Some new research indicates that teaching kids all these word problems about speeding trains and slices of pie may be a mistake, and that children actually learn math better if you just teach them abstract equations from the get-go. Obviously, I'm in no position to judge (my boring guess is that different approaches work best for different people and there should be some diversity of classroom methods available for different students), but it's a bit bizarre how little effort we put into developing serious research-based pedagogical methods. You'd think this would be a major component of federal education policy, but it's really not.
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Bad Math
26 Apr 2008 02:45 pm
Comments (84)
One thing that always helps is if the teacher somehow manages to make math fun. A third-grade teacher Miss Burujy used to let the boys jump out of their seats like leapfrogs to be the first to answer an equation. Plus, she was cute.
If Candidate B has 1,721 delegates and Candidate H has 1,579 delegates, how much B.S. does Candidate H need in order to get the nomination? Circle your answer.
Sorry, I couldn't resist.
What HIIJSM said. The point is to learn when and how to apply the math.
Also, I think your statement "how little effort we put into developing serious research-based pedagogical methods" shows a bit of Harvard bias. At most state universities, there is a College of Education that's anything from a tenth to a fourth of the entire faculty, their research activity is almost entirely on pedagogical methods, and is mostly federally funded. The NSF and Dept of Education sink huge amounts of money into this.
This may or may not be right, but I am sure that modus ponens and modus tollens must be taught in Grade 1, and proficiency in these concepts must be a n absolute requirment for entry into the second grade.
The students who learned the math abstractly did well with figuring out the rules of the game. Those who had learned through examples using measuring cups or tennis balls [didn't]
So the kids who learned abstract rules of math did well at inferring the same abstract rules from the game. But kids who learned other concrete examples of the math didn't do well at inferring the rules from the game. Which kids would have done better at applying the rules to other concrete examples, like measuring sticks or beach balls? I think it's pretty obvious that kids should be taught the abstract rules of math and also how the rules can be applied to concrete examples.
I think there's a simple explanation for why these problems are being taught, and why they are not successful.
What the eventual goal of math education is for students to see real world problems, recognized the mathematical principles behind them, and then solve the mathematical equations that correspond to the real world problems.
But there are two steps here -- recognize the math problem, and solve the math problem. These two steps need to be taught independently, as they involve very different skills. Incidentally, this is why physics is frequently very hard for people who are good at math -- the central difficulty in physics is figuring out exactly what the math is underlying the physical problem. The rest is just figuring out how to solve the corresponding math problem, which is (comparatively) straightforward.
Based on this research, first students need to be taught the abstract concepts -- THEN figure out how to solve story problems.
There are two things that would need to happen to get the kind of innovation you want:
1) A lot fewer federal dollars (because unless the dollars drop, the federal rules that are the problem won't go away)
2) With a less heavy hand from on high, teachers unions will also be less national in nature, and more willing to work with local control
You want change? Get the feds out of the system. You want the status quo? Give the Feds more influence.
My sister - who's taught middle school math in CA and TX for the last two decades - could have told you as much. As she has been forced to teach to the test, rather than basic skills, she's been terrifically frustrated by the focus on story problems rather than basic equations. Her students are not acquiring mathematical skills that are a given overseas. Of course, that leaves American students far less competitive upon graduation.
The problem is that curriculum is increasingly created by individuals who've spent zero time in the classroom. Decisions are made based on educational theories, not on proven protocols.
A lot fewer federal dollars (because unless the dollars drop, the federal rules that are the problem won't go away)
And the fact that the vast majority of funding in cognitive science, incuding this project, comes from federal sources supports this argument how. . .?
At most state universities, there is a College of Education that's anything from a tenth to a fourth of the entire faculty, their research activity is almost entirely on pedagogical methods, and is mostly federally funded. The NSF and Dept of Education sink huge amounts of money into this.
Actually, that's more the biggest *problem* rather than the solution. Schools of Ed mostly contain the stupidest faculty and students---and they're also extremely gullible and "excitable" as well. The "research" they produce is pure junk that has done untold damage to all our public schools over the last half-century.
Suppose we took all the stupidest, most gullible and "excitable" people and set them up as "public health advocacy researchers". The result would probably be something like a wave of lunatic hysteria against all vaccinations, and (eventually) a deadly and totally mysterious national outbreak of measles and polio....Gee, that's exactly what's now starting to happen!
Hmm... if every parent got a voucher for, say, 80% of the per-student cost of the local public school, maybe lots of different, small schools would be established to compete for students, each with its own pedagogical methods? Sounds like a great way to provide different solutions for different types of kids.
The biggest problem with math education in this country is that we lie to students.
First, we teach them that math is hard.
Then we teach them that you can't take big numbers out of small numbers.
Then we teach them how to take big numbers out of small numbers.
Then we continue to teach them that there's only one way to do things and then show them the new way. it's terrible.
The second biggest problem with math education in this country is that we let education majors teach math to children.
if every parent got a voucher for, say, 80% of the per-student cost of the local public school, maybe lots of different, small schools would be established to compete for students, each with its own pedagogical methods?
My guess is that because every person not only learns differently, but learns different material in a different way (i.e, some students will learn physics through method A and spelling though method B--others will be the opposite) that trying to track students into schools with a variety of pedagogical methods is not necessarily going to be any better than forcing them to attend a single school.
In my experience, I've learned best when the teacher was able to tailor his/her teaching method to my own particular learning style--which would suggest that smaller class sizes (where instructors have more time to individualize their teaching ) would be a more desirable policy goal.
