A better way to teach math
opinionator.blogs.nytimes.com
opinionator.blogs.nytimes.com
This is basically how Law School Confidential thinks law school manages to keep the supply of good lawyers low: fool the students into thinking that talent and intuition teach you law, and let a few students who outline and memorize like crazy "fall up" through the cracks.
I say "designed", but in that mostly subconscious way that we "choose" to do most things -- it feels right, it is "obvious" because it is how we learned originally, and it doesnt lead to weird results (a world full of talented math users which might cheapen the supposed "talent" of the lucky ones).
In my experience, public school teachers absolutely love to categorize kids into stupid and smart, with disastrous results unless you are one of the lucky ones (which probably has as much to do with good looks, high-prestige parents, and social skills). I wouldn't say there is NO bell curve, but it could be much, much flatter if it we wanted it to be -- but it's no fun to be an officer without a bunch of stupid enlisted men to boss around, and public school wastes a lot of taxpayers money to create those stupid people.
In my experience, most peoples' experience with public school teachers is formed around their own educational experiences, which is essentially no experience at all. Being the parent of a student in public school isn't necessarily a useful experience, either, as the teacher may be trying to tell you that little Johnny isn't as bright as you think he is, but you're still insisting that he's the smartest kid in the class.
My wife has had students that simply don't want to learn. They're smart kids, but they spend all their energy coming up with excuses for why they're not doing their schoolwork. My wife has had students that want to learn but are so angry at the world because of what's going on in their home life (abusive parents or siblings or just living in the projects). My wife has had a fourth grader who understood 'quarant' (French for 40) in terms of 'quarantine' and made connections that way.
Public school teachers, by and large, love to teach and love to see kids learn. Yes, there's some really bad teachers out there, but there are also really bad designers and really bad software developers. For the most part, though, we (as designers and software developers) don't have to work in thoroughly dysfunctional workplaces.
My experience of public school teachers has been from volunteering regularly as a tutor, being in the periphery of the educational system as an Upward Bound teacher and tutor. The public school teachers I have met generally taught to the "talented and driven" in their class, did a decent job, but were incredibly naive about the social system in which they did their supposedly idealistic work. And to a one, they all believed in the myth of talent and a wide bell curve (and had never heard of the pygmalion effect...)
Teaching 20 or 30 students in a class – whether you have the same students all day as in elementary school, or whether they’re in 5 hour-long batches as in middle/high school – is a tremendously demanding job under the best conditions. If most of those students are dramatically behind where the standard curriculum says they’re supposed to be (discouraged because they’re failing all the official tests), with heart-breaking home lives or time-consuming jobs outside school. Add administrative overhead and so forth, and teachers in many cases just don’t have the time and energy to give every student what he needs. It’s a risky career move and a heck of a lot of work for a teacher to decide that the state mandated curriculum is a load of crap and independently push off in a new direction.
Learning math requires intense focus, and a class of 30 people who don’t want to be there makes everything really really difficult, not at all comparable to a 1-on-1 or small group tutoring situation. The “talented and driven” get some attention because they are easier to teach. But I don’t think your generalization is fair to teachers. The ones I know dump their hearts and souls into reaching the worst-off students, and go home and cry about it when they don’t feel like they’re getting through.
HS level math is vary simple. You can write a single textbook that covered the full range of math from preschool to calculus, but the focus on 3 days of instruction, a day of review, and then a quiz or test slows things down. It can be easier to skip ahead and then go back and review than try and approach math in tinny nibbles. It's like spending a full year going over cement foundations before you mention that the goal is to put a house on top of that flat slab.
Personally, I used to do other classes homework assignments in my math classes. I can recall getting in about 4 seconds a new topic that the class spend a full week going though in minute detail. It was so bad I once accidentally did the next chapters review vs the assigned homework and did not even notice at the time.
PS: I am all for better instruction, but perhaps we could consider going a little further. We could probably get the average 10th grader to really understand Calculus, but I think the goal should be to dive into DifEq and number theory etc.
