged girls affected US scores.
The key to math education is practice. Even drilling, maybe.
Other fields deal with concepts more or less mapped to the real world. Physics is about real world, more or less (right until you get to quants, then the level of abstraction rises dramatically). Same goes for biology, and even computer science in general. There, you can rely on words, which usually convey meaning.
In math, you can't rely on words. You'll never understand even relatively simple things like complex analysis or Fourier transform just by reading about it -- words are never enough to transfer the knowledge to you. You need to play with it, solve actual problems, understand in practice how various "moving parts" are related to each other, and then accept the naming convention (which is almost an afterthought, born as a mean of reference, not as a way to describe things). Therefore, relentless practice and solving abstract problems (a lot of them) is the only way to teach (or learn) mathematical concepts.
Some teachers want to make math more accessible with bringing it "down to earth", mapping mathematical concepts to more concrete problems. It is theoretically possible, and I was a supporter of this approach until very recently. However, math just doesn't work this way. Math is pure abstraction; linking the abstractions to earthly affairs too early shuts down mathematical thinking (creates biases that prevents applying mathematical insights to other fields that are different from the one learned).
Mathematical abstractions cannot be transferred by words and formulas alone; they need to be internalized by practice and drilling.
Yes, and notation is key. If there is even just one symbol that the student can't understand, then the entire understanding of the paragraph, or even entire book, is at stake.
However, I do think that examples and applications help a great deal. Sometimes, a mere description of first principles doesn't do it for me, and then, when I see an actual example, I suddenly understand it and the theory along with it.
> The key to math education is practice. Even drilling, maybe.
Do you have any research to support that? Because from what I've seen we currently think that drilling does nothing to help with understanding, and that one of the problems people have with maths is applying the wrong technique to a problem because they don't understand the problem.
> You'll never understand even relatively simple things like complex analysis or Fourier transform just by reading about it
Very few 8 year olds are grappling with fourier transforms.
Whitehead and Russell famously took several hundred pages in Principia Mathematica to prove the validity of the proposition 1+1=2. What is it that kids are understanding that W&R took much pain over? Understanding equates to learning the rules and when and where they can be applied. That's where drilling comes in.
> Whitehead and Russell famously took several hundred pages in Principia Mathematica to prove the validity of the proposition 1+1=2. What is it that kids are understanding that W&R took much pain over?
W&R proved 1+1=2 in their axiomatic system. It wasn't done to prove once and for all that 1+1 is in fact 2, but to show that their system produced mathematical truths. The truth of 1+1=2 was already assumed and understood since they were kids, and that's why it was necessary that their system also produce it.
Of course PM was obliterated by Godel shortly after.
Godel did not destroy formalization. He showed that there would always be true things that cannot be proved. It's true that the axioms used by Whitehead and Russell are not the axioms usually used today, but PM is still an important book that has influenced much work through today and Beyond. You might find this website interesting, which formalizes axioms and then proves many things from them: http://us.metamath.org/index.html
It's interesting that you mention 8 year olds, because 8 year olds learning basic maths need to go through the same process.
First kids need to be drilled to pick up the basics such as tables of multiplication, the understanding comes afterwards (for some). This is how maths has been taught for ages, but not so long ago common perception under teachers changed: people started thinking, wouldn't it be better to teach kids the ideas and the reasons why basic math rules work, even better, let them discover those rules them by themselves, and get rid of those mind numbing drills?
After a couple of decades, the common perception has switched again: the answer is 'no' [1]. To be able to grasp the ideas behind math rules, your brains have to get familiar with the basic concepts first. And that just takes time and practice.
When those basic concepts are engraved in your brain, only then are you able to start playing with them and build higher abstractions.
Learning complex mathematics is not any different.
Did you even read the article? Because what you say is a direct contradiction of the article. The article even explains why the reforms didn't work in the US(TL;DR; they weren't properly implemented) while they did work elsewhere, like Japan which is the example the article uses. Given that their statements are backed by evidence while yours are not, I think I'd rather believe them.
As an aside, no one is saying that there is no place for practice in math teaching. What they're saying is that it needs to be balanced with time spent actually learning the concepts and exploring on your own rather than just practicing and drilling. From the article:
"Similarly, 96 percent of American students’ work fell into the category of “practice,” while Japanese students spent only 41 percent of their time practicing"
Those students still spend just under half their time practicing, which is still significant, just not the overwhelming majority of their time as in the case of American schools.
i don't agree completely. I think math also has a point where, until you you pass it, can make the connection to real-life. Adding, Subtracting, simple equations. But then you have to let go and resist the urge to "justify" the material by providing real-life assignments.
But i agree that the key is practice. At university, where the pace picks up dramatically, you figure out that going to the lecture is not as important as finishing the assignments. Always finish the assignments.
This very much reminds me of learning to play a musical instrument. You don't need to learn a bunch of music theory to get better. You just need to practice a lot. The theory does no good if you can't actually play, and it makes much more sense to learn after you can play since you know what it already sounds like when reading about it.
The same goes for youth sports. You go to practice for a couple of hours a couple of times a week, and do the same thing or very close variations of it over and over and over.
Eventually you get pretty good, but there is just no substitute for good repetitions with a good coach correcting your form and technique.
Then you scrimmage where you put those into practice perhaps at a slower pace and get a chance to be creative, and then you put them into a game where things happen full speed. And after the game you evaluate what went well and what didn't.
It baffles me that anyone thinks mathematical learning (or any other) can be done any other way. Reps matter.
I always considered myself bad at math in school. For me the drilling as the problem. I have trouble focusing on arbitrary tasks. After I left high school and started to teach myself programming I became much better at it. Having an actual problem to solve, instead of a worksheet created by a random number generator, is key.
Reminds me of this interview by Knuth [1], where he says, because of his initial insecurity that he was not good enough, he started by out doing twice as many math exercises as his classmates.
He at first put in a lot of effort, but eventually he was just coasting through ahead of his classmates. He attributes it to the drilling he put in initially.
But doing twice as many problems is not the same as drilling. When people say "drilling" in American education, they generally mean doing simple and mechanical problems for speed. For instance, the multiplication drills at some schools involve playing a voice on a CD or mp4 that speaks multiplication problems at a certain rate.
Now, doing many exercises is a different thing! one that I encourage! In most textbooks, exercises range in difficulty and approach. I write math problem sets for students, and I build them to go from mechanical to sophisticated. The last few problems involve proofs, applications, links between different areas of mathematics and statistics.
Students in my class who just do probability drills will not be able to do the last problem on the problem set. They don't have any practice at problem-solving in this context. Students who do all the exercises, and more, can do amazing things.
I think they key is being able to jump between the narrative ( story ) domain and the formal symbols domain. At first, it is learning a language. Later, it is learning how to learn that language dynamically.
I agree with your statements about needing to 'play with' mathematics to properly understand it. I don't think that's a good explanation for why Americans suck at it, though, because the same goes for software development and it's really hard to argue that you guys suck at software.
The key is "you need to play with it". The way to get this to work in practice with elementary mathematics is something like Guesstimation. To my knowledge, this is not taught in most math classes. And the reason is simple: it is messy.
They might try something that looks similar with word problems, but that is a far cry from the majesty of investigating the world with simple arithmetic. The problems they come up are contrived and students are no fools. Problems must come from within, not be handed to people.