"And the fact that the vast majority of funding in cognitive science, including this project, comes from federal sources supports this argument how. . .?"
The answer to that question, RSA, is simple: even a stopped clock is right twice a day. Federal dollars fuel federal rules. Federal rules, by their nature, are remote, and "one size fits all".
If you want something else, you have to limit the federal role. Will that have beneficial effects everywhere? Probably not. Will it produce more good than bad? I believe so.
Thinking that there's a perfect for all, Federal level solution is just nonsense. In software development, I like to call this "letting the perfect be the enemy of the good enough" - with the understanding that "the perfect" is unattainable.
but it's a bit bizarre how little effort we put into developing serious research-based pedagogical methods. You'd think this would be a major component of federal education policy, but it's really not.
I don't carry any water for NCLB, but this is an elementary failure to do one's homework.
The real problem is that educational research doesn't really rise to the standards of research in the hard sciences, and may not be able to.
So any outcome you want -- often for ideological reasons -- is probably supportable by 'the research'.
In any even, to paraphrase Neil Postman in The end of Education, a student given a sufficently powerful 'why' to learn is going to survive almost any 'how' to learn. It is the former that we as a country have not got sorted out.
L.A. Times: At L.A. school, Singapore math has added value It's about an L.A. inner city school that saw score soar (rhymes, don't it?) after using books from Singapore , which like a lot of countries in Asia and Eastern Europe use a system developed in the idyllic Soviet Union of the 1930s.
The official site is singaporebooks.com, and the cheapest place I found to buy them (I get weirdly obsessive about things sometimes) was called I think christianbooks.com, something like that.
singaporemath.com, that is, not singaporebooks.com
As mentioned by several others, the issue here is that (a) word problems require transformations in order to render them useful for extant equations, (b) it is difficult for children to create abstract, mathematical representations from word problems, and (c) transfer between word problems and equations is very difficult. That being said, I agree that Colleges of Education do little to promote rigorous, empirical research that leads to instructional innovations.
The research, although not conclusive, is very interesting. The basic idea, as I understand it, is that students learn to apply math best if they learn the generic general rule before trying to apply the rule to specific, concrete problems. The idea is not the moving train problems are bad, per se, but they have to be introduced at the right time.
From what I can gather, this is very good and valuable research. For somewhat mysterious reasons, we are doing a poor job in education, especially k-12 education. The solution won't be simple, and taking a scientific approach should be a major component. Although personal experience is a good first step, we have learned in the example of medicine, that "case studies" can be extremely misleading.
With the caveat of the previous statement, the conclusions of the research are consistent with my experience. I teach in a medical school. About 10 years ago case-based learning was the rage a and a number of med schools, including Harvard, jumped in whole hog. Things have cooled down a good bit. Learning basic principles through example has many problems, context-specific learning being one of them. Learning to apply abstract concepts to specific real-world situations is a difficult process, a process that needs explorations.
There isn't much good education research because all the good research shows:
A. Different teaching techniques only account for a modest part of the variance in learning outcomes compared to the family backgrounds of the students.
B. Much of the difference in family background is accounted for by race.
Thus, the quality of education research in modern America resembles the quality of astronomical research in Italy following Galileo's conviction.
The problem is that curriculum is increasingly created by individuals who've spent zero time in the classroom. Decisions are made based on educational theories, not on proven protocols.
Judy pretty much nailed it. Plus, teaching theories are typically fads informed by political/sociological beliefs. Actual facts about what works with real kids are not allowed to enter the mix. The thought behind No Child Left Behind was to inject so actual data into the mix -- but like most Federal "solutions" the implementation was screwed up.
But hey, political positions on both sides have been supported. Who cares what happens to the kids??? And the economy is even less important than that.
Hmm... if every parent got a voucher for, say, 80% of the per-student cost of the local public school, maybe lots of different, small schools would be established to compete for students, each with its own pedagogical methods? Sounds like a great way to provide different solutions for different types of kids.
80% of the per-student cost of public schools will pay exactly not one single private school's fees in the country. It'll be a nice gift to rich familes who are already sending their kids to private schools, though.
No, no, no, no, no, no, no.
Cognitively Guided Instruction. Look it up.
If my two daughters are any indication, it works.
"80% of the per-student cost of public schools will pay exactly not one single private school's fees in the country."
It would pay the tuitions at Catholic schools currently, but if every public school student's parent got a voucher, that dramatic increase in demand for alternative schools would lead to numerous new schools entering the market. It would be easy for educational entrepreneurs to profitably offer a quality education in a small school to a targeted population for 80% of the per-student cost of a public school.
The most obvious example of the insanity reigning in American education thinking, the No Child Left Behind act put together by President Bush and Senator Kennedy, in response to Bush's theory that minority students were being held back by "the soft bigotry of low expectations," mandates that every single student in American public schools who isn't clinically retarded would reach "Proficient" (the second highest level, above Basic, Below Basic, and Far Below Basic)in math and reading by 2014.
We're living in a country where to hold a position of responsibility in education, you have to, in effect, publicly proclaim that the sun goes around the earth, or you can be James Watsoned.
Is it any surprise, then, that rationality and evidence plays little role in schooling?
Steve Sailer:
It's just one school, of course, but the case described in the link I provided strikes me as contradicting your assertions.
Here it is, again.
For somewhat mysterious reasons, we are doing a poor job in education, especially k-12 education.