It would be very problematic, and this happens, if teaching consisting of showing micro steps is followed up by homework and assessments that also focus narrowly on micro steps. That serves up exercises for students, but it doesn't allow students the learning opportunities developed by working on actual problems. The rest of this comment is a FAQ file I send to families of new students in my math classes about the distinction between "exercises" and "problems."
PROBLEMS VERSUS EXERCISES
I frequently encounter discussions among parents about repetitive school math lessons, so a few years ago I prepared this Frequently Asked Question (FAQ) document about the distinction between math exercises (good in sufficient but not excessive amount) and math problems (always good in any amount).
Most books about mathematics have what are called "exercises" in them, questions that prompt a learner to practice the concepts discussed in the mathematics book. By reading one mathematics book, and then several more, I learned that some mathematicians draw a distinction between "exercises" and "problems" (which is the terminology generally used by the mathematicians who draw this distinction). I think this distinction is useful for teachers and learners to consider while selecting materials for studying mathematics, so I'll share the quotations from which I learned this distinction here. I first read about the distinction between exercises and problems in a Taiwan reprint of a book by Howard Eves.
"It is perhaps pertinent to make a comment or two here about the problems of the text. There is a distinction between what may be called a PROBLEM and what may be considered an EXERCISE. The latter serves to drill a student in some technique or procedure, and requires little, if any, original thought. Thus, after a student beginning algebra has encountered the quadratic formula, he should undoubtedly be given a set of exercises in the form of specific quadratic equations to be solved by the newly acquired tool. The working of these exercises will help clinch his grasp of the formula and will assure his ability to use the formula. An exercise, then, can always be done with reasonable dispatch and with a minimum of creative thinking. In contrast to an exercise, a problem, if it is a good one for its level, should require thought on the part of the student. The student must devise strategic attacks, some of which may fail, others of which may partially or completely carry him through. He may need to look up some procedure or some associated material in texts, so that he can push his plan through. Having successfully solved a problem, the student should consider it to see if he can devise a different and perhaps better solution. He should look for further deductions, generalizations, applications, and allied results. In short, he should live with the thing for a time, and examine it carefully in all lights. To be suitable, a problem must be such that the student cannot solve it immediately. One does not complain about a problem being too difficult, but rather too easy.
"It is impossible to overstate the importance of problems in mathematics. It is by means of of problems that mathematics develops and actually lifts itself by its own bootstraps. Every research article, every doctoral thesis, every new discovery in mathematics, results from an attempt to solve some problem. The posing of appropriate problems, then, appears to be a very suitable way to introduce the student to mathematical research. And it is worth noting, the more problems one plays with, the more problems one may be able to pose on one's own. The ability to propose significant problems is one requirement to be a creative mathematician."
Eves, Howard (1963). A Survey of Geometry volume 1. Boston: Allyn and Bacon, page ix.
I have since read about this distinction in several other books.
"Before going any further, let's digress a minute to discuss different levels of problems that might appear in a book about mathematics:
Level 1. Given an explicit object x and an explicit property P(x), prove that P(x) is true. . . .
Level 2. Given an explicit set X and an explicit property P(x), prove that P(x) is true for FOR ALL x [existing in] X. . . .
Level 3. Given an explicit set X and an explicit property P(x), prove OR DISPROVE that P(x) is true for for all x [existing in] X. . . .
Level 4. Given an explicit set X and an explicit property P(x), find a NECESSARY AND SUFFICIENT CONDITION Q(x) that P(x) is true. . . .
Level 5. Given an explicit set X, find an INTERESTING PROPERTY P(x) of its elements. Now we're in the scary domain of pure research, where students might think that total chaos reigns. This is real mathematics. Authors of textbooks rarely dare to pose level 5 problems."
Graham, Ronald, Knuth, Donald, and Patashnik, Oren (1994). Concrete Mathematics Second Edition. Boston: Addison-Wesley, pages 72-73.
This digression becomes the subject of a, um, problem in Exercise 4 of Chapter 3: "The text describes problems at levels 1 through 5. What is a level 0 problem? (This, by the way, is NOT a level 0 problem.)"
"First, what is a PROBLEM? We distinguish between PROBLEMS and EXERCISES. An exercise is a question that you know how to resolve immediately. Whether you get it right or not depends on how expertly you apply specific techniques, but you don't need to puzzle out what techniques to use. In contrast, a problem demands much thought and resourcefulness before the right approach is found. . . .