Drilling vs understanding is not an either..or. Most learning starts with messing around, trying to figure something out. It is slow and will be painful if the interest is not there. At some point, understanding is sufficient with a foundation in place and then drilling comes in to get a mastery, but what "drilling" is and needs to be will vary from person to person. Too little, and the speed fails to materialize inhibiting understanding of higher level concepts. Too much and it gets boring shutting off critical interest.
All of this suggests that teaching mathematics in a one-size-fits-all is difficult. Also, imagine trying to grade 120 guesstimations for weeks on questions the students come up with. This is the essential difficulty of our current teaching mindset, namely the single authoritarian figure blessing or cursing the work of others.
The key to true learning is excitement and motivation and little external judgement. Interested people will largely self-correct with a few subtle pointers here and there.
Even at the level of Fourier transforms, most students probably fail to understand the problem. So how can it possibly make sense? Math is about working around hard problems, but the teaching of mathematics leads one to believe that there are simple algorithms to apply and we are done.
A lower level example of this is Newton's method. Solving f(x) = 0 in their experience (quadratic formula) is easy and graphing shows the zeros anyway. What is the problem it solves? But what if you just give them a graphical piece of the curve and ask where they think a zero is? Then the problem becomes clear and the solution can make sense. Then it gets translated into steps (draw a line that fits well, find its root, find out what it looks like around there, repeat). And then to master it, one can do a number of drills.
Good luck doing this, however, with people who do not care one iota about it and live in a culture where the majority of Americans have disdain for math (probably born out of a mixture of shame of how poorly they did in math class and bitterness that this most wonderfully human tool for exploring the world has been denied them).
I agree with you completely. Has anyone really good practice books and other literature to practice and drill? That's something I find really hard to find.
I disagree with this. I remember when I was in primary school and I was confused about how to multiply decimals. At that level I didn't even know what it would mean to multiply a decimal: how can you have 5, 0.2 times? I asked the teacher and instead of explaining the concept properly, i.e. 0.5 is half etc., I was just told to multiply the other way around, by adding up 0.2, 5 times - and do drills with that.
It's this kind of teaching that makes good robots and terrible mathematicians. (Thankfully I'm neither the former not the latter.)
Well properly understanding why certain number systems work (fractions, negative numbers) requires some understanding of set theory, and the great mathematicians of the past struggled with a lot of these concepts we take for granted (negative numbers, complex numbers). It's unrealistic to expect high school teachers to understand these foundations.
A.) For the obvious not everyone learns the same way
B.)Drills and practice are important, but being able to actually apply your math skills is also important. I remember in high school when there were math word problems so many people HATED them. These were people in honors/college level classes and they struggled to actually apply the math to situations. Find how fast the train is moving, how long the ladder is, and Geometry proofs all made many of them get confused. They didn't understand the point of the math they knew.
Personally, I got SO bored with drills/memorization. And personally for me, that if that's the -only- method of learning used it's awful for long term retention. The math skills that I've retained the longest were taught in several different ways-- a mix of drills/practice, visualization, mixing it with other types of problems so that I could see how the puzzle pieces fit together, etc. I understand the point of what I'm doing, it's not just some abstracted concept that I happened to write 50 times.
I agree that past a certain point you'll have to accept abstraction, but especially in early years up until probably Algebra, it's a useful skill to be able to use math in the real world. Seriously, I've met so many people who can't calculate a tip, understand fractions enough for simple cooking, do their budgeting, or begin to understand economics and taxes enough to be even decently informed citizens.
That's because you're lazy. I know you really genuinely think you aren't, but you are... I'm dealing with this same mindset in my 13-year-old son right now. That's like saying you want to get in shape, but the repetitious method of lifting weights up and down is just SO boring.
That's like saying I'm lazy because I prefer sometimes doing a dance class over just running on the treadmill.
There's more than one way to do things. That said, drills are important. You do have to practice at some point. But if it's the -only- thing that's used, as it is in some classes I've taken, it's boring and harder to retain the information. And in my personal experience, I've seen that sticking with -only- that method can sometimes be counterproductive. It's avoiding the depth of understanding needed for actually applying the math.
In a way I don't disagree. It depends on the person though. For a person that struggles with the abstractions, drilling is necessary because it forces one to spend enough time with the subject until it "clicks."
This comment suggests you did not read the article. The article directly contradicts your assertions.
In talking about how Japanese teaching methods are better than American teaching methods, it states: "Similarly, 96 percent of American students’ work fell into the category of “practice,” while Japanese students spent only 41 percent of their time practicing."
Modern math classes tend to be very applied. I.e. the primary objective is to get the student to be able to solve real world problems with math. (A farmer sells 3 potatoes for a dollar. How much do 8 potatoes cost?). There seems to be some indication however, that abstract math is more accessible to some children, especially if they aren't supported by parents during their homework, etc. Because of this real-world to mathematical-world translation step. By exploring ways to make math more accessible, "friendly" and useful, teachers might actually make it harder for students to pick up math.
This is an interesting thought for me, because I tutored kids in math who were not doing good in school. I kind of ended up with a typical scheme to get them from "risk of failing the class" to Bs and occasionally As.
First I would introduce a a few techniques (equation solving in every case and the mathematical topic of their class - logarithms, binomial terms, etc. - also) and then give them very simple drill exercises. And a lot of them that we would solve together. I.e. simplifying exponentials $exp(3) * exp(5) = exp(8)$ etc. I always made sure that they were able to solve these drill exercises eventually, and they were all able to, because they picked up the scheme.
As a next step, I gave them the applied problems their teachers would ask them to solve, and made it into a translation problem. I.e. I explicitly told them that this was now just a translation. They could often identify with this because they self-identified often as language persons ("I like the literature class best", or "I like french class most"). I wouldn't ask them to solve the translation result right away. But they often just naturally did because it was not so different from the drill exercises.
Point is, I do not at all understand why I was needed for this. The teachers consistently failed their students in class, by not providing them with the very basic math skills and confidence that they needed to solve more complex problems.
> The teachers consistently failed their students in class, by not providing them with the very basic math skills and confidence that they needed to solve more complex problems.
Finding a good math teacher in America is difficult because the good math students rarely become teachers.
I've latched onto this as my explanation for why English students are worse with math (English counts numbers weirdly, but Asian languages do math within their language to perform counting):
Oh sure. The best education in the US is awesome, possibly the best anywhere. But the average is pretty sad, particularly compared to the rest of the first world.
Guy I went thru undergrad with went to a high school where the 10th grade algebra class requires naming the principle to be applied for each step in a problem, as if it were a proof.
This is painful. But he'd learned algebra properly. I hadn't. His math grades were much better than mine for a long time.
256 comments
[ 2.6 ms ] story [ 588 ms ] threadSo two questions:
1) Why doesn't the preeminence of the US math knowledge appear to seep into the primary and secondary school education?
2) If the primary and secondary education in the ROW produces such a high level of capability relative to the US population, why aren't their universities better represented in the rankings?
[1] http://www.usnews.com/education/best-global-universities/mat...
[2] http://www.topuniversities.com/university-rankings/universit...