This is actually not completely correct. Ten years ago Gerald Bracey broke down scores by states and came up with this result from the 2nd International Math and Science Study
Top Scorers
Asian students, U.S. schools 287
Taiwan 285
Iowa 284
Top third of U.S. schools 284
Korea 283
Advantaged urban students, U.S. 283
Hungary 277
White students, U.S. schools 277
Bottom
Jordan 246
Mississippi 246
Hispanic students, U.S. schools 245
Bottom third of U.S. schools 240
Disadvantaged urban students 239
Black students, U.S. schools 236
I think it is fair to amend the quote to say the US is doing a poor job in k-12 education for poor minorities which, of course, was the impetus of NCLB.
To have an intelligent national conversation about this, we have to ask ourselves why this is. The right will argue that imposing a right-to-work school system with no tenure and publicly funded school choice is the answer for this population. We have such a system in Ohio - the charter school system is hardly a gem.
While schools that serve minority and poor populations are undoubtedly accountable, let's keep in mind that there is a larger responsibility toward the poor and minorities that our society and its government have largely abdicated. Until that is addressed, we are likely to see little improvement.
it's a bit bizarre how little effort we put into developing serious research-based pedagogical methods
I totally agree with that statement, but it is also important that people (children) understand that those "abstract equations" actually solve real-world problems. I ended up graduating college with a math major, and one of the questions that came up again and gain in the college math courses I took was "what is the real-world application of this?" It makes the bullshit somewhat more tolerable to wade though.
Matt, blame your fellow "Progressives." The teachers' unions don't want an approach which directs teachers to do specific things in a specific way, and Bill Ayers types want to use the classroom to teach "social justice."
It's quite clear that phonics is far better than "whole language," for example, but "progressive" educators fight it tooth and nail, at every opportunity.
Sluggo, I don't understand what those numbers mean.
In response to Fred, I think there are two basic reasons it's highly unlikely that schools would compete for federal voucher dollars by pursuing innovative research into improving pedagogy.
Reason #1 - there are many ways in which schools can try to gain an edge in a competition for students, and increasing educational quality is not necessarily the most cost-effective one. Private colleges also compete for student dollars, and lately a lot of that competition seems to be going into fancy dorms, athletic facilities, dining options, etc. - the kind of stuff that impresses students and their parents while on a tour, and has zero impact on educational quality. Information asymmetry, always an issue.
Reason #2 - even if a school did respond to competitive pressures by pursuing educational innovation, there'd be no reason that other schools couldn't simply copy any innovation (and we'd want the other schools to do so - offering intellectual property protection for an effective teaching methods is clearly not a good idea). Thus, there's no incentive for an individual school to devote resources to educational research, as their competitors could simply free-ride on any advancements.
In general, there are many areas in which the automatic assumption that "private competition will lead to more innovation and effectiveness" is justified, there are some areas in which that assumption is questionable, and there are some areas in which that assumption is obviously extremely flawed. I'd put education in the third category.
I'm very skeptical of these conclusions, and wonder how they measured the students' performance. If they measure how well the students who practices with lots of word problems did on a test that is written in an abstract way, that skews the result, likewise for the opposite.
I recall my experience as a teaching assistant for electrical engineering at UC Berkeley. The students were excellent at mathematics, but their knowledge of how to apply the math to the application was far more limited, so their strategy was to get the TAs to provide enough hints that they could write down an equation, and crunch on it to produce the result.
But it is only with word problems that a student learns how to apply mathematics to real life: subtraction tells us how much money we have left; division tells us how to share fairly, multiplication tells us how much it costs to buy 23 of something, and so on. If the student just learns x*y, they might be set up well to pass a standardized test, but the whole exercise is then just a waste of time.
Of course, my perspective is that of an electrical engineer; academic mathematicians might see it differently. Same with a teacher who's forced to teach to the test.
"Much of the difference in family background is accounted for by race."
In the sense that couples are usually of the same race, and thus so are their offspring, differences in family background are mostly accounted for by race. But I fail to see how this is relevant to the quality of education research. Rather, it appears to me to be a racist non sequitur.
I'm very skeptical of these conclusions, and wonder how they measured the students' performance. If they measure how well the students who practices with lots of word problems did on a test that is written in an abstract way, that skews the result, likewise for the opposite.
I recall my experience as a teaching assistant for electrical engineering at UC Berkeley. The students were excellent at mathematics, but their knowledge of how to apply the math to the application was far more limited, so their strategy was to get the TAs to provide enough hints that they could write down an equation, and crunch on it to produce the result.
But it is only with word problems that a student learns how to apply mathematics to real life: subtraction tells us how much money we have left; division tells us how to share fairly, multiplication tells us how much it costs to buy 23 of something, and so on. If the student just learns x*y, they might be set up well to pass a standardized test, but the whole exercise is then just a waste of time.
Of course, my perspective is that of an electrical engineer; academic mathematicians might see it differently. Same with a teacher who's forced to teach to the test.
Though he and I disagree a great deal about much of the surrounding concerns, Steve Sailer is absolutely right that any educational model that suggests that you can bring 100% of students to any set benchmark is total lunacy. Here's an amazing fact for all of you: not all students are equally smart, not all of them have equal social and familial conditions that would allow them to reach similar standards, not all of them are ever going to reach arbitrary standards. A one size fits all approach is precisely what needs to be avoided to have quality education. No educational fiction can succeed if it's predicated on the pleasant fantasy that everyone is equally able to achieve academic success.
