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.
>>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.
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.
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.
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.)
> 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.
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.
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.
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.
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.
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.
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.
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?
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.
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.
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.
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.
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.
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.
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.
[+] [-] jimhefferon|9 years ago|reply
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.
[+] [-] whack|9 years ago|reply
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.
[+] [-] WhitneyLand|9 years ago|reply
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.
[+] [-] clifanatic|9 years ago|reply
Americans stink at reading, too.
[+] [-] jandrese|9 years ago|reply
[+] [-] js2|9 years ago|reply
- 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:
http://mckinseyonsociety.com/how-the-worlds-best-performing-...
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.
[+] [-] agumonkey|9 years ago|reply
[+] [-] unknown|9 years ago|reply
[deleted]
[+] [-] qntty|9 years ago|reply
[1] https://en.wikipedia.org/wiki/Proof_School
[+] [-] Pitarou|9 years ago|reply
(just kidding)
[+] [-] dzdt|9 years ago|reply
[+] [-] njharman|9 years ago|reply
[+] [-] 11thEarlOfMar|9 years ago|reply
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?
[1] http://www.usnews.com/education/best-global-universities/mat...
[2] http://www.topuniversities.com/university-rankings/universit...
[+] [-] mathattack|9 years ago|reply
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.
[+] [-] atemerev|9 years ago|reply
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.
[+] [-] krmboya|9 years ago|reply
He at first put in a lot of effort, but eventually he was just coasting through ahead of his classmates. He attributes it to the drilling he put in initially.
[1] http://www.webofstories.com/play/donald.knuth/9
[+] [-] libeclipse|9 years ago|reply
It's this kind of teaching that makes good robots and terrible mathematicians. (Thankfully I'm neither the former not the latter.)
[+] [-] DanBC|9 years ago|reply
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.
[+] [-] snarfy|9 years ago|reply
[+] [-] Jemmeh|9 years ago|reply
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.
[+] [-] Kenji|9 years ago|reply
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.
[+] [-] ArkyBeagle|9 years ago|reply
[+] [-] LeanderK|9 years ago|reply
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.
[+] [-] taneq|9 years ago|reply
[+] [-] alistproducer2|9 years ago|reply
[+] [-] dleibovic|9 years ago|reply
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."
[+] [-] cmdrfred|9 years ago|reply
[+] [-] Fluid_Mechanics|9 years ago|reply
[+] [-] nav123|9 years ago|reply
[+] [-] jostylr|9 years ago|reply
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).
[+] [-] Swizec|9 years ago|reply
[+] [-] rdtsc|9 years ago|reply
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.
[+] [-] nemo44x|9 years ago|reply
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.
[+] [-] rabboRubble|9 years ago|reply
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.
[+] [-] krapht|9 years ago|reply
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.
[+] [-] wirrbel|9 years ago|reply
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.
[+] [-] yazaddaruvala|9 years ago|reply
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.
[+] [-] tomc1985|9 years ago|reply
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
[+] [-] Frogolocalypse|9 years ago|reply
Don't the countries that rank better use the rote method even more?
[+] [-] increment_i|9 years ago|reply
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.
[+] [-] pessimizer|9 years ago|reply
[+] [-] saretired|9 years ago|reply
[+] [-] ausjke|9 years ago|reply
[+] [-] danharaj|9 years ago|reply
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.
[+] [-] 58|9 years ago|reply
https://www.washingtonpost.com/local/education/majority-of-u...
I'm no fan of the American education system, having suffered through it a full 12 years, but I have to believe it's not the primary cause here. Math is hard, and near impossible if you're stressed. I excelled at math, despite relatively boring math curricula. Why? Because I wasn't stressed as a kid, my family was stable and did not suffer from any serious physical and mental illness, and one parent always made enough money so that the other could stay at home throughout my entire childhood. I had an enormous advantage, and most all of the kids I knew through advanced math classes and math competitions had a similarly charmed existence.
[+] [-] HarryHirsch|9 years ago|reply
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!
[+] [-] yanjuk|9 years ago|reply
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.
[+] [-] droithomme|9 years ago|reply
And yet at the same time we have these sorts of track records:
http://www.popsci.com/us-dominates-at-sending-stuff-to-mars
It's not just Mars or the moon, or decades ago, the US also is where most operating systems and software and CPUS are conceived of and designed. As well as countless other things that people who are so ignorant would not reasonably be expected to be able to do.
[+] [-] daodedickinson|9 years ago|reply
[+] [-] jason_slack|9 years ago|reply
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.