There are a lot of reasons why the US does well in Universities and poorer up until then relative to the rest of the world:
1 - In much of the world, the school you get into matters more than what you did there. (The lowest University of Tokyo graduate is considered higher than the top grad of any other school - so getting in there is the hard part)
2 - In the US we invest more in higher ed than K-12 relative to the rest of the world on a per-pupil basis - especially at the top schools. (Look at the Harvard or Yale endowment on a per-student basis)
3 - In the US, college professors are at the top of their peer group academically. It's a mixed bad in K-12.
Your typo is very accurate :-)
I would say that our problem in K-12 is a self-perpetuating one. The teachers were taught math badly and so never really learned it (and learned to hate it into the bargain). So then they teach it badly.
I'm not sure there's any solution except for tuning the students in to Khan Academy and suchlike programs.
I think parents have an enormous role to play in the effectiveness of K-12 education. If they are not very much involved (e.g., enforcing, participating, encouraging) then school isn't valued or prioritized, nor will the average student see how fun many subjects can be to learn (assuming the teacher may not be effective at this).
> were taught math badly
> teach it badly
Just an aside, wouldn't this be expressed as "taught poorly"?
Blame the English teacher who thunk it badly. :-)
> Just an aside, wouldn't this be expressed as "taught poorly"?
I don't think there's anything wrong with "badly" here, and at least one dictionary [0] seems to support me; see esp. sense 2. But "poorly" works just as well.
[0] http://www.dictionary.com/browse/badly
Agreed, but that doesn't mean parents are not able to be encouraging. Their support, I believe, is more valuable than their level of absolute education on the common subjects.
If the parent can get the child to explain it, that also helps. AKA rubber duck debugging.
Yes, this is what I meant to convey.
Ultimately it's on the parents.
I'm a college math professor, and I've talked to people who've taught the required "math for elementary ed majors" class.
From everything I've heard, it's bleakly depressing. The students (i.e., the future teachers) show little aptitude, curiosity, or work ethic. They just want to be shown algorithms that will always lead them to the correct answer.
I haven't taught such a course myself. I hope what I've heard is exaggerated. Liking kids is well and good, but if you're going to be a teacher then you should also like learning. My own elementary school teachers did, and everyone deserves an education as good as the one I got.
Your elementary teacher may not be a math major because they are expected to teach all subjects. There is also probably some bias against deep content knowledge because "it's elementary school after all".
Secondary Education degrees are subject specific, so secondary education math students would, in fact, be mathematics majors.
[1] https://nces.ed.gov/fastfacts/display.asp?id=58
In general they won't be, and I wouldn't expect them to be. The subject matter of these classes is usually much simpler than freshman calculus.
It's not math per se that I care about here. If, for example, these future elementary teachers disliked reading and displayed the same attitude towards being asked to write critically about a novel, then I would consider that equally disqualifying.
Outside of people who explicitly want to teach math, the math skills of most K-8 folks I've seen is abysmal.
That's why it's such a weird contrast for us egalitarian European schmucks when we get to know US colleagues in academia, who are extremely professional and well educated, while watching the daily news makes us think that the majority of US citizens must be mentally retarded and suffers from chronic lead poisoning.
You probably think we only pay attention to news from the US but the Brexit vote is merely the most recent example to show that Europeans can be just as manipulable and unintelligent as Americans.
That's because it's a very easy job. Once you've filtered out 95% of students, the rest would thrive if you threw them in a closet with a book and a flashlight.
I think you just described my kid, who would be happiest if we did that with him.
But that hope is in vain :-(
For example, maybe if you are a very good mathematician in the US you can get high-paying jobs for intelligence agencies, data analytics, that sort of thing. But if you are a mathematical genius in another country, maybe they don't have the same job opportunities so you have to go into math teaching.
That would cause the US to have very good mathematicians and at the same time terrible math teachers.
Because preeminence of top tier institutions (which are kind of global centres anyhow) - has absolutely nothing to do with teaching math to the commons.
Here's a hint:
+++ Americans don't suck at Math +++
There's a very un-PC but very large elephant in the room that people won't discuss.
+ European American and Asian American 'testing scores' are actually pretty good - and have been holding steady for a very long time. (Asians do a little better). Nothing has changed.
+ Latino American and African Americans fare poorly, but having been getting better since we've been measuring by standards (i.e. 1950's-1970's).
Here's the trick:
+ European Americans actually do better than Europeans - on average. + Asian Americans to better than Asians - on average. + Latino Americans do better than Central/South American Latinos + African Americans do better than Africans.
The key correlating factor here is 'ethnicity'. 'Ethnicity' is the broad, generalist predictor of educational outcomes. This definitely not 'race' and it's not even 'IQ' (those things are plausible but controversial) - it's a series of behaviours, social norms, examples, attitudes towards work, success, access to services, social networks, mentors, role models, etc. etc. etc..
Educational outcomes (and crime stats, income stats) break down along ethnic lines. In a manner of speaking - America can be thought of as 'four nations' - White, Black, Asian and Latino. Obviously - it's very crude and generalist, and policy based on this would probably be racist - nevertheless - you pretty much have to look at the data given this.
In the end: American test score results have more to do with the changing ethnic composition of the American population than they do anything else. Again: White people and Asians in America have performed consistently he same for decades. Teaching methods haven't changed much, students habits haven't changed much - so the outcome is naturally consistent.
More economic prosperity, access to services and different attitudes + deeper integration have meant Latino A. and African A.'s are doing a little bit better - but because there are so many more Americans of those groups - particularly Latino Americans - it changes the outcome of the 'average american test score'.
Analyzing educational results does not make sense until you break it down along ethnic lines. Once you do - it becomes crystal clear. It's the absolute #1 most important thing about the educational data that turns 'paradox' about educational investment (teaching has remained largely the same) and outcomes into 'perfect sense'.
Unfortunately, it's so sensitive few will want to talk about it - for fear that the general public equates educational outcomes to 'intelligence' and try to strongly correlate ethnicity + race to this, which would be fodder for racist/KKK types, which wouldn't really help the overall social situation in America.
Anyhow - America is actually doing pretty well overall.
I've always heard this explained as a sort of "selection bias". Since immigration to the US (particularly for university education) is often seen as desirable, the people who manage to pull it off tend to be above the mean. Do you feel that explanation rings false?
I suggest your theory is probably very true for Asian Americans - the one's who came here are 'la creme do la creme'.
But not for Europeans. Europeans that came here were the poorest, the least educated, criminals, fringe religious types etc.. Europe was 100x more civilized than America during early history - why would anyone with any social status leave London in 1800 - to go and live a million miles away, a very, very hard, back-breaking life?
Well - those tenant farmers who could get cheap land and get out from under the thumb of their landlords etc..
But it's a good point.
I think this is an explanation that could only be come up with by the descendants of those who have emigrated.
Thinking here in Scotland, the people who emigrated were not necessarily the most able or genetically superior somehow. Often, they were simply the most desperate. People who were cleared off their farms by landowners, people who had no other options available to them but to roll the dice and go abroad to Canada or Australia or the USA.
Most folk don't want to emigrate, certainly not in the 19th century. It is a last resort that you do if you are out of options. But perhaps the most capable and able have other options to take advantage of?
"Collective emigration is, therefore, the removal of a diseased and damaged part of our population. It is a relief to the rest of the population to be rid of this part."
https://en.wikipedia.org/wiki/Highland_Clearances
Europeans in those statistics include all ethnicities living in Europe - e.g. about 1/3 of students in Germany are not ethnically German.
https://en.wikipedia.org/wiki/Academic_achievement_among_dif...