Hmm... if every parent got a voucher for, say, 80% of the per-student cost of the local public school, maybe lots of different, small schools would be established to compete for students, each with its own pedagogical methods? Sounds like a great way to provide different solutions for different types of kids.
The truly amazing thing about the voucher movement is that it has neither a deductive case for why vouchers should improve education-- no one can explain how, exactly, vouchers would improve education, besides vague and ill-defined notions of "competition" or "the power of markets", especially considering pedagogical technique is identical between private and public school; nor is there empirical evidence to demonstrate that private schools do a better job of teaching. The selection bias in evaluating private schools is massive; they overwhelmingly exclude every type of student who has difficulty academically, and so of course have students with higher grades and test scores. (As I've said before, it's like only allowing students over a certain height into your schools, then bragging about how tall your student body is.) In the aggregate, private schools don't teach students better, the teach better students. (Of course, the higher functioning students at public schools are still higher functioning, and in fact the top students from public schools routinely outperform the top students from private schools.)
Individual students moving from public schools to private schools in voucher programs do not see significant gains in the limited data that we do have. Voucher proponents want us to gamble millions of dollars and the academic futures of our children on a system, when they are incapable of actually demonstrating the specific mechanism through which it is suppose to improve schooling, or of providing empirical evidence that it would be an improvement.
Sluggo, thanks for the data. And of, course, If you added in achievement test scores for, say, Mexico and Nigeria, you'd see that the rank order of performance in U.S. schools by different ethnic groups is pretty similar to the rank order of performance by their distant cousins in their ancestral homelands.
That's the basic reality that will get you James Watsoned out of the American education business for pointing it out.
So, our institutions focus on the practically impossible problem of eliminating the racial gaps in American students' performance, instead of the much more achievable goal of helping students improve to closer to their individual potentials.
That's why the education industry is so anti-rational, so swept up by constant fads, by the search for magic solutions that will square the circle. Because all thinking is devoted to making the sun go around the earth. You wouldn't want to end up like Galileo, would you?
I think there are two basic reasons it's highly unlikely that schools would compete for federal voucher dollars by pursuing innovative research into improving pedagogy. . . Thus, there's no incentive for an individual school to devote resources to educational research, as their competitors could simply free-ride on any advancements.
I agree entirely. So we have federally funded research on cognitive development that, in this case, compares pedagogical strategies without being invested in policy decisions. That's a good thing; otherwise (as you point out) there'd be little incentive to share results or even do the research in the first place.
Listen to Freddie. He knows what he's talking about.
For example, after badgering by the Bill Gates Foundation to "raise standards" so that every student in the Los Angeles Unified School District would be eligible to attend the University of California, the LAUSD, the nation's second largest school district, has now set a rule that beginning with 9th graders entering this September, in order to graduate from high school, they must pass not just Algebra I and Geometry, but also Algebra II.
The dropout rate in LAUSD is already about 55%. Algebra II is really, really hard for kids who lack the capacity for high level abstract thought. So, this new policy will condemn kids who will never in their lives need to use anything taught in Algebra II to go through their lives as "High School Dropouts" just because they weren't born smart enough to pass Algebra II.
This is an atrocity being perpetrated on innocent young people by the Great and Good of our society. But ... nobody gets it because if you want to remain a member of the Great and Good, you aren't allowed to publicly talk about the logical implications of central facts like differences in IQ.
Ha, Ha!
Why don't our public schools *also* require that all students pass an exam in Algebraic Topology to receive a H.S. Diploma? They're just refusing the see the mental ability in *all* children, not just the ones who eventually become CalTech math Ph.D's.
Still, that wouldn't be a sufficiently high level of expectation---we're still trapped in the depths of "soft bigotry".
What we really need is to require that each and every American H.S. student independently produce a consistent theory of fractional (not partial!!) derivatives in order to graduate. After all, if Richard Feynman managed to do that while still in high school, why can't every American student be held to the same high standard?
I know this is a regular hobby horse of mine, but to me, it's vital that this country redevelop industries that can provide a living wage to non-college educated workers. There was a time in this country where large numbers of workers who did not go to college could reasonably expect to get a job that provided them with a wage capable of raising a family; to a large degree, that has disappeared in the last 40 years. This isn't just about being smart, either-- not everyone has the kind of intelligence that thrives in the college environment, or is the kind of person who would most benefit a company in an "information economy" field. I furthermore think that this collapse in wages for uneducated workers has had profound effects on family and social structures in this country. Perhaps I'm overly optimistic, but I do think a lot of the social problems conservatives decry could be ameliorated by a regrowth in a living wage for non-college educated workers.
This post nicely illustrates why research in education is a mess (and it's not because dept. of education people are dummies).
First, as folks have very correctly pointed out, learning mathematical operations is different from learning how and when to use them. What is a student who has "learned math better" able to do - solve equations? set equations in real world situations? both? some other thing? You've got to answer those questions first, before you can find out which teaching methods are preferable.
Then, there's the problem of application. The study in this post was done on college students. It wouldn't be surprising to me to discover that college students are better at learning from abstract concepts than young children are - I can't imagine that it would be surprising to anybody. The findings may have something to say about how young children learn, but they also may not. More research is necessary before you could say. People (educators and laypeople alike) all too often jump on things studies like this to make big changes based on research that doesn't necessarily apply to the population being taught.
