Chad is not happy with my previous post where I consider that we shouldn't expect that everyone should be able to pass algebra conditional upon a deep understanding of the subject. First, let me state that my post was in part operating outside what I will call the "Cohen narrative." Rather, I wanted to interject the opinion that variation is a contingent fact of human history (otherwise, we wouldn't have been shaped by natural selection). I was attempting to offer that the alternatives are not black and white in that everyone should learn algebra or that everyone need not learn algebra. Granted, many of the observers qualified that any educated person needed to know algebra. I simply suggest that not everyone is educable to the same extent. If basic literacy and arithmetic are the standard for being educated, then everyone is probably educable. If algebra and geometry are the standard for being educated, I suspect a large minority are not educable. If basic diffential and integral calculus is the standard for being educated (18th century math) than only a small minority are educable, and excluding Matthew Yglesias (Harvard, philosophy, 2003 magna cum laude). If the ability to learn multiple languages with a reasonable level of fluency after initiation of puberty is the standard for being educable, then I must withdraw my name from the pool based upon induction.
This goes to another part of Chad's comment, where he states: it seems a little too Steven Pinker-- "our ape-like ancestors on the savannah didn't need algebra, so we never evolved the brain module for it.... Well, I don't have that much of a problem with Steven Pinker, though I disagree on a lot of details with him. But in the case of algebra Chad's point is pretty orthogonal, if you may excuse a mathematical analogy, to what I'm getting at. I don't think higher cognitive functions are really "hard wired" as tightly integrated modules which were subject to selection during our Pleistocene past. A bit off topic, I suspect humans are still subject to selection on cognitive traits, and that