> Unfortunately, it's so sensitive few will want to talk about it - for fear that the general public equates educational outcomes to 'intelligence' and try to strongly correlate ethnicity + race to this, which would be fodder for racist/KKK types, which wouldn't really help the overall social situation in America.
Fortunately we have you.
If he did go looking for statistics, he would find that the effect he describes is better explained by the expanding lower class in america, regardless of race.
http://blogs.wsj.com/economics/2014/10/07/sat-scores-and-inc...
It's widely known that Asian American and European American students outscore Latino and African American students. Nobody would contend that.
For you to indicate that there is a very specific issue i.e. 'poverty' that is the 'driving factor' is wildly speculative and outlandish - a fetish of your ideology - and and the 'impetus' is upon you to prove it one way or another.
My presumption of 'ethnicity' is much better because it would include 'average income' as a natural outcome of an ethnic groups situation regarding education, racism, poverty etc.. And FYI - 'average income' would be a function of ethnic attitudes overall.
Nobody would deny that 'income' has some effect on test scores - but my friend - you have to grasp the obvious reality that cultures who don't value 'education' will have significantly lower incomes, and more likely to be in poverty. 'Poverty' is the result of bad education, as much as the cause.
If you've travelled the world then you know how crazy different people are.
Many of those differences will yield different attitudes to school, to homework, to respect for authority, to criminality, to family etc.. All of those things will affect educational and life outcomes.
By the way - here is PISA scoring across national groups (breakdown for USA students):
http://web.archive.org/web/20111211103448im_/http://www.vdar...
Is it 'racist' to say that Sub Saharan Africa is a very, very different place than North Africa / Maghreb? They are like Venus and Mars they are so different. And it's not just 'income' or 'the weather'. They are entirely different cultures.
The world is very diverse. Different values = different behaviours = different outcomes.
It's not ok to comment like this here.
Several of your recent posts have crossed the line into incivility. Addressing people as "buddy", telling them about the "fetish of [their] ideology", etc.: all this is patronizing and rude. Please don't do any more of it on Hacker News.
If you're going to engage in difficult topics, you have a responsibility to do so with extra respect, not less.
My comments were reasonable and thoughtful - if borderline controversial - and yet it was directly ad-hominem to call me 'racist' and the comments 'wildly speculative'.
I call you 'Buddy'? Someone (you?) called me 'racist'. Which is more offensive in reasonable discussion?
'Buddy' is a colloquialism, it's not offensive or attacking.
At the same time you offer no insight, rebuttal or points of your own. You're attacking people and chiding them 'not to speak anymore' - while I have no problem with any of the topical discussion.
Your attempt to shut down the discussion while not offering countervailing arguments to is exactly the antithesis of HN, in my opinion.
So if you have some relevant comments about education and potentially relevant factors, however speculative - do so. I'd suggest adding to the discussion instead of trolling and attacking.
This is not a 'safe space' wherein normal ideas are deemed unspeakable because someone, somewhere, might feel uncomfortable.
There is no need to bring up race or ethnicity to describe this effect of scores getting worse. Everything you attribute to being Black or Hispanic can more simply and more accurately be described by being poor. SAT score is most directly tied to family income, not race. http://blogs.wsj.com/economics/2014/10/07/sat-scores-and-inc...
Instead of worse scores being the result of changing ethnic composition, they are the result of changing income distribution, and the expansion of the lower class.
+ This is not true +
Even when normalized for income - there are still large variations in outcomes.
Again, I'm not really saying it's 'race' or 'genetic' or 'IQ' so be careful with your disdain :).
My friend - do you know the stereotype of the 'Asian who works hard in school because his parents compel him to' - well, it's not just a stereotype, it's true. Some cultures value education more than others. That will show up in the results.
Asian Americans are 600% over-represented in tech (i.e. 5% of population but usually 30% of tech companies). That's not some fluke - and it's not because 'Asians are super rich' - it's obviously a cultural preference. They are choosing STEM and tech for whatever reasons.
It doesn't make some people better or worse than others as human beings, it just means some variations in outcomes whenever you measure something.
This this is what actual 'diversity' really means, though ironically I think most 'pro diversity' advocates really intend to have a hyper-egalitarian situation wherein there is little diversity beyond skin tone :)
The "immigrant advantage" fades in three generations, at which point achievement by children reflects the US average. This is the influence, if you like, of American "ethnicity", which is often quite anti-intellectual.
The intro here is an interesting read: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3442927/
It's a fallacy. US is a large country, so those talented students concentrate in fewer universities. In Europe for instance, due to language and culture barriers, talented students from Czech Republic do not very often go to Cambridge. You need to look at mean or median if you want accurate assessment.
The European Allied soldiers were so disciplined that they just kept climbing up the ladders and getting killed one by one, following their orders to their deaths. The Americans saw this and said, "fuck that, I'm not climbing up there."
I think most Americans are pragmatic and they won't do something unless it makes sense. And to be honest, most people don't need to study math. Or at least it's not obvious that they do. I think most of the math professors I've talked to would agree. They view math, as it's taught in core curricula, more as an art than as having vocational value.
> The European Allied soldiers were so disciplined that they just kept climbing up the ladders and getting killed one by one, following their orders to their deaths. The Americans saw this and said, "fuck that, I'm not climbing up there."
Wow, that anecdote explains American supremacy better than anything to date. /s
Without a citation I'm going to have to call bull-shit on that one, I'm just trying to imagine their CO standing there with an ever mounting heap of corpses at the bottom of the ladder and not once thinking 'this doesn't seem to work'.
Some googling does not turn up any evidence for your story either.
war is an exercise in stupidity to begin with, it shouldn't be surprising there are pockets of even worse. But to claim that structurally Americans refused to get out of the trenches in a certain way in order not to get killed whereas docile Europeans were led like lambs to the slaughter is not something I've found in any history of World War I (or II for that matter).
Both wars had extremely heavy casualties on both sides, and in both wars there were quite a few instances of CO's treating their men like disposables. The Christmas Truce is a beautiful story about such behavior. Even so, both sides were desperately trying to win the war and the rule would have been to not take action hastening the demise of the men on one's own side.
I've yet to come across any substantiation of the anecdote related above, if it was structurally true you'd think it would be more than a mere anecdote.
It would if it were true.
> I'm not even American
That's immaterial.
> and I never claimed that the anecdote is true.
Well, you didn't claim that it wasn't true either, but the whole thing hinges on whether or not the anecdote is true so if you bring it up I'm going to assume that you at least believe it to be true and that the conclusion is supported by the anecdote.
If we're all just going to make stuff up to prove some point then it becomes very hard to reach conclusions.
The final assault on the remains of the entrenched opposition were without exception extremely bloody and the side that would take the others trench never did so without significant losses.
You can see this especially strongly in graduate programs, where US citizens can often get a sort of "affirmative action" because they are the only ones eligible for NSF student fellowships (incentivizing universities to admit them). There's just very little domestic interest in math and physics, despite both feeding into rather favorable job markets (as long as you aren't dead set on being in academia long-term).
As for your second question, it's an extension of US universities' preeminent standing in most fields (on average, there are of course exceptions). I don't think there are really any math specific effects going on there.
