I teach math. I am impressed that when you read these comments, very few of them show evidence of having read the article.
The article is about experience with actual students, with ones who have a real knack for the subject and would learn it despite what any teacher did, and with ones who waver. There proves to be a better way, which the article describes in various ways, including as "sense-making."
One of the problems here is that everybody has a pre-formed opinion and does not seem to want to risk having to modify it with the evidence of experience.
> "One of the problems here is that everybody has a pre-formed opinion and does not seem to want to risk having to modify it with the evidence of experience."
That's probably the most accurate description of HN comments I've seen. People see a headline related to something they already have pre-existing beliefs about, they read a few paragraphs just to make sure they're on point, and then they jump straight to the comments to argue in favor of what they already know to be true.
I'm willing to wager $10000 that a significant majority of HN commenters in any thread do not read the article in full, and even among those who do, a smaller minority actually post comments addressing the meat of what the article is saying.
To be fair, most times articles coming from mainstream news organizations do not offer anything new to learn for the professional readers of HN, readers likely to have already encountered the arguments presented therein from years past of reading and learning. That is, NYTimes articles and the like frequently make weak arguments that support a preexisting opinion or agenda, and are meant for consumption by the non-professional general populace.
There was a time, for example, that NPR and PBS provided informative information to me, but as I grew I began to realize that I knew what they were going to say before they said it and that further growth required better information sources.
> I'm willing to wager $10000 that a significant majority of HN commenters in any thread do not read the article in full, and even among those who do, a smaller minority actually post comments addressing the meat of what the article is saying.
I'm curious... did you read the article in full? I'm not saying you're wrong but it would be funny if you didn't read it as well :)
One of the things I like about Scientific America is they have an Executive Summary at the start of the article. So while folks may not have the details they at least have the correct gist.
I skimmed the article. Do I have pre-formed opinions... I do. Would they have changed after reading the article in detail ... they might have if there was some hard evidence other than the obvious stat that the US is bad in math.
My question (I'm having my first child with in a month) is what do I do as a parent other than be patient:
"Training teachers in a new way of thinking will take time, and American parents will need to be patient. In Japan, the transition did not happen overnight. When Takahashi began teaching in the new style, parents initially complained about the young instructor experimenting on their children."
My father forced my brother and I into Kumon. Kumon was hardly imaginative. Did it work? I think a little bit.
Education techniques particularly math has changed so many times that it is beginning to look like diet and exercises. We are all a little jaded so I can understand the plethora of pre-formed opinions.
> My question (I'm having my first child with in a month) is what do I do as a parent other than be patient:
You could look into how your local public school system teaches. Taking to the people who teach there, for instance, is helpful. Expressing support for an NCTM-like approach, if you find you do support it, would also be great.
> Education techniques particularly math has changed so many times that it is beginning to look like diet and exercises. We are all a little jaded so I can understand the plethora of pre-formed opinions.
I understand and as a person who teaches math, and teaches teachers too, I feel the same impatience.
I want to know why we don't just go out and see, on actual students, what actually makes them better at understanding and being able to solve problems. Try different things and see what works. Review video tapes. Give before and after tests, and focus groups, and see what the people in the class learn the most from. Train teachers in the details of doing the job, as an engineer or doctor would be trained in details.
The thing is, that's what the folks described here are doing. Have they got the final answer? I doubt it, particularly as the impacts of tech is so little understood even now. But they are making progress, at least it seems that way to me. (If you are interested, Building a Better Teacher is a source that would give you more of an understanding of what the discussion is. This article was part of the marketing people's plants for the book.)
This comment is related to your muse starting with "I want to know why..." I'm simply pondering and looking for feedback from someone who is actually a teaching professional unlike myself.
"There [my comment: Finland], as in Japan, teachers teach for 600 or fewer hours each school year, leaving them ample time to prepare, revise and learn. By contrast, American teachers spend nearly 1,100 hours with little feedback."