"A good problem is mysterious and interesting. It is mysterious, because at first you don't know how to solve it. If it is not interesting, you won't think about it much. If it is interesting, though, you will want to put a lot of time and effort into understanding it."
Zeitz, Paul (1999). The Art and Craft of Problem Solving. New York: Wiley, pages 3 and 4.
". . . . As Paul Halmos said, 'Problems are the heart of mathematics,' so we should 'emphasize them more and more in the classroom, in seminars, and in the books and articles we write, to train our students to be better problem-posers and problem-solvers than we are.'
"The problems we have selected are definitely not exercises. Our definition of an exercise is that you look at it and know immediately how to complete it. It is just a question of doing the work, whereas by a problem, we mean a more intricate question for which at first one has probably no clue to how to approach it, but by perseverance and inspired effort one can transform it into a sequence of exercises."
Andreescu, Titu & Gelca, Razvan (2000), Mathematical Olympiad Challenges. Boston: Birkhäuser, page xiii.
"It is easier to advance in one topic by going ahead with the more elementary parts of another topic, where the first one is applied. The brain much prefers to work that way, rather than to concentrate on ugly technical formulas which are obviously unrelated to anything except artificial drilling. Of course, some rote drilling is necessary. The problem is how to strike a balance."
Lang, Serge (1988), Basic Mathematics. New York: Springer-Verlag, p. xi.
I don't think so. Everyone builds on a solid foundation and microsteps. I think what tends to happen though, and is discussed in the article, is that by the time kids reach HS they either have a very solid foundation, or an extremely poor one. The kids with the poor foundation will have trouble understanding things regardless of how long you spend on it. They've developed gaps over the past nine years of math education.
I think if we address the gaps upfront in elementary school that in HS math will move much quicker. Not because the kids are smarter, but simply because you aren't compensating for 30 kids in a class who all struggle with a variety of concepts they should have learned in elementary school.
You can even go the other way. If you say 4a + 8b + 4 = x, and a = 10 * 10; b = 10. Now divide by 2b. Want to explain another base, say a = 16 * 16; b = 16. etc.
PS: My father even taught a first grader long division that way. (Note: I think he used x1; x2, x3 and he soon move to the traditional form.)
Edit: My point is if you want to build a house you can start with a foundation, or the windows. Keep studding math and there are plenty of opportunity's to review older concepts with new insights. Buy trying to focus on each little nibble in isolation it's harder to link concepts. EX: The sign rules are really just the associative property of addition and multiplication in another form. {-b + a = a + -b = a - b} {-b + -a = - ( a + b) } {-a * -b = (-1 * -1) * (a * b) = 1 * a * b}
But I did SAT tutoring when I was an undergrad for disadvantaged kids and I used to have serious trouble with a lot of basic concepts that made it really hard to go further.
For example:
1) Some students didn't understand greater than vs less than symbols. They didn't know which direction indicated greater than.
2) Some students didn't know the relationship between the numerator and denominator. 3/2 and 2/3 would be routinely confused. Couple this confusion with (1) above and things get really weird.
3) Area, volume, circumference -- while they knew the concepts they didn't map the terms to the concepts.
4) The relationship between remainders and fractional parts was often completely unknown.
5) Percent to values often wasn't known. "60% off sales" often meant a bit more than half off. The right intuition, but you need an actual answer for the SAT.
You're right, math is not linear. But if you only understand 50% of what is taught in elementary and junior high school, you're going to understand even less in high school.
On that we can more or less agree. Perhaps we could salvage some of that time and put it to more productive use. And then perhaps after a few years we can go back and look at it again. People think of math as a sort of tree structure where you need this long chain of tools to get to understand each new concept. I am simply suggesting it's closer to a graph where multiple approaches can reach the same concept and reinforce each other.
As to the SAT it's mostly a middle school math test. I suspect after a while you build a little crib sheet of the basic concepts that people needed to understand. And yet considering people "studded" this stuff for ~5 hours a week, 36 weeks a year, for ~10 years I suspect the crib sheet was not all that long.