Oh yeah? What jobs are there for a physics PhD to do physics in that isn't academia? Government research lab? Military aerospace contractor?
As long as you are willing to go into industry, you're looking at a 90-100k starting salary[1]. That's pretty damn good considering that in the US nobody pays for a physics PhD; the average TA stipend is around $30k with tuition and medical insurance covered. You also have a decent chance of getting a research assistanship to provide the stipend once you start working seriously with an advisor.
The high number of physics PhDs getting postdocs off the bat is driven by people who want to go into academia/research, not by a lack of industry options. Admitidely if you're dead set on going into research (especially right off the bat) you better be ready to play the same low-paid tenure lottery as everybody else, but if you're just looking for a good career a physics PhD is a pretty good bet.
[1]: https://www.aip.org/sites/default/files/statistics/employmen...
If getting an art degree meant you could try and become an artist (with the same astronomically low chances of success you see today) and then immediately fall back on a solid career if you fail, don't you think it would represent a pretty good option? If you enjoy physics or math, you already live in a world with that possibility.
The bad effects of such societal level bias against math can be seen in public spending in schools also. Enormous amounts of money are spent on football coaches/teams[2] in schools whereas not enough money for math teachers/students.
The society seems to equate success only with one's ability to earn money. Then you can see that many such popular public figures "getting successful" without math as they can be seen to earn large amounts of money.
This is sad as such people even if are successful on money front, cannot understand any of the complex aspects of modern societal/business/political structures (be it, things like privacy issues, or many public policy issues like taxes) and of course cannot understand any of the modern technological/scientific discussions (be it discussions regarding global warming or genetically modified food or statistical significance of some tests).
[1] https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1968613
[2] http://www.theatlantic.com/magazine/archive/2013/10/the-case...
I am generally not a fan of speaking in terms of "privilege" and "identity", but this is precisely an example of "developed Western country privilege". The attitude goes, "See, I can still be successful/well-off without having to hunch down and study math like those third-world FOB immigrants do." It comes down to status signalling: not having to do math becomes a status signal. It also appears it is mostly this attitude and not the widely blamed "bro culture" that is the major reason why women (in the West, because this didn't happen elsewhere) abandoned CS sometime in the late 80s/90s.
In the 1980's, consumer level computers were being marketed as "boy toys" and american culture internalized the idea.
That is absolutely not true of anything the average non-math-major has ever been exposed to under the heading "math."
It's rote symbol manipulation requiring diligence, practice, and attention to detail. American K12 math education asks for fast and reliable algorithm execution, not insight.
Only math majors and attendees of a few exceptional private high schools will ever seriously engage with proofs.
Relative to the US population, the primary and secondary education do not produce a high level of capability.
Looking at the fields medals per capita[1] you can see that the US doesn't have as much as the UK, Russia, or France. You can also see that the university with Fields medals recipients [2] for more details, and indeed you can see that of the mathematicians associated with Princeton are not Americans.
As a specific example, in [1] you can see that France is generating more than 4 times more fields medalists per capita than the U.S. Why is that?
It probably has to do with the more rigorous Math education. Look for instance at this translated Math final[3] exam for French high school students. This is for the "Literary" students, those who focus more with the worst level of Math. You can see examples of the "Scientific" math test here [4]. It's in French, but it's Math, just by looking at the symbols it's possible to understand. It has some differential equations solving, probabilities, geometry with vectors/planes in 3D spaces, Series analysis, etc. Integrals and derivatives are also part of the program. In Physics these concepts are applied to calculate velocity/speed/radioactive decay etc.
[1] http://stats.areppim.com/stats/stats_fieldsxcapita.htm
[2] http://mathworld.wolfram.com/FieldsMedal.html
[3] https://gfbrandenburg.wordpress.com/2011/06/19/a-look-at-one...
[4] http://www.letudiant.fr/bac/bac-s/corriges-et-sujets-du-bac-...
The best American student mathematicians are the equal of anywhere in the world. But when people talk about poor math performance in the US, they're talking about the average—average math performance is terrible. This is because the U.S., more than any other advanced country, tolerates a high level of poverty and economic inequality. This inequality is reproduced within the educational system, and brings down our averages.
The average is lower not because the U.S. has lower performance across the board, but because so many more students lack the most basic numeracy. Too many students are going to school hungry, come from families torn apart by joblessness, abuse, and addiction, or where the single breadwinner has to work 80 hours/week in order to achieve a higher level of poverty, and simply doesn't have time to help the kids with their school work or ensure they are disciplined students, much less ferry them to all sorts of enrichment activities.
The US does have the richest mathematics knowledge in the world because the US can buy it.
Timber sometimes gets called 2x4 even though it's 50x100, but that seems less common now.
Oddly just about everyone still uses feet and inches for their height. I hear kilos more and more often for weight, I don't think I've ever heard cm for height in the UK.
Returned milk is (you know, in dem bottles). But milk you buy in the store is 1L, 2L, etc.
> Oddly just about everyone still uses feet and inches for their height.
True. Probably because of the recognition of the 6' centering measurement. People know that if you're 6', that's kind of the cut-off for what is referred to as 'tall enough'. Over that, you're generally 'tall'. Under that, and you're average height.
Only time I see litres is some of the niche brands (like Cravendale and branded organic) in bigger supermarkets, or 500ml in corner shops and some garages. So they can charge more. Presumably people compare 2l price with 2.3l / 4 pint instead of £/l.
Sure beer is served by the pint, but no one's ever going to ask for "10 ounces" when they want a half.
Pretty much the only conversions people ever do are between inches and feet when talking about height.
It just doesn't add up.
I wish in hindsight that instead of taking endless years of calculus there was a high school version of real analysis, and exercise problems rooted in real-life to motivate the learning.
Leave the program execution to the computers, give the children critical thinking skills.
Most people "in the trenches" writing tools that would make use of the fundamentals in their field (either in the physical sciences or engineering disciplines) tend to agree that linear algebra has disproportionately little representation compared to calculus. To be fair, that is somewhat of a downstream problem from mathematics departments, and speaks more about the possibility of stagnation in the undergraduate science and engineering curricula.
fair caveat though: my PhD is in numerical linear algebra...
I think you have to contextualize mathematical in the broader problem with american public schools: they're awful, awful places for many students including me. I ended up studying higher mathematics in college and still study on my own for my own pleasure (and I get to apply some really high powered ideas to programming once in a while which grants a satisfaction that lingers). I think my math education could have been advanced 5 or even 10 years if public school weren't such a soul-crushing stultifying nightmare of procedure and compliance.
Mind sharing what some of these ideas are?
This is all described and implemented here: https://hackage.haskell.org/package/kan-extensions-5.0.1/doc...
In particular, the type:
Let's unpack this. The following type represents a state machine whose nodes are labeled with a continuation demanding an `a` and executing an effect: The forall quantification in CoT means that it does not care what the result value is. It will take a state machine, manipulate it for a bit, produce an `a`, pick a continuation from the state machine and execute it.That's a lot going on! I don't have time or space to explain how this is actually useful but here's a handy approximation:
- `w` is an interface
- `w s` is a model labeled with a value of `s` for each state it can occupy. These labels are a sort of view.
- `CoT w m a` is a controller that can manipulate a model and compute in some effectful context.