This line took my notice. I have to wonder if their school systems are such that teachers do not have to be in front of students 6 continuous hours daily as my daughter's teachers do. (My daughter is in elementary school.) By the time her teachers finish with students, attend after-school meetings and grade, I can see there being precious little mental energy left for professional development.
My friend Scott made a career change a decade ago. After being a software developer for 22 years, he switched to teaching tech in a middle school. His comment to me after his first year was that teaching is much more exhausting because you have to be "on" 5-6 hours daily. If you're "off", students pick up on it and reflect it. His opinion hasn't changed after a decade.
My only comment would be that I find what your friend Scott finds. I don't teach middle school, I teach in college, so my day is more mixed than his. But my training was as a high school teacher and my wife is a high school teacher, and I do find that her routine is exhausting.
I hear you about having little mental energy left. There are lots of intensely dedicated teachers but it is unreasonable to expect that they will be on 24-7. They get to have families, to have hobbies, etc., just like everybody else. Just like engineers, police officers, and any other highly trained professional.
(It is especially discouraging, for me, when someone writes a letter to the editor seeming to assert that teaching is something women do so they get summers off with their kids, kind of a hobby. Those make me cringe.)
IMO a culture developed in the US in the past few decades - I would be hard-pressed to pinpoint when but it's been within my professional life in IT - that we should minimize any activities in a work environment that do not have a direct tie to revenue. The result is that professional development is all but non-existent unless you do it on your own time.
The US is scrutinizing its education system right now. Unfortunately, we're looking for simple answers and pinning blame on teachers when the problems are more systemic. My neighbor is a visiting professor from Germany. His wife is a teacher in grades 5-8 in Germany. My wife and I had dinner with them a while back. They made the observation that blaming teachers took place in Germany about 10 years ago. When all was said and done the problems were more systemic, they were corrected, and the system is considerably better now in their opinion. Hopefully we'll follow suit.
> I want to know why we don't just go out and see, on actual students, what actually makes them better at understanding and being able to solve problems. Try different things and see what works. Review video tapes. Give before and after tests, and focus groups, and see what the people in the class learn the most from.
Probably because parents don't want their kids to be one on the receiving end of an experimental technique that doesn't work and sets back their child's learning and development by an unknown amount.
This goes especially for bike shed topics where everyone thinks they have it figured out. HN's favorites seem to me nutrition and schooling. Those threads never fail to get 300 comments.
Hmm, I wonder at what ratio of readers to non-readers you'd be able to get action at even odds... 1:5? 1:10? At what ratio would you be willing to wager against this bet?
You set the ratio at 1:2 which I would not take even odds against. If you'd said 1:20 I'd think about it depending on how strictly "in full" were defined and how we decided to operationalize it.
Isn't that the problem with Internet discussions in general? It's so rare to find a person whose opinion has changed after reading a different perspective on the Internet.
>>everybody [on HN] has a pre-formed opinion and does
>>not seem to want to risk having to modify it
Jim, this is seems like a crazy generalization. I assume you haven't tried to quantify this and it's just anecdotal opinion right? No problem, let's continue:
1). All my beliefs are falsifiable. Since your claim includes "everybody", I guess this counterexample alone proves your conjecture false.
2). If the most open minded community on the Internet scores 100, what score would you give HN? Can you name a single one that's better? I'd like to check it out.
3). How often does a response here randomly adopt the retort "Its Obama's fault" or an ad hominem attack? When it does happen it's almost immediately killed or downvoted off.
On a separate note, the article plays up how bad Americans are at math. Is this data controlled for household income and education level of parents? The factors are so significant it's hard to accept the data without them.
Lastly, my personal experience mostly matches the article. So many math classes were rote, boring, and uninspiring. I first learned it didn't have to be this way during discrete math because it required less algorithms, more creativity in proofs, and pulled back the curtain on what really made things tick.