This is exactly the reason why most people don't get math. If you start with the windows, the student asks "whats the point in learning this?". They can't see the house being built when you start with the windows. Without the foundation there is no intuitive understanding thus everything seems like meaningless rules. And in fact they are just that-meaningless.
No, the micro-steps are the solution, not the problem. Those who "get" math intuitively break problems down into smaller steps, often unconsciously. For those that can't intuitively do this (most students), they need to be taught explicitly the micro-steps. The micro-steps build the foundation for the higher level steps and the problem solving.
Learning the micro-steps well is like learning an abstraction in a program. Once you learn the abstraction to the point that your understanding of it is unconscious, then your thought processes are lifted to a higher level. This is true understanding of math.
Exactly. The argument is not to "do things in micro-steps", it's to do things in appropriately-sized steps. As your understanding of math improves, you can (and should!) take bigger and bigger steps.
The point is that, for presumably a variety of reasons, a really large percentage of students don't drop down a level in abstraction when they get stuck--they get depressed and give up. I see a large part of education's role being the inculcation of good habits, and this is just a sucky, sucky habit.
I think one of the main things that separates the quick learners from the slow learners is how much breaking steps into micro-steps can be done unconsciously. The data lends weight to this. Teaching a concept in micro-steps doesn't help the smartest people learn it better, but it brings the slow learners up to speed. The difference is how small the steps have to be for the students' unconscious to make the connections and thus reach "understanding".
Case in point, I was made a very proud dad today. My first-grade daughter and I were going over her math homework, which was flash cards with addition equations on them. This week the focus in her class is the numbers 4 and 5, so on each flash card, at least one of the terms was 4 or 5.
I was drilling her on them, and every time there was an equation the sum of which was >10, she'd mutter under her breath something about subtraction that I couldn't quite discern. I finally stopped at the "5+7" flash card, and asked her to walk me through her process for solving that particular problem. Her response:
"Well, I know 7 is minus three [from ten], so I do minus 3 from 5 which is 2, then I put a 1 in front of that, and that's 12."
In other words, she's doing | (10 - 7) - 5 | + 10. Or, basically, 7+3 = 10, 5-3 = 2, 10+2 = 12. I asked her, "Emma, is that how they teach you to add at school?"
"No," she said, "I just think about it like that."
Is that the absolute easiest way to solve that problem? Well, maybe; I don't know. I'm like 99% sure, however, I do that exact same process when I do mental calculations. That doesn't matter though. What matters is that she's analyzing problems, identifying and generalizing patterns, then applying those patterns to NEW problems. She is numerically literate! (http://en.wikipedia.org/wiki/Numeracy) This is especially gratifying for me, since we spend a few hours at home every week going over more advanced math topics like fractions and multiplication. She doesn't quite grok it yet, but at least when she sees it in school it won't be a total surprise.
In the end, she figured out a system to solve problems on her own that works 100% of the time. Critical thinking skills are paramount. Learn those and everything else comes eventually.
While breaking things up into micro steps is the easiest way to solve a particular mathematical exercise (and important in solving problems), joining things up into a single overarching entity is how you actually understand the subject. So breaking stuff up is really useful within a problem, but on its own doesn't give you the long-term preparation you need, and is inappropriate over an entire curriculum. I think any mathematics curriculum needs to have both, but it's the joined-up approach that's more often lacking in schools.
I haven't been able to get the JUMP curriculum (site down?), but I'd really like to see how it deals with these issues.
I do not know how you intended this to sound, but it comes off in a way that is horribly inaccurate of both officers and enlisted men.
The majority of officers I know have gone out of their way to help the enlisted learn and did what they could to ensure they had access to colleges. Many officers were ready to help any enlisted person with a real desire to do so to get their education finished and get a commission themselves.
On the other side of it, most of the enlisted men I knew were at or above average intelligence. Especially in certain fields such a communications and intelligence it was very common to find people who already had degrees when they enlisted or finished them quickly after enlisting.
You mean "narrower" or "skinnier", not "flatter". A flatter bell curve would mean the opposite.