The same thing happens with scale as well as sample - a method proven effective in a remedial class of 8-10 students might be worse than useless in a regular class of 32 (and vice-versa). Or what works in one school might be a disaster district- or state-wide.
Hate to beat a dead horse, but everybody's ignoring me! Waahh!
I see sailer's made a whole bunch of comments, so I want to clarify that the assertions (ok, on second look, that should have been singular. I said that my article contradicted were "A. Different teaching techniques only account for a modest part of the variance in learning outcomes compared to the family backgrounds of the students."
Again, a school used the methods that are being used successfully in Asia, and quickly found a huge increase in its students' abilities. It seems to me that this is something worth exploring more.
Ask anybody who has transferred from a school in the Far East to America and they'll tell you that can't believe how "easy" the math courses here are...This one case study, at least, suggests that methodology may have something to do with this, and the obvious fact that family background, etc. are important to academic success isn't a reason to dismiss the importance of methodology.
BTW, the method I'm harping about was designed by mathematicians together with developmental psychologists. It seems that the this sort of collaboration should obviously be necessary to designing an effective method (unless, of course, you think there cannot possibly be such a thing).
I have more thoughts, but...um, there's other stuff I should be doing.
oh, and the reason I brought article up was that that the methodology as I understand it is consistent with the findings Matt brought up.
Anyway I've done a bit of NCLB tutoring (English mainly), and, while I've been able quickly teach kids skills they'd missed or struggled with in school (not, I think, due to any fault of the teachers, but just because tutoring can make this possible), but the "standards" that we are supposed to teach to seem completely arbitrary to me.
BTW, I was always placed in "honors" math classes when I was a kid (I imagine I would have been a bit below average in Taiwan...did I mention I did the slacker English teacher thing in Taiwan for 4 years?), but I tended to find word problems a bit confusing until I took algebra, which really opened the door for me.
If effectiveness were a real, consistently-utilized government measure, neither No Child Left behind nor abstinence education would ever have been considered, yet see what we have.
And again, the "Remember personal info" check box continues to be a cruel joke.
Joe Buck,
While I agree that Berkeley undergrads (physics, in my case) do have difficulty translating problems into a set of equations, my experience is that the better the student was at the pure math part, the better they were at the initial stage. That seems consistent with the study we're discussing.
In upper division classes, though, there was a different problem: many students refused to adopt the mathematical abstractions that make (say) E&M problems easier, even as they are willing to dive into extremely tedious and detailed calculations. In lower division, we gave them the false impression that underneath every physics problem is an easy math problem. Not so.
There are lots of ways to improve schools, but the entire field of k-12 education theory has come to dominated by charlatans because the basic fact -- that some kids are smarter than others -- is taboo. And the reason it's taboo is that when you try to objectively measure performance, you get massive "disparate outcomes" by race.
So, contemporary educational theory resembles 16th Century alchemy, with constant fads and high-priced gurus preaching contradictory techniques because the basic fact -- you can't turn lead into gold -- is not allowed to be mentioned.
The research, although not conclusive, is very interesting. The basic idea, as I understand it, is that students learn to apply math best if they learn the generic general rule before trying to apply the rule to specific, concrete problems. The idea is not the moving train problems are bad, per se, but they have to be introduced at the right time.
Yes. I believe that math instruction went through a phase where the idea was that kids would learn it more easily if they were taught through "real world examples," and then the abstract principles would become clear-- this was the idea that you could learn the abstract principles "by doing." As many other people have pointed out in this thread, it's generally considered to be better to learn the abstract principles by teaching the abstract principles and then learn to decompose practical problems into the abstract principle, which you can then solve.
I find myself reluctant to make sweeping statements about how math should be taught because I am good at learning math. It was straightforward to me to learn lots and lots of rules and equations and methods and figure out how to apply them. Most math scholars are going to come from the same background, and it is at least worth considering that people who aren't good at math might pick it up better if different teaching methods are used... but that requires extensive testing and drafting of protocols rather than just coming up with an "exciting idea" about how to teach math and convincing everyone to follow the methods, as a fad.
A few points:
Once people gave up on the idea of turning lead into gold, they found there was a tremendous amount you could methodically do with lead and gold and all the other elements. The age of scientific chemistry had begun, to the great benefit of humanity.
In educational theory, we're stuck in the Alchemy Age.
Went through this as my kids progressed through school. At some point math became a literary subject. Class dicussion of word problems centered around "problem solving strategies" forget the actual arithmetic operations to solve the problem. And it was verboeten for the teacher to point out that some strategies were shit because that would discourage children.
I think part of the ugly secret behind this, as pointed out by the mother of a girl in my son's 5th grade class about 8 years ago, was that by teaching math as a literary subject ... girls would do better relative to boys than they had been. I'm agnostic on the genetic vs social-cultural origin of US boys traditionally testing higher in math than US girls. But I would speculate that the agp has declined as (1) math has assumed more of a literary element of word problems vs. mathematical operations, and (2) as overall US scores have declined relative to other nations.
Sailer proves once again that he's a racist retard. Everyone--EVERYONE--recognizes that blacks and hispanics are way, way, behind in the U.S. education system. This isn't a taboo topic. Everyone also recognizes that some kids are smarter than other kids, we need a better educational farming system, etc. Everyone also recognizes that U.S. public education generally can be greatly improved, as seen by various other countries that have much more rigorous standards than the U.S. as well as lower dropout rates. Finally everyone recognizes that actually systematically improving public education is one of the hardest and most intractable problems that we face as a society.