People on my Facebook feed are still venting over "Obama" trying to teach some number sense to kids. Like if you don't do the subtraction exactly how the parents were taught in school it's going to ruin the kids for life.
To me, the article isn't really about math... math is just used to illustrate how poorly we teach in America, and then talks about the reasons why, primarily that we do a really poor job of teaching teachers. Some key quotes:
- It wasn’t the first time that Americans had dreamed up a better way to teach math and then failed to implement it. …The trouble always starts when teachers are told to put innovative ideas into practice without much guidance on how to do it. In the hands of unprepared teachers, the reforms turn to nonsense, perplexing students more than helping them.
- American institutions charged with training teachers in new approaches to math have proved largely unable to do it. At most education schools, the professors with the research budgets and deanships have little interest in the science of teaching.
- Without the right training, most teachers do not understand math well enough to teach it the way Lampert does. “Remember,” Lampert says, “American teachers are only a subset of Americans.” As graduates of American schools, they are no more likely to display numeracy than the rest of us.
- Left to their own devices, teachers are once again trying to incorporate new ideas into old scripts, often botching them in the process.
- In Japan, teachers had always depended on jugyokenkyu, which translates literally as “lesson study,” a set of practices that Japanese teachers use to hone their craft. A teacher first plans lessons, then teaches in front of an audience of students and other teachers along with at least one university observer. Then the observers talk with the teacher about what has just taken place. Each public lesson poses a hypothesis, a new idea about how to help children learn. And each discussion offers a chance to determine whether it worked. Without jugyokenkyu, it was no wonder the American teachers’ work fell short of the model set by their best thinkers.
And the most important two paragraphs:
The other shift Americans will have to make extends beyond just math. Across all school subjects, teachers receive a pale imitation of the preparation, support and tools they need. And across all subjects, the neglect shows in students’ work. In addition to misunderstanding math, American students also, on average, write weakly, read poorly, think unscientifically and grasp history only superficially. Examining nearly 3,000 teachers in six school districts, the Bill & Melinda Gates Foundation recently found that nearly two-thirds scored less than “proficient” in the areas of “intellectual challenge” and “classroom discourse.” Odds-defying individual teachers can be found in every state, but the overall picture is of a profession struggling to make the best of an impossible hand.
Most policies aimed at improving teaching conceive of the job not as a craft that needs to be taught but as a natural-born talent that teachers either decide to muster or don’t possess. Instead of acknowledging that changes like the new math are something teachers must learn over time, we mandate them as “standards” that teachers are expected to simply “adopt.” We shouldn’t be surprised, then, that their students don’t improve.
But this isn't news (even if the article is from 2014). I've linked to this report from 2007 many times before and I'll do so again:
To find out why some schools succeed where others do not, McKinsey studied 25 of the world’s school systems, including 10 of the top performers. The experience of these top school systems suggest that three things matter most:
Getting the right people to become teachers; Developing them into effective instructors; and Ensuring the system is available to deliver the best possible instruction for every child.
In the US, we fall down on all three of these points.
I know this is off topic, but I wonder why we use Japanese like this. jugyo = lesson, kenkyu = research. If we absolutely must make it into a proper noun, some kind of brand to sell, how about just "Lesson Research" instead of Jugyokenkyu? But really, it seems it would be better to just say it plain: "Japan's educators research their lesson plans through a process where..." "Without performing research around their lesson plans, it was no wonder the American teachers..."
In the same way we don't say "signboard" or "billboard" but say "Kanban"? Or for the same reason we say "jujitsu" but not "gentle, soft, supple, flexible, pliable, or yielding art or technique".(see also Judo: "gentle way").
I'll take a stab at this though. The use of "jugyokenkyu" most likely expresses not just a "lesson plan", but also a philosophy and system that a literal translation to "lesson plan" can't express.