High powered MVC. The upshot is that because we're leaning on Functors, Monads, and Comonads there are extremely well behaved and natural composition operations. For example, `fmap` allows us to change the view of a model by applying a pure function. The fact that `CoT` is a monad means that we can run controllers in serial, as well as combine our controller languages in sensible ways. The fact that `w` is a comonad means we can reason about the behavior of models equationally. This lets us transform, compose, and compare models with mathematical precision.
This is a project I only seldom work on in my spare time. I have already applied it to writing GUI's and game logic to good effect but I'm still exploring the design space regarding reactivity and temporal behavior. This has led me to reading the mathematical literature pertaining to products of comonads/sums of monads (they're dual).
Looking at my kids maths lessons, especially in late Junior, so much effort was spent to actually hide the maths that I wonder they learnt anything!
Was always my favourite subject, but that was despite school rather than because of.
Everyone? I love math.
Especially word problems -- most of them felt like they were written for students wearing intellectual blinders... if you had any modicum of relevant knowledge outside of the lesson oftentimes word problems were impossible to solve
Don't the countries that rank better use the rote method even more?
>Though lesson study is pervasive in elementary and middle school, it is less so in high school, where the emphasis is on cramming for college entrance exams
I think the implication is that kids are supposed to be taught to play with math in addition to the rote memorization, not that rote memorization is evil.
Many of the more "fun" math techniques rely on knowledge acquired through rote memorization, just like dynamic programming uses the results of inefficient calculations to speed up subsequent calculations.
> Many of the more "fun" math techniques rely on knowledge acquired through rote memorization
I wasn't aware of this. I have a nine-year-old. Are there any examples of this?
For example, take a look at this lesson: http://www.homeschoolmath.net/teaching/md/distributive.php
Ignore the stupid rectangles, and ignore the part where it says "They actually use the distributive property, but we do not need to explain that to 4th grade students." (seriously, wtf)
A child who practices lots of sums, such that she knows how to add 420 and 56 without counting with their fingers or writing things down, will be able to learn this kind of multiplication with ease. Once they learn that and practice a lot, they will be able to generalize the method to multiply any two two-digit numbers.
Memorizing things is important because if you have to count with your fingers to calculate 3x3, you will never be able to calculate, for example, 23 squared in your head. But if you know the times tables your thought process might look like this:
20x20 = 400
23x20 = 400 + 60
23x23 = 460 + 23x3 = 460 + 69 = 529
(23 times 3 is "the hard part" where most kids who know all the theory (distributive property) but are out of practice (not enough rote memorization) will lose track)
A more advanced example is what engineers used to do a lot before they had calculators: They memorized log tables, so when they wanted to multiply big numbers they just added their logs together, because log(a x b) = log(a) + log(b).
An unrelated example is converting miles to kilometers. The official relationship is that a mile equals 1.609 kilometers, an ugly number that doesn't work well with mental arithmetic. But that number is kinda close to the golden ratio, don't you think? So if you are the kind of weirdo who memorizes the Fibonacci sequence, you can quickly calculate that 13 miles should be about 21 kilometers (and 21 miles is about 34km, and so on), because the ratio between a Fibonacci number and the next approaches phi.
It's not about "fun", it's about experimentation. Free-form intellectual play.
But students are never told why they should care about abstractions in the first place, which is unfortunate. Many high schools simply refuse to speak the students' language.
I suspect this will change in this century. IMO, if a high school really wanted to be progressive, they would totally reform their math curriculum to include more exposure to applied math and computer sciences. Young folks should be using math to build their own Instagram or Minecraft clones that they can deploy to their devices that VERY DAY - using the concepts they've been introduced to in mathematics.
My high school math teacher majored in math and then got a certificate in teaching. She wasn't a teacher who was told to teach math among other classes. She knew all the advanced stuff (beyond what a high school curriculum required), she was excited about it, she could explain things in various ways, give analogies, was available after class to ask questions and so on. And that is post-collapse Soviet Union full of corruption, poverty and other crap like that. Surely if we can spend trillions of dollars on F-35 we can get us some good math teachers...?
In this country I see a large disconnect between words "Oh kids are so very very important, they are our future, they can't play outside too far because they will be abducted and we care so much for them" and deeds: I see large classrooms, not enough teachers, teacher are underpaid, not interested in math. Funding comes from local property taxes so rich neighborhood get more money, poor ones get less.
Another thing is I remember teachers were respected. Imagine how we react to someone saying "Oh they are doctor. And then everyone nods, right, they are very successful. Or lawyer, or works for Google and so on". Why isn't teaching like that? My mom tells me someone back home thought she was a teacher, because she at her age spoke some English. And she took it as a great complement. In this country you tell someone you thought they were a high-school teacher and they might get offended. Something is very wrong here...
As for making it more exciting -- initially I studies math by repetition and it worked pretty well for me. I think works because kids are amazing at memorizing stuff. Why not first take advantage of that? Starting them out with set theory sounds all cool on paper that is not how humans learn. Multiplication table, basic patterns, even operations are fine to memorize first. Later on it makes most sense to introduce proofs, word problems (I remember doing lots of word problems, our teachers were crazy for them) and so on.
I know people will throw stones at me for this, but the reason teachers are not the top 1% of the society is because functionally their job is a commodity. (We're talking about elementary/middle/high school teachers, not professors here).
It is not hard to find someone who knows high school math, for example. I'm not discounting the fact that there are sometimes really outstanding teachers, but the thing is, it's hard to objectively measure their performance since their teaching talent is not directly related to how well their students do. On the other hand, the "teachers" at universities are well respected since their talent is not only limited to how well they teach but the quality of their own research--the value is much easier to quantify.
Sure if you ask people, they respect teachers in the abstract sense that they're people doing a very tough job for very little money, but it's hardly a career people aspire to, and certainly most teachers I know will admit it was their second or third choice that they kind of fell into.
Teaching in Canada is along the lines of nursing or something: it's professional, requires a degree of competence and certification, there is social value. It's not exactly 'doctor' but it's respected for what it is, as it is in most nations.
I think it's because people assume American teachers are not paid much, but the one's I know are paid reasonably.
I envy you this experience. I still remember telling a math teacher in 8th or 9th grade that I wanted more information on why a formula worked, how it was originally derived, or any different way of looking at it to better understand it. She essentially said, "you don't need to worry about all that, just memorize it." It was intensely frustrating for me, and contributed to me losing interest in academic pursuits. I don't know how you can train teachers to be more open to this kind of thing, but it sure would help.
I used to think so too, but I don't think that is exactly the case. School districts usually span many neighborhoods, rich and poor. Within school districts, there are school zones (school boundaries), or mappings of residential addresses to assigned schools. While it's true that richer neighborhoods tend to have better schools, the property tax money is collected centrally at school district level, and then distributed across schools, both rich and poor. I haven't seen evidence that districts distribute more money to schools in richer neighborhoods.
If anyone is more knowledgeable on this, I would love to learn more.
I was very unprepared for the math part of it and it was hard. I tested into a remedial math level and when I looked at the CS requirement I knew that wouldn't be a good way to start. So I bought a used math book and spent a few weeks studying hard so I could get into a decent math placement so I could be where I needed for CS. This was just the start.