Now what is arguably taboo is the assertion that blacks/hispanics are somehow innately stupider than whites. It's taboo because this assertion is both completely wrong and completely racist, and rightfully relegated to the shadowy corners of polite society (which, unfortunately, seems to include the Yglesias blog commenting section).
Beyond the broad findings in this article, I think there's a case to be made that there is something uniquely intimidating and problematic about the two trains passing example. It's become the quintessential convoluted word problem in pop culture, usually referenced with the punchline that nobody remembers how to solve it. And on top of that, isn't this "real world scenario" actually totally devoid of real world application? I mean, when has anyone ever REALLY needed to calculate when two trains traveling in opposite directions would pass? If so, for what purpose?
It's sort of like asking a word problem about how much time it would take to torture the location of a ticking bomb out of a terrorist. Just because it's a conceivable, non-abstract situation doesn't mean it actually happens.
Although I find some aspects of this thread way off-subject and offensive (ahem, Steve Sailor), others have kept some focus.
The article is interesting. It seems to be a real experiment that shows a clear but somewhat surprising result. As with virtually all good experiments, it has been done on a narrow sample (college students) with a restricted problem set (a certain type of math problem). But, from what I can tell, the results are clear. Is this the end of the story? No. And the authors don't contend that it is. But it is how a serious investigation should proceed. In the meantime, we have to develop public policy based on our best guess about what will work. Currently, best-guesses are that students need a mix of instruction of principal with real-world examples, rather than discovering principles on their own. But I strongly applaud the use of the scientific method, rather than selective statistics or wished beliefs to guide public policy.
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Sluggo presents interesting statistics about US student performance relative to other counties. I was referring to a variety of studies comparing performance on science tests. One, for example, is comparison of the performance of US students in advanced physics courses compared to students in other counties on physics exams, where US students do poorly. Young (can't find the reference) has suggested that science teaching in high school lags, but, as adults, science literacy is better in the US than other counties due to better teaching in US colleges. Our per-student-per year spending is MUCH higher in college than high school, and I think the results are in line with this. Overall, the style, mode and culture of k-12 education is radically different from college education. I feel the two groups have a lot to teach each other.
I agree wholeheartedly with gregor. It's amazing how many people get confused with those.
In an earlier post I said "young", when I meant "jon miller". Here is a NY Times article from 2005.
http://www.nytimes.com/2005/08/30/science/30profile.html
From the article:
". . .40 (t + 1) = 400 - 50t, where t is the travel time in hours of the second train. (The answer is below.)"
"The answer is below"?? I'm told Japanese newspapers assume a college level readership. Not that we'd want that, but come on, can't they assume their readers know 7th grade math?
matt: they could assume that NYT readers know how to do 9th grade math (only nerds like us do algebra in 7th grade), but I'm reminded of a geometry teacher who liked to quote the old saying "to assume makes an a$$ of you and me". I would guess that less than 50% of NYT readers could solve that problem accurately.
I teach and tutor a bit of math, and I see students blow simple algebra like this all the time. On top of student ignorance, consider all the adults who haven't looked at the stuff since 9th grade. Why would they remember how to do something they sort of knew how to do in high school?
cgaros, I can see people blowing it, after all I did do some TAing at Berkeley. Still, you don't see the Times littering their space with footnoted definitions of 9th grade vocabulary words. I say leave those few people who can't solve the problem, but are somehow still interested in the answer, to ask their friends how to solve it.
Not much use for algebra in daily life.
Another thing. Teaching methods are reseached endlessly. It's very diffficult to make the studies absoloutly rigerous becasue so much about school learning depends on the individual teacher and the make up of the class. Teaching is an art. One teacher could much better results that another using the same methods. The next year, becasue the pesonalities in the classroom have changed, the results could be worse with the same teacher and same method. That doesn't mean that over time, and in many diffferent classrooms, certain methods can be shown to be more effective.
PS. Steve Sailer go away. We've heard it all a thousand times before. We get it. Black people are stupid.
Thank you, Korha.
Steve, if the rest of the world realy were saying that everything boils down to nurture, then you might have a point. Except that your point is that everything is nature.
Bear with me here, but it seems you feel that everyone is afraid to agree with you because they fear being branded as a racist. In actuality it is because you're looking at disparate results across the races and concluding that this proves that different races are inherently more or less intelligent. This strikes most people as absurd because you're making a big leap in logic there. It is evident that through the exigencies of history, the various forces that determine a family's intellectual and social background, as well as how society treats the individual, have not been applied evenly across the races. Therefore a tremendous amount of the forces acting upon a person's station in life, including their performance in school and on intelligence tests, are social in nature, and that it is not solely one's genetic makeup or in-womb development that determine one's performance in these matters.
Isolating the nature vs. nurture effects has thus far proven intractable because there's no way to conduct controlled experiments on lifetime development of intellectual capabilities without raising people for years in completely isolated Skinner Boxes or something similar. And no, adoption or controlling for household income or the like is not sufficient because a person reacts to how they are perceived by society and society's expectations throughout their lives.
So yes, some people are smarter than others. Very few people would dispute this. Yes, blacks and Hispanics have low test scores. Making the leap to say that this is somehow genetically innate not only strikes people as absurd because of the racist conclusions, but also because in effect the implication is saying that the centuries of colonization, slavery, and racially-based power structures had no effect on people living today, that your entire lot in life depends on your genes, and that we can easily put people into categories of good and bad genes based a few external physical markers. That's what people find so offensive about your views.