Definitely. But if the Japanese usage is as a proper noun, you can translate it and still use it as such. We do this with native words all the time - lower-case agile vs capital Agile, for instance. The question isn't whether or not we should proper-noun-ify things as much as whether or not we are served by doing so with a foreign word -- the meaning of which isn't immediately clear to an uninitiated audience.
That's a really good point. Though I do think the article is specifically about math, it generalizes to the point you're making.
I thought the article did a particularly good job explaining how the "new math" is very dependent on high quality teaching rather than adhering to a new curriculum.
Unfortunately, I think this is where the US has fallen down where it comes to math (and, as you've pointed out, other subjects). We seem to think we need to find the magic approach that will work.
My kids are in school, and they are doing common core math. In many ways I do think it's much better than the old way (put the big number in the house, put the little number outside the house... except if you're dividing the little number by the big number...). There are a lot of good ways to add fractions, or do long division. It's not only math, but math is actually an unusually good subject to teach this kind of creative problem solving.
Here's what I see as the problem: we observe a teacher approaching a problem a different way. When students are asked to subtract 4 3/4 from 6 2/5, you could do the ol' algorithmic cross multiply trick. It's long, and boring, but it does work. Or, you could write the numbers down on a number line, notice that there are distances between the numbers that amount to integers, and small additional bits of remaining distances that amount to fractions. Add those up and you've got your solution.
That's just one of many possible approaches. My guess is that a talented teacher might do this, or something else.
Here's what I feel common core does: it notices one particular creative approach and concludes, "oh, the way to teach this is Step 1: create a number line, step 2: mark off the whole numbers, Step 3:..."
Essentially, they're looking to reduce the creative approach to another mechanical set of steps. This may be an overstatement and overuse of this phrase, but it's sort of a "cargo cult" of math instruction. The point never was the number line, it was that a teacher was talented enough to see a better approach for this problem, and that, after repeated examples and exercises, the students develop this ability as well.
I do think this article gets at this - that the point isn't the particular "creative" approach, it's the creative approach itself.
Unfortunately, this will never work without talented teachers. They need to be drawn from the top tier of math grads, they need to be very good at teaching and connecting with students, they need to be fluid and creative in their approach, and they need excellent training and experience.
Almost nothing about the US educational system, outside a few very selective and rare programs, would draw this sort of person into teaching (I hope it's obvious I'm not talking about research universities).
The irony is, once you have these teachers, I suspect don't really need to formalize this approach! "Common Core Math", if done well, is probably what talented, creative math teachers would naturally do on their own. You can't get it from a set of mechanical steps, and if you're doing what you should (drawing in top teachers), you don't need the mechanical steps anyway.
Two things: firstly I think most teaching devolves to a cargo cult, unless the culture explicitly tries to keep this from happening.
Secondly, this quote: "Admittedly, a tenacious commitment to improvement seems to be part of the Japanese national heritage."
I wonder if a missing phrase might be "collective and individual improvement".
Anglo cultures seem to lack this, because they seem to be more about authoritarian displays of power than about striving for collaborative competence.
It's not that individual teachers are necessarily authoritarian - although many school boards seem to be. It's more the political subtext is 'Shut up, listen to my authority, obey, and repeat."
A culture based on "Think, explore, explain, discuss, debate, experiment, and fact-check" works very differently.
When teaching looks more like the first than the second, *it's not really teaching STEM fundamentals" - it's teaching a robotised cargo cult version of STEM that doesn't truly promote STEM principles.
Kids have three options - they can either obey, which will get them good grades but no deep understanding. They can fail - which is a popular choice, because rote learning is harder than learning by deep understanding. Or they can fail and rebel and decide that math is a waste of everyone's time, because who the hell cares, and would you like fries with that?
So math teaching is a symptom of a deeper problem, which is a culture that's more geared towards top-down competitive hierarchical classroom dynamics - with the teacher as top dog - than to promoting non-competitive collaboration and sharing of individual insights, with the teacher as inspiring facilitator.