As an example, I was a physics major in college, so I'm used to thinking of myself as being pretty numerate. And yet, I've noticed that I'm not a very efficient learner of higher math; I enjoy it, which keeps me chugging along, but I often find myself getting discouraged when my brain doesn't automagically internalize new abstractions. Instead of approaching a new abstraction as a bundled collection of less-abstract micro-steps, I think "hmm, if I were really smart this would just sink in." Sometimes the new abstraction does just sink in, but it often doesn't, and then I feel briefly bummed about not being the radiant genius I thought I was.
This is a dumb attitude! An abstraction is like a steak; you can eat it, just maybe not in one bite. And the incredible, amazing thing about abstraction is that as you get better at it, you get to take bigger bites!
This all reminds me of something Kent Beck says in his TDD book:
It is not necessary to work in such tiny steps as these. Once you've mastered TDD, you will be able to work in much bigger leaps of functionality between test cases. However, to master TDD you need to be able to work in such tiny steps when they are called for.
Being able to do things in tiny steps is a skill, and being willing to do things in small steps when you choke on a big step, rather than feeling dumb and giving up, is the key to learning just about anything.
This was a global decision born out of necessity. Because there aren't as many teachers as students we had to make time static and mastery variable. You take a class for a set period of time and then you get a grade based on how much of it you understood. A B means you understand the topic about 80%. In an ideal world every student would always get 100% but simply move at a different pace with the best students simply consuming more material throughout their school career (calculus, linear algebra, etc).
The tremendous news is that technology can and will turn this on it's head. The Kahn Academy does this already with tremendous success and it's the single most important thing that has happend to education in a long time.
For getting a full idea of the scope and vision, whats Sal's ted talk: http://www.ted.com/talks/salman_khan_let_s_use_video_to_rein...
(My comment sonly apply to standardized education, not to streaming students into different programs such as specialprograms for gifted students.)
So I guess my points are:
1. I'm doubtful that the "paint by numbers" approach is what the young Picasso needs.
2. Maybe I'm wrong about (1), but we certainly shouldn't subject the advanced students to this method until higher end performance has been measured.
3. I find the attitude of the author, who used the phrase "even things out" as if it was a good thing, concerning. Evening things out should be an explicit non-goal of education.
Some advanced students might even get even better and increase the variance! I can't comment on this program, I know nothing about tutoring. But I know that I personally like breaking things down into little steps, and I think I would have enjoyed a program running on these lines if I was allowed to move at my own pace.
Consider Round 2, where sigma=1.2%, mu=98%. Suppose hypothetically that 90% of students are clustered below 100 (an absolute measure of performance), 96.8% of students are below 150 pts, and 99.2% of students are below 300 pts.
In this case, the absolute variation is huge (300 vs 150). But because only 2.4% of students score between 150 and 300 pts, on a percentile graph, it looks like sigma has been reduced.
I think it's unlikely that this has occurred, but the graphs given don't preclude it.
Furthermore, where do you read that the objective is "low variance"? I read "higher average".
Perhaps one side-effect of the overall improvement is that the teacher doesn't have to spend as much time with struggling students and can devote more time helping high end students unlock their potential.
Teachers tell me that when they begin using Jump they are surprised to discover that what they were teaching as one step may contain as many as seven micro steps
It's about finding out how to simplify what some may see as the easiest step. Think about the lowest common denominator and build from there.
What took so long with each student was, based on their original question, systematically figuring out what their real misunderstanding was. This could be very time consuming, and took enormous patience on my part. It required much back and forth with the student so I could build a mental model of their mental model. Then I had to figure out how to build a bridge from their mental model to the correct one. That also was time consuming, because I sometimes had to build several bridges before finding one that clicked with the student.
Usually helping the Nth student on a project was quicker than helping the first because by that time I recognized what were common problems and built a bag of tricks to explain them. For example, it took me a surprisingly long time to realize the students had no practice running through an algorithm on paper, then translating that to code. (This was partially because the algorithm was so simple I hadn't even realized it was an algorithm.) So one technique I used was to, on paper, set up what was needed for the algorithm to work (list of numbers, table of results, etc.), then make them tell me what to do to get the correct result. I was acting as, basically, an intelligent computer that could be programmed in English. After doing this a few times, the students could finally "see" the algorithm, but it took a lot of time and effort with each student.