Not much use for algebra in daily life.
The 'subprime' mess disagrees.
Just to expand on this subject, for all we know, maybe what Steve Sailer is saying is true. It might be. However, I don't believe we're anywhere near being able to prove that one way or another. Here is what we would have to be able to prove for it to be true:
1. "Intelligence" is a definable human quality measurable on a linear scale.
2. IQ tests accurately measure this quality.
3. Different populations perform differently on these tests to significantly significant degrees.
4. These differences persist under all controls for social influence, showing that the differences truly are genetic.
5. These different populations correspond accurately with our simplified model of "race."
I don't believe we're close to proving any of these five conditions necessary for theory to be true. I'm not saying you've been proven wrong, which you haven't. But what you're asserting is, under current knowledge, essentially unknowable.
For all we know, black people may be *more* innately intelligent than white people. The differences due to social effects, and the limits of our knowledge of what exactly defines "intelligence" and "race" create way too much variance for us to be able to make any kind of meaningful statements on innate differences in intelligence between races.
Adam Villani,
This post by Sailer may answer some of your questions. A couple of excerpts:
The size of the black-white racial gap in IQ has been studied for over 80 years. Of course, the answer is purely probabilistic. To say that blacks on average have lower IQs than Asians or whites is not to say that all blacks have lower IQs. In fact, about six million African-Americans possess higher IQs than the average white.The most comprehensive investigation of the size of the white-black IQ gap was carried out by Philip L. Roth of Clemson and colleagues in a 2001 article, "Ethnic Group Differences in Cognitive Ability in Employment and Educational Settings: A Meta-Analysis," in the academic journal Personnel Psychology.
They looked at 105 different studies covering 6,246,729 individuals and found an overall average difference between whites and blacks of 16.5 IQ points, or 1.1 standard deviations. The 95 percent confidence interval runs merely from 1.06 to 1.15 standard deviations (in other words, there is strong agreement among the 105 studies).
The black-white IQ gap is not the end of the story. I've been arguing for close to a decade, IQ tests probably do not measure well certain cognitive skills that blacks may tend to be better at than are whites and East Asians, such as improvisation. Life consists of trade-offs, so perhaps it's not startling that New Orleans, home to the great black improvisatory art form of jazz, did not display tremendous talent at planning ahead for this inevitable disaster.But, nevertheless, the IQ gap does matter.
You thought the IQ test has been officially discredited by somebody or other? Not according to the Supreme Court and the military.
In the 2002 case Atkins v. Virginia, the Supreme Court, in effect, abolished the death penalty for killers with IQs below 70. Liberals applauded. As Andrew Sullivan blogged:
"It's an article of faith among many liberals that I.Q. has no meaning, it's culturally constructed, and should never be used to judge people's intellectual ability. But suddenly, when I.Q. is the means by which to rescue retarded criminals on death row, I.Q. is just fine, thank you very much."
Many on the right as well claim, at least in public, that IQ is meaningless. As the rightist blogger Tacitus alleged:"I tested with a ridiculously high IQ as a child, and I was pretty proud of that till I got to the Army and found it didn't count for anything... we should not pretend [IQ tests] are an objective basis for science."
Well, go tell it to the military, which has been giving IQ tests to enlistment applicants since the First World War.
Almost nobody in the media is aware of the vast investment the U.S. military has made over the last 88 years in IQ testing of potential recruits, and the huge number of correlation studies they have done comparing soldiers' IQ with their actual performance. I was only barely aware of it myself until I spent hours last fall interviewing military psychometricians for my article showing that John F. Kerry scored a bit lower on his officer application IQ test than George W. Bush did. (This was the report that Tom Brokaw asked Kerry about on the NBC Nightly News.)
Because the U.S. military knows that bad things tend to happen to low IQ soldiers—and to their comrades who have the misfortune to be standing nearby—since 1991 only about one percent of new enlistees have IQs below the 30th percentile (i.e., an IQ of about 92). (See Table 2.8 in this Defense Department report.)
Last year, the Army announced that because of tribulations in meeting recruitment quotas due to the Iraq War, it would up its share of new soldiers scoring below the 30th percentile all the way to … 2 percent.
You know, I got Steve Sailer confused with "Petey," for some reason. Of course they've both developed a level of quasi-fame as ubiquitous blog commentatorizers, but I somehow forgot which one was the eugenicist, otherwise I wouldn't have entangled myself in verbal intercourse with the dude. Just felt like mentioning that.
Godoggo,
Thanks for the Singapore Math link. That's really very interesting, its depressing that every math expert the LA Times spoke to loved the approach but was skeptical that schools would adopt it. Apparently its too big a hassle for the teachers to be retrained to teach in a new way.
I agree with Freddie's point about the lack of blue collar opportunities. Likewise, Gene's speculation that how math is taught perhaps having disparate results by gender is probably right on. I just finished Leonard Sax's (best known for advocacy of single-sex schools) new book Boys Adrift-- Sax covers both of these topics in some detail. http://www.boysadrift.com/
And I would point out to Korha that people who use "retard" as an insult shouldn't throw stones.
Note: Fred's comment hadn't appeared when I wrote the previous.
Just a quick response:
1. I acknowledged in my post that blacks, as a larger population group, tend to score low on tests; while not explicitly stated previously, this does include IQ tests.