Do you have an opinion on math-circle inspired schools like Proof School[1]? To me it seems obvious that this is the direction that math education should go in for kids who are passionate about math.
Well, I clicked on the link, and it just said, "You have to log in to view this article." Are you seriously suggesting I have to register with the New York Times before I'm permitted to share an opinion?
Couldn't read article. Hit paywall or something, didn't bother checking what just clicked "close tab" after the tenths of second it took to recognize it was not article. A reflex I've developed.
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.
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".
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'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.
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.
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.
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.
> 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.
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.
> 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.
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.
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).
>> 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.
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
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.
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.
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.
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.)
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.
"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.
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.
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.
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.
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.
"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?
> 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?
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.
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."
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.
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..
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.
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 :)
> 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.
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.
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.
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 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.
> 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.
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.
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
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.
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.
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.
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 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.
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.
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."
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.
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.
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).
> 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.
> 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.
> 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.
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.
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.
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.
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.
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.
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.
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?
Looks like the answer is buried at the end: "Finland, meanwhile, made the shift by carving out time for teachers to spend learning. There, as in Japan, teachers teach for 600 or fewer hours each school year, leaving them ample time to prepare, revise and learn. By contrast, American teachers spend nearly 1,100 hours with little feedback."
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.
The new SAT etc now puts more weighs on reading comprehension and make math less important, which further weakens the willingness to study math, so the Asians(who are traditionally better at math) will not dominate the SAT high scores. As a country this will only make math education much worse, it should be another way around, otherwise, you can not compete in the STEM field, which is crucial for future.
Chinese Tsinghua Univ beats MIT to take the top seat at USNews for engineering. I think this is the second time in a row.
As far as I know, Chinese universities took the best students without considerations of gender at all. They do have AA for races but it is minimal.
USA is moving towards to AA-for-school-and-workplace, I hope this will make more people happier with high self-esteem, in the meantime you lose your competitiveness quickly.
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).
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.
Since no one has brought up "Nix the Tricks" yet, here's the link to the ebook: http://nixthetricks.com/
Foreigners are likely unaware that these are the algorithms routinely taught to US students in math class. There are no formal proofs, no emphasis on making sense - the only purpose is to have something memorizeable to have students pass the next test (and possibly the No-Child-Left-Behind test at the end of the year). There is no generalization, no focus on understanding. When you tell an American kid at university level "this is why concept XY makes sense" they don't understand why you might say this, they are only interested in the algorithm and the solution (and passing the next test, of course).
Small wonder that anyone exposed to that curriculum sucks at math and on top of that us turned off the subject. It's like Feynman in Brasil!
I taught my two kids long division in about ten 40-minute sessions spread over 5 days. I sat with each child as we learnt the procedure. To begin with I held the pencil and asked questions, then the child took over the writing and I monitored to fix mistakes until finally the task could be performed/practiced solo.
In this one-on-one practice approach misconceptions are eliminated quickly at the start. It could not easily be replicated in a large group. Instead the approach in the article seems to be about groups of people identifying each other's misconceptions. Either way the effectiveness lies in avoiding bad habit formation.
If you look at YouTube each method of arithmetic has variants and you can pick the one that looks best. e.g. I chose a method of multiplication with consistent placing for the carries which reduced error considerably over what I was taught at school.
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.