Throughout my grade school and high school years (born mid-80s, Canadian), near everything was memorization. Memorize multiplication tables instead of learn to do arbitrary multiplications quickly. Memorize your table of elements instead of emphasis on what it means. Memorize how to find the roots of a quadratic function, the rules for arithmetic with fractions, physics equations... (I'm pretty bad at memorizing, so this doubly infuriated me because I had to look for the underlying relationships to do well). Even after my province introduced a new curriculum which was supposed to stop all that and focus away from memorization.
Anecdotally, this is only slowly changing if at all, as new teachers come in and old ones who won't give up on rote memorization retire. I know this is all just anecdotal, but in my (ongoing) university education, I still see a lot of students put their emphasis on memorization, which suggests to me that they were taught that is what it means to 'learn.'
How can you do arbitrary multiplications quickly without memorizing your times table? You can't quickly compute 47*36 without being able to reel off "six times seven is forty-two, seven times three is twenty-one, four times six is twenty-four, four times three is twelve", can you?
I hated learning my times table (in fact I hated mathematics when mathematics was just arithmetic) but a certain amount of memorization is necessary and unavoidable if you're going to be able to do basic mental arithmetic, which remains an important life skill even if you do have a calculator.
My mental process for solving your example equation was to reduce it to 50x36-3x36 which is IMHO a much simpler process.
I don't know where I learned that process (probably from my dad), but simplifying the values to easily calculated approximate values and then adjusting that value for more precision seems to make much more sense than what was taught to me in school. It also caused a lot of frustration between myself and my teachers who demanded that all of the micro steps be written out instead of making what seemed to me as logical jumps.
Edit: formatting
Just saying.
Our daily life seems to rely on literacy much more than it relies on numeracy. You can only know how to add and subtract (mostly money) and you will still function all right in your daily life and a great variety of jobs. Not so much if you have difficulty reading. This seems to be slowly changing in the future, but if you consider the learning curve, mathematics has a much steeper one compared to reading and writing well.
Does anyone have a better explanation?
http://timss.bc.edu/PDF/t03_download/T03_M_Chap1.pdf
a chart of mathematics achievement levels in various countries. Although the United States is above the international average score among the countries surveyed, as we would expect from the level of economic development in the United States, the United States is well below the top country listed, which is Singapore. An average United States student is at the bottom quartile level for Singapore, or from another point of view, a top quartile student in the United States is only at the level of an average student in Singapore.
That the UPPER range of students in the United States is poorly served by current school mathematics instruction in the United States is shown by a careful analysis of the PISA studies of developed countries around the world. PISA's own analysis refers to specific instructional practices in different countries and other differences in country conditions that make a difference in educational outcomes.
http://www.oecd.org/document/2/0,3343,en_32252351_32236191_3...
Some bloggers in the United States persist in blaming these outcomes on the ethnic diversity of the United States (ignoring the ethnic diversity of Singapore and other countries that outperform the United States). Eric A. Hanushek, Paul E. Peterson, and Ludger Woessmann point out in their analysis of the PISA data, "U. S. Math Performance in Global Perspective: How well does each state do at producing high-achieving students?"
http://www.oecd.org/document/2/0,3343,en_32252351_32236191_3...
that the real problem in United States mathematics education is leaving behind too many of the high-ability students, of whatever ethnicity, compared to many other countries. A specific response about what is wrong with mathematics teaching in United States classrooms comes from Patricia Clark Kenschaft in the Notices of the American Mathematical Society volume 52, number 2 (February 2005).
http://www.ams.org/notices/200502/fea-kenschaft.pdf
Most elementary school teachers in the United States, repeated studies of the issue have shown,
http://www.nctq.org/resources/math/
have poor mathematics preparation in their own higher education and little mathematics knowledge when they enter the classroom. They then are directed by their school districts to use textbooks that are ineffective for primary mathematics instruction, so it is no wonder that most pupils in the United States (and the same applies to Canada) finish primary schooling with poor preparation for higher mathematics study. I speak and read Chinese and have lived in various parts of the Chinese-speaking world. I have Chinese-language textbooks of mathematics at home from more than one country. I am confident that young people of any ethnicity in North America can learn math well if they are taught with materials like those, because I am a math teacher by occupation and my classes include a very ethnically diverse group of students, who thrive in the classes and far exceed the meager expectations of United States classrooms.