2. I never said that IQ had no meaning. I've scored pretty high on IQ tests myself, and think I'm pretty smart, and I believe this helped me score well on the tests. But I'm also aware that even if I take one of those tests today, I can find a lot to criticize on it as I'm taking it, noting ambiguities, cultural bias, etc. IQ tests do have value, but they're not rock-solid, unbiased measurements of pure intelligence, either.
3. We also don't really have much of a better gauge of intelligence now, so I don't doubt that the courts and military use them and I don't criticize them for doing so. But the fact that the tests can be useful measurements in no way means that they lack significant flaws. Moreover, the Supreme Court and the military may be very good at what they do, but their institutional success doesn't make them authorities on science.
4. I read the Sailer page you linked to, and while there are some interesting points (I hadn't been aware that the military set its bar at the 30th percentile, for example), it doesn't contain any information at all to help reach any kind of conclusions about how to separate genetic from societal differences in measured intelligence.
Incidentally, somebody made a snotty little remark about my "personal information." Now, I realize that for a lot of blog commenters this sort of thing is kind of like farting, but I just want to mention that if you've spent much time doing the slacker English teacher thing, it's very likely you've run into some really, really bright people. Drunken yobs, perhaps, but bright ones!
"I read the Sailer page you linked to, and while there are some interesting points (I hadn't been aware that the military set its bar at the 30th percentile, for example), it doesn't contain any information at all to help reach any kind of conclusions about how to separate genetic from societal differences in measured intelligence."
Adam,
Sailer doesn't claim to know exactly what percentage of the IQ gap is due to hereditary factors and what percent is due to environmental factors (although some researchers have speculated that the hereditary component is about 70%). He does anticipate your objection in this post, though:
A common stratagem, I've found, is to assume that IQ differences matter only if they are genetic in origin. Since no decent, civilized, right-thinking person could possibly believe that racial differences in IQ have any genetic basis, then racial and national differences in average IQ can't possibly exist.Except—they do exist.
And, as I will show that—no matter what their origin, whether in nature or nurture or both—these IQ gaps will continue to exist for many decades.
See the chart in the post showing the stability of average IQ scores of different races across decades.
You may also be interested in this statement by 52 leading experts on intelligence that was published in the WSJ at the time of the controversy over The Bell Curve. A few excerpts:
The Meaning and Measurement of Intelligence
1. Intelligence is a very general mental capability that, among other things, involves the ability to reason, plan, solve problems, think abstractly, comprehend complex ideas, learn quickly and learn from experience. It is not merely book learning, a narrow academic skill, or test-taking smarts. Rather, it reflects a broader and deeper capability for comprehending our surroundings--"catching on," "making sense" of things, or "figuring out" what to do.2. Intelligence, so defined, can be measured, and intelligence tests measure it well. They are among the most accurate (in technical terms, reliable and valid) of all psychological tests and assessments. They do not measure creativity, character personality, or other important differences among individuals, nor are they intended to.
5. Intelligence tests are not culturally biased against American blacks or other native-born, English-speaking peoples in the U.S. Rather, IQ scores predict equally accurately for all such Americans, regardless of race and social class. Individuals who do not understand English well can be given either a nonverbal test or one in their native language.
Practical Importance
9. IQ is strongly related, probably more so than any other single measurable human trait, to many important educational, occupational, economic, and social outcomes. Its relation to the welfare and performance of individuals is very strong in some arenas in life (education, military training), moderate but robust in others (social competence), and modest but consistent in others (law-abidingness). Whatever IQ tests measure, it is of great practical and social importance.10. A high IQ is an advantage in life because virtually all activities require some reasoning and decision-making. Conversely, a low IQ is often a disadvantage, especially in disorganized environments. Of course, a high IQ no more guarantees success than a low IQ guarantees failure in life. There are many exceptions, but the odds for success in our society greatly favor individuals with higher IQs.
Source and Stability of Within-Group Differences
14. Individuals differ in intelligence due to differences in both their environments and genetic heritage. Heritability estimates range from 0.4 to 0.8 (on a scale from 0 to 1), most thereby indicating that genetics plays a bigger role than does environment in creating IQ differences among individuals. (Heritability is the squared correlation of phenotype with genotype.) [snip]
If you educated the masses, they might catch on! We can't have that.
This study is a no-brainer. Contrary to multiple assertions made in the comments above, teachers are not all idiots. We do realize that the best way to teach skills is to have students practice them, and as such no algebra teacher worth her salt will teach symbolic manipulation "by using concrete examples."
Having said that, most state exams and the SAT WILL ask students "word problem" type questions, and so decoding them is another, separate (and much harder) skill that also must be taught.
I will also point out that though most of the comments here have taken the opportunity to bash public schoolteachers, the study focused on teaching methods the authors found widespread at colleges and universities.
This is in keeping with my experience--public schoolteachers know far more about teaching and learning AND are more current about research techniques than university teachers, who often have had no pedagogical training and simply lecture the way they remember they college teachers doing.
Emily, JK, and Gustav picked up on what Matt failed to: this study has NOTHING to do with K-12 teaching methods, and there ARE studies in K-12, despite what the article claimed. Read more, but... sigh. I suppose I shouldn't be surprised that only 3 of 75 comments here picked up on the fact that the setting was in college, not K-12.
Isn't the point of word problems to promote abstract thought and the application of math, and not to teach math per se? Obviously abstract thought and creativity is not something that is going to be able to be measured qualitatively, which is one argument against basing educational policy solely on measurable metrics.
Posted by How Insane Is John McCain? | April 26, 2008 3:11 PM