I had a long road with a lot of frustration but I made it. I made it because I never worked so hard at something up to that point in my life. There were times I thought it was hopeless but I just continued to do rote math problems, over and over again. And slowly the concepts started to sink in more and more. But there was always the frustration of missing the little details and forgetting a concept or not having enough experience honing a certain skill. But if I kept at it, over and over again, I would learn it and it suddenly became easy.
I was "bad at math". It would have been so easy to quit and try something else. But I knew I wanted to not only pass my tests but really understand the calculus and differentials, etc I had to do. And it was the best thing I ever did for myself.
Not only does working so hard at something prove you can work hard and achieve something but it shapes the way you see the world.
Math is hard. Some of us just have to work harder. People should realize that failing is normal and if you keep at it you will eventually get it. For some people it takes longer than others. I could never create math. But with enough time I believe I could eventually truly learn anything because of this experience.
I'm always disappointed when people say "some people just can't learn X".
It isn't that they cannot learn X, they just cannot learn it in the same time frame, and they shouldn't be expected to!
This is exactly the argument I use when seeing people criticizing or laugh at others over their profession or education. I also think that the time frame one needs to "learn X" greatly depends on what they already know, i.e everyone is born a genius; training is what makes the difference.
[...] and they shouldn't be expected to!. Hats off for this too. Probably the biggest flaw in most of the world's educational systems.
https://www.washingtonpost.com/local/education/majority-of-u...
I'm no fan of the American education system, having suffered through it a full 12 years, but I have to believe it's not the primary cause here. Math is hard, and near impossible if you're stressed. I excelled at math, despite relatively boring math curricula. Why? Because I wasn't stressed as a kid, my family was stable and did not suffer from any serious physical and mental illness, and one parent always made enough money so that the other could stay at home throughout my entire childhood. I had an enormous advantage, and most all of the kids I knew through advanced math classes and math competitions had a similarly charmed existence.
http://marginalrevolution.com/marginalrevolution/2015/01/no-...
edit:
[1] http://www.cps.edu/About_CPS/At-a-glance/Pages/Stats_and_fac...
[2] http://www.census.gov/quickfacts/table/PST045215/1714000
One of my friends graduated with an engineering degree from MIT. I once asked him if he could have traded that to be better at Football, would he? His response was that if there was even a remote chance he was good enough to play in the NFL he would have traded education for that chance.
Humans innately crave fame, and the lack of scarcity that is perceived to come with it. The USA has sadly, and unknowingly groomed that romanticism of fame, towards consumables (this might be the long term effects of Capitalism). Rather than grooming that romanticism of fame towards research/intelligence/creation (the renaissance).
Even local to the film industry, almost 100% of 10 year olds want to grow up to be Brad Pitt or Salma Hayek, not George Lucas or Stephen Spielberg[0]. Even fewer want to be the writer.
Even local to football, everyone knows the names of the player that caught that hail marry, got that big hit, recovered that fumble. Few know the names of the people who create those plays.
Only in recent years have the masses truly started to recognize creators/intelligence with the title famous (Gates, Jobs, Zuck). But again for the wrong reasons, i.e. for their money. Even artists Picasso, Beethoven (adored for generations) have only truly been appreciated by the aristocracy, i.e. the rich and "mathematically" acute.
Which way do you suspect the causal link lies? Are we first intelligent, therefore we appreciate the intelligent? Do we first appreciate the intelligent, therefore we become intelligent?
[0] I mean no disrespect to Brad Pitt or Salma Hayek, or actors in general. Good Actors need to be extremely intelligent, but this isn't why they are adored. I'd argue this part of them is even sadly shunned by the media and populace.
https://youtu.be/XnVcaHMsYqM
And yet at the same time we have these sorts of track records:
http://www.popsci.com/us-dominates-at-sending-stuff-to-mars
It's not just Mars or the moon, or decades ago, the US also is where most operating systems and software and CPUS are conceived of and designed. As well as countless other things that people who are so ignorant would not reasonably be expected to be able to do.
"Americans don't want their kids to have too much homework
this wasn't the case for me. i'm american. maybe you're wrong?
"Americans don't want their kids to have too much homework
this wasn't the case for me. i'm american. maybe you're wrong?
Yeah, so there's this as a reason. I definitely wasn't a good student until high school. I definitely had a bad teacher who gave me an excuse for YEARS to accept bad results in math. My mother still bitches about that guy to this day. I suspect I have a slight learning disorder that affected my ability to process numbers. I mix 6/9s. 7/9s when transcribing numbers. Made homework difficult. Once I got to the point where letters started replacing numbers in homework, I went from a C student to an A student in math and was competing for the top grade in my math classes. Once I learned that I could do the work, I did the work and excelled.
I suspect that this type of bias affected more girls than just me, and these discouraged girls affected US scores.
Other fields deal with concepts more or less mapped to the real world. Physics is about real world, more or less (right until you get to quants, then the level of abstraction rises dramatically). Same goes for biology, and even computer science in general. There, you can rely on words, which usually convey meaning.
In math, you can't rely on words. You'll never understand even relatively simple things like complex analysis or Fourier transform just by reading about it -- words are never enough to transfer the knowledge to you. You need to play with it, solve actual problems, understand in practice how various "moving parts" are related to each other, and then accept the naming convention (which is almost an afterthought, born as a mean of reference, not as a way to describe things). Therefore, relentless practice and solving abstract problems (a lot of them) is the only way to teach (or learn) mathematical concepts.
Some teachers want to make math more accessible with bringing it "down to earth", mapping mathematical concepts to more concrete problems. It is theoretically possible, and I was a supporter of this approach until very recently. However, math just doesn't work this way. Math is pure abstraction; linking the abstractions to earthly affairs too early shuts down mathematical thinking (creates biases that prevents applying mathematical insights to other fields that are different from the one learned).
Mathematical abstractions cannot be transferred by words and formulas alone; they need to be internalized by practice and drilling.
However, I do think that examples and applications help a great deal. Sometimes, a mere description of first principles doesn't do it for me, and then, when I see an actual example, I suddenly understand it and the theory along with it.
Do you have any research to support that? Because from what I've seen we currently think that drilling does nothing to help with understanding, and that one of the problems people have with maths is applying the wrong technique to a problem because they don't understand the problem.
> You'll never understand even relatively simple things like complex analysis or Fourier transform just by reading about it
Very few 8 year olds are grappling with fourier transforms.
W&R proved 1+1=2 in their axiomatic system. It wasn't done to prove once and for all that 1+1 is in fact 2, but to show that their system produced mathematical truths. The truth of 1+1=2 was already assumed and understood since they were kids, and that's why it was necessary that their system also produce it.
Of course PM was obliterated by Godel shortly after.
First kids need to be drilled to pick up the basics such as tables of multiplication, the understanding comes afterwards (for some). This is how maths has been taught for ages, but not so long ago common perception under teachers changed: people started thinking, wouldn't it be better to teach kids the ideas and the reasons why basic math rules work, even better, let them discover those rules them by themselves, and get rid of those mind numbing drills?
After a couple of decades, the common perception has switched again: the answer is 'no' [1]. To be able to grasp the ideas behind math rules, your brains have to get familiar with the basic concepts first. And that just takes time and practice. When those basic concepts are engraved in your brain, only then are you able to start playing with them and build higher abstractions.
Learning complex mathematics is not any different.