Which bloggers?
Incidentally, I think "ethnic diversity" is a red herring. Any bloggers who talk about "ethnic diversity" are merely trying to couch their conversation in PC language to avoid ad-hominem accusations of racism.
The issue is not "ethnic diversity". The issue is that certain ethnicities underperform. Specifically, African Americans and Hispanics (30% of the US, 0% of Singapore) underperform. Nonhispanic whites (about 65% of the US and close to 0% of Singapore) achieve mid-level performance. Asians (about 4% of the US and close to 100% of Singapore) overperform.
These effects are HUGE in comparison to inter-country effects. Asian Americans score about 10 pts below Singapore. The EU15 scores about 50 pts below Asian Americans, and a couple of points above All Americans.
http://super-economy.blogspot.com/2011/01/how-well-do-above-...
http://super-economy.blogspot.com/2010/12/amazing-truth-abou...
It is simply incorrect to pretend that ethnic gaps in education do not exist, or to pretend that they do not explain a large portion of the gap between the US and the rest of the [edit: first] world.
Also, it would be helpful if you were a little more specific in your citations. Citing a gigantic PISA reports is much less useful than citing a specific table or figure. I'd love to learn more, but I don't have time to read the whole thing, and I have no idea which parts you are referring to.
[edit: wanted to clarify that I don't think the gaps between the US and poor locations, e.g., rural inland China or India, are primarily due to ethnicity. But gaps between wealthy first world countries do seem well explained by such factors.]
Ethnic gaps may exist, but they probably are symptomatic of gaps in wealth more than anything else. But who konws. There are so many other factors at play for people of darker skin in general in society that I'd hesitate to subscribe to one theory.
With that said, it does point to the fact that US public schools are probably better than most believe them to be.
I haven't looked at it extensively, but I don't believe this is true.
This study shows that SES (Socio Economic Status) doesn't explain much of the gap.
http://www.umich.edu/~rdytolrn/pathwaysconference/presentati...
This study (sorry, can't find a non-paywall version) shows that even holding income constant, blacks do worse than whites at the college level. In particular, low income whites perform as well as high income blacks.
http://www.jstor.org/pss/2963200
Also, even if you could explain the Black/white income gap via wealth, racism or other environmental factors, how would you explain the White/Asian gap?
http://en.wikipedia.org/wiki/File:Personal_income_race.png
And how would you explain that in spite of wildly different environmental factors (compare Singapore to Texas), there is a relatively small gap between Asian Americans and Asians? And similarly, there is only a small gap between white Americans and white Europeans?
(By the way, I'm not trying to claim there are no other factors at play. I'm just pointing out that race does seem to be a biggie.)
See: http://collegepuzzle.stanford.edu/?p=1590
"“The Differential Impact of Wealth vs. Income in the College-Going Process” finds that wealth and income affect the college choice process differently, with wealth consistently being more significant in predicting who enrolls in college, and the type of college they attend – even after controlling for student differences in academic achievement, habitus, social capital, and cultural capital."
More research clearly needs to be done here, but from what I've seen its clear that wealth looks to dominate income.
And don't get me wrong, I'm not saying there aren't other differences. In fact I suspect there are societal and cultural differences that will manifest in one way or another. If you spend a lot of time in Asian households in the US, you'll see they often resemble, culturally, canonical Asian homes more than "US" homes.
And there's data about the success of African immigrants to the US. Who often don't come wealthy (although some are), obviously are Black, but still do well. So there's clearly a culture issue here too, but I don't think we can really tease this apart without more data.
But this doesn't change my original point - US schools do only slightly worse than Singaporean schools at educating people with an Asian-style home environment, Asian genes, or whatever else being Asian is a proxy for. And cultural factors in black America may cause black Americans to perform worse than whites and African immigrants. Whatever the underlying factor is, it has nothing to do with the school system. That's the only point I'm trying to make.