Well, a lot of people came to America hoping they would need math less. Americans stink at it because they feel they don't need it and when they feel otherwise it will be too late. It's easy enough to encourage seduction too math as the article suggests, but the real challenge is somehow blocking out more and more distractions (other options) when you can't reduce them because freedom and democracy are national ideals. Math can be interesting but how do you keep it more interesting than video games when you don't have a culture willing to regulate them as harshly as South Korea or China? How do you keep people from following Howard Hughes and having fewer friends, staying mostly home, and watching an astonishing amount of recorded video? Howard Hughes was hardly someone who could do nothing with math, but there are a lot of pitfalls in America to which the schools are blinded or can't do anything about. How do the schools, with such little authority, reduce problem behavior AND help something more difficult and at least supposediy more healthful fill the void. This is the problem with the democratic subject that Plato described: in democracy so many behaviors must be acceptable / not disqualifying for power and respect that people cycle through a great diversity of them. This is far more a pressing issue than whether teachers already trying to teach math do it this way or this other way: if they still have to compete with more and more cell phones and games and other entertainments and Babel, they will get diminishing returns.
To put it more succinctly, when you have this American phenomena often labeled "ADHD", whatever you think is really happening there, education must be viewed in light of the whole attention economy of the child's life.
I am 39, American, white and I am taking a math class at my local college. I am the oldest person in the class by far. I get A's, so do a few others. Most get C's.
I do every homework problem. I do variations of the homework problems. I spend at least 6 hours outside of our class time (3 hours) doing the math.
When we have tests the Instructor gives us problems that are not like the homework where she can see if we can evaluate and apply the concepts to things we haven't yet seen.
Americans who need to be good at math, are good at math. If you don't use it, you lose it.
Hell, I'm an EE, so I'm probably one of those who would be considered "good at math" (at least by layman opinion), yet I may do at most a couple of integrals a year. I'm mostly just doing basic algebra on an RPN calculator. Though I did ace all that EM theory, random proceses, etc in grad school; do I use the math behind it? Rarely.
Can't read article, paywall or adblock or something.
Is it because our popular culture ridicules being good at math as nerdy?
Is it because almost every american can name at least one sports or pop singer who makes multiple millions but probably has never even heard of a mathematician let alone be able to name one.
The root cause is we don't value math as a society. Until we do, we won't spend the effort needed to figure out how to teach individuals better.
"Seventy percent of all children attend preschool for three years. Young children are gradually trained in important school-related behavior, practicing routines that will be used throughout elementary school."
"Two traits pervade the culture and are taught by both parents and teachers: effort and persistence."
As far as I'm concerned, higher national IQ, more discipline in the classroom and possibly a longer school day is why America 'sucks' at Math compared to Japan. May be more of one and less of another, and let's not ignore the negative relationship between mean class IQ and the frequency of disruptions. This is only up to a point - potted plants have very low IQ but aren't disruptive, but like American students they also suck at maths. Anyway, I can say to nytimes, without reading the article that panflutes and sandals can only help America's children so much.
American dont stink at Math, American stinks at training kids in general, you cant give too much home work, kids need time to play, and the Chinese kid in China is at school till 9 pm constantly doing and practice math. yeah, an avg American kid is not going to be as good as avg Chinese kid.
However, the top 3-5% of the American kid is going to be just as good if not equal to their Chinese counter parts.
In China, education made all kids above avg regardless if you are below or above intellectually. In America, an idiot is going to be an idiot, and a genius will take that step to be a genius.
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.
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):
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?
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.
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.
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.
I hear Americans all the time telling themselves that they are bad at math. These are my friends and family. I tell people that I'm into math. >50% of the time the first thing someone says is "I'm bad at math."
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.
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'm an American, and I don't "stink at math". I will compare my math SAT and GRE scores with any nation in the world. Similarly for my Ph.D. dissertation in applied math and my peer-reviewed, published papers in applied math. Opps, those paper were already reviewed as "new, correct, and significant" according to international standards.
Next, uh, I do have some opinions about the NYT, e.g., their track record on getting and publishing good, correct information. Hint: That opinion could not go much lower.
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.
The article is about experience with actual students, with ones who have a real knack for the subject and would learn it despite what any teacher did, and with ones who waver. There proves to be a better way, which the article describes in various ways, including as "sense-making."
One of the problems here is that everybody has a pre-formed opinion and does not seem to want to risk having to modify it with the evidence of experience.