[1] http://educationbythenumbers.org/content/kumon-worksheet-sty...
As an aside, no one is saying that there is no place for practice in math teaching. What they're saying is that it needs to be balanced with time spent actually learning the concepts and exploring on your own rather than just practicing and drilling. From the article:
"Similarly, 96 percent of American students’ work fell into the category of “practice,” while Japanese students spent only 41 percent of their time practicing"
Those students still spend just under half their time practicing, which is still significant, just not the overwhelming majority of their time as in the case of American schools.
But i agree that the key is practice. At university, where the pace picks up dramatically, you figure out that going to the lecture is not as important as finishing the assignments. Always finish the assignments.
Eventually you get pretty good, but there is just no substitute for good repetitions with a good coach correcting your form and technique.
Then you scrimmage where you put those into practice perhaps at a slower pace and get a chance to be creative, and then you put them into a game where things happen full speed. And after the game you evaluate what went well and what didn't.
It baffles me that anyone thinks mathematical learning (or any other) can be done any other way. Reps matter.
He at first put in a lot of effort, but eventually he was just coasting through ahead of his classmates. He attributes it to the drilling he put in initially.
[1] http://www.webofstories.com/play/donald.knuth/9
Now, doing many exercises is a different thing! one that I encourage! In most textbooks, exercises range in difficulty and approach. I write math problem sets for students, and I build them to go from mechanical to sophisticated. The last few problems involve proofs, applications, links between different areas of mathematics and statistics.
Students in my class who just do probability drills will not be able to do the last problem on the problem set. They don't have any practice at problem-solving in this context. Students who do all the exercises, and more, can do amazing things.
regions where math levels are higher due to parents pushing for it.
"proper" americans are great in design, marketing and sales. mix the two systems and you get SV.
They might try something that looks similar with word problems, but that is a far cry from the majesty of investigating the world with simple arithmetic. The problems they come up are contrived and students are no fools. Problems must come from within, not be handed to people.
Drilling vs understanding is not an either..or. Most learning starts with messing around, trying to figure something out. It is slow and will be painful if the interest is not there. At some point, understanding is sufficient with a foundation in place and then drilling comes in to get a mastery, but what "drilling" is and needs to be will vary from person to person. Too little, and the speed fails to materialize inhibiting understanding of higher level concepts. Too much and it gets boring shutting off critical interest.
All of this suggests that teaching mathematics in a one-size-fits-all is difficult. Also, imagine trying to grade 120 guesstimations for weeks on questions the students come up with. This is the essential difficulty of our current teaching mindset, namely the single authoritarian figure blessing or cursing the work of others.
The key to true learning is excitement and motivation and little external judgement. Interested people will largely self-correct with a few subtle pointers here and there.
Even at the level of Fourier transforms, most students probably fail to understand the problem. So how can it possibly make sense? Math is about working around hard problems, but the teaching of mathematics leads one to believe that there are simple algorithms to apply and we are done.
A lower level example of this is Newton's method. Solving f(x) = 0 in their experience (quadratic formula) is easy and graphing shows the zeros anyway. What is the problem it solves? But what if you just give them a graphical piece of the curve and ask where they think a zero is? Then the problem becomes clear and the solution can make sense. Then it gets translated into steps (draw a line that fits well, find its root, find out what it looks like around there, repeat). And then to master it, one can do a number of drills.
Good luck doing this, however, with people who do not care one iota about it and live in a culture where the majority of Americans have disdain for math (probably born out of a mixture of shame of how poorly they did in math class and bitterness that this most wonderfully human tool for exploring the world has been denied them).
It's this kind of teaching that makes good robots and terrible mathematicians. (Thankfully I'm neither the former not the latter.)
A.) For the obvious not everyone learns the same way
B.)Drills and practice are important, but being able to actually apply your math skills is also important. I remember in high school when there were math word problems so many people HATED them. These were people in honors/college level classes and they struggled to actually apply the math to situations. Find how fast the train is moving, how long the ladder is, and Geometry proofs all made many of them get confused. They didn't understand the point of the math they knew.
Personally, I got SO bored with drills/memorization. And personally for me, that if that's the -only- method of learning used it's awful for long term retention. The math skills that I've retained the longest were taught in several different ways-- a mix of drills/practice, visualization, mixing it with other types of problems so that I could see how the puzzle pieces fit together, etc. I understand the point of what I'm doing, it's not just some abstracted concept that I happened to write 50 times.
I agree that past a certain point you'll have to accept abstraction, but especially in early years up until probably Algebra, it's a useful skill to be able to use math in the real world. Seriously, I've met so many people who can't calculate a tip, understand fractions enough for simple cooking, do their budgeting, or begin to understand economics and taxes enough to be even decently informed citizens.
That's because you're lazy. I know you really genuinely think you aren't, but you are... I'm dealing with this same mindset in my 13-year-old son right now. That's like saying you want to get in shape, but the repetitious method of lifting weights up and down is just SO boring.
There's more than one way to do things. That said, drills are important. You do have to practice at some point. But if it's the -only- thing that's used, as it is in some classes I've taken, it's boring and harder to retain the information. And in my personal experience, I've seen that sticking with -only- that method can sometimes be counterproductive. It's avoiding the depth of understanding needed for actually applying the math.
In talking about how Japanese teaching methods are better than American teaching methods, it states: "Similarly, 96 percent of American students’ work fell into the category of “practice,” while Japanese students spent only 41 percent of their time practicing."
Modern math classes tend to be very applied. I.e. the primary objective is to get the student to be able to solve real world problems with math. (A farmer sells 3 potatoes for a dollar. How much do 8 potatoes cost?). There seems to be some indication however, that abstract math is more accessible to some children, especially if they aren't supported by parents during their homework, etc. Because of this real-world to mathematical-world translation step. By exploring ways to make math more accessible, "friendly" and useful, teachers might actually make it harder for students to pick up math.
This is an interesting thought for me, because I tutored kids in math who were not doing good in school. I kind of ended up with a typical scheme to get them from "risk of failing the class" to Bs and occasionally As.
First I would introduce a a few techniques (equation solving in every case and the mathematical topic of their class - logarithms, binomial terms, etc. - also) and then give them very simple drill exercises. And a lot of them that we would solve together. I.e. simplifying exponentials $exp(3) * exp(5) = exp(8)$ etc. I always made sure that they were able to solve these drill exercises eventually, and they were all able to, because they picked up the scheme.
As a next step, I gave them the applied problems their teachers would ask them to solve, and made it into a translation problem. I.e. I explicitly told them that this was now just a translation. They could often identify with this because they self-identified often as language persons ("I like the literature class best", or "I like french class most"). I wouldn't ask them to solve the translation result right away. But they often just naturally did because it was not so different from the drill exercises.
Point is, I do not at all understand why I was needed for this. The teachers consistently failed their students in class, by not providing them with the very basic math skills and confidence that they needed to solve more complex problems.
Finding a good math teacher in America is difficult because the good math students rarely become teachers.
http://www.wsj.com/articles/the-best-language-for-math-14103...
http://www.imo-official.org/
Guy I went thru undergrad with went to a high school where the 10th grade algebra class requires naming the principle to be applied for each step in a problem, as if it were a proof.
This is painful. But he'd learned algebra properly. I hadn't. His math grades were much better than mine for a long time.