This cannot possibly be true. School system are artificial constructions created by men to serve men. They are not given laws of nature. The only thing we can tell from the above is that the current school system does not serve well black Americans. School system should adapt to whatever the "cultural factors" of the populace they serve.
Whether this should be through prolonging schooling hours; providing stronger and more persistent emphasis in personal hard work; correcting a harmful self-image; getting rid of institutional racism; using different instruction techniques; or any other method that may addressed the "underlying factor" can only be discovered through research, experimentation and a willingness to change and adapt.
It is easy to find examples of a schooling that adapts to the unique circumstances of some children, for example: intense language training for children from immigrant families, schools at hospitals or schools that follow a circus around the world (http://www.guardian.co.uk/education/2007/jan/26/schools.uk).
It is irrelevant when considering the question of whether Singaporean schools do a better job than US schools. For all we know, Singaporean schools might be worse than US schools - i.e., they could education Asians equally well, but blacks/whites/hispanics vastly worse [1]. But it wouldn't matter since Singapore doesn't have any blacks/whites/hispanics.
[1] You actually see this effect (Simpson's paradox) when comparing US states. Texas is better at educating blacks, whites and hispanics than Wisconsin, but appears worse when averaging over the entire population.
http://iowahawk.typepad.com/iowahawk/2011/03/longhorns-17-ba...
Looking across those examples, parental income and access to good schools seems to be a more powerful indicator than ethnicity. It just so happens that the 2 are correlated in America.
...or at least an employer willing to ship them across the ocean in return for indentured servitude as manual laborers in the 1850-1900 era. And it's not so much parents as great great great grandparents.
Asians got to the US the same way as Italians and Irish, and pretty much at the same time.
http://en.wikipedia.org/wiki/Immigration_and_Nationality_Act...
Especially look at the section called "Long-Term Results". I haven't studied this personally, but my history professor in college claimed that this act is essentially responsible for the fact that America has the ethnic diversity it does today.
Basically, before 1965 only a very small number of non-white people were allowed to come to the United States. (Partly because of immigration restrictions, partly because of the Gentleman's Agreement of 1907 in which Japan agreed to restrict immigration for us.) After 1965, the quotas were much more equal for different regions of the world.
Wish I could've upvoted this 100,000,000 times.
The problem is that this approach requires patience, persistence and hard work. Do this help students in the long-term? If you learn math without struggling, will you learn to think in the same way? Is this teaching to the test over teaching you how to think?
The struggle mostly comes from situations where a) people just can't follow the explanations at all, since the reasoning is not explained well enough, or there are jumps too large, or b) people who have trouble adapting past strategies and concepts to more novel situations. For example, they might know how to solve some class A of word problems, but once you switch the word problems to solving the equation in the other direction, they just get lost.
I think most of the 'left as an exercise for the reader' stuff appears once you get to a high enough level that you a) assume that the reader is proficient enough and b) that the reader actually cares enough about the subject to be able to do so. For example, my friends taking a bunch of pure math subjects get that crap all the time. "So we've proved X theorem for Y case, Z will be left as an exercise", and they eat it up (partly because it really is an exercise).
Now, I still think a lot of the time, it's inappropriate, and just used cause the writer is lazy, or has used up too much space on diagrams (especially in physics textbooks...).
There are actual exercises, where solving it is just a matter of applying some concept. However, a lot of math books have things where you have beat your head against the wall to figure them out. Although, the internet has made it easier to lookup, nowadays.
School textbooks often try to make things simpler. However, when these texts are written based off of older texts, which skip steps, they may inherit the style of the older works.
I hope that the Jump system is real and that it solves the problems outlined in the article.
Emphasize the positive.
If a child gets three out of four questions wrong, I will mark the question that is correct first and praise them for getting the correct answer. Then I will say, “I think you didn’t understand something with these other questions” or “You may have been going too fast,” and then I will point out their mistake – or ask them to find it themselves! I’ve found that if I start by mentioning the mistakes, a weaker student will sometimes simply give up or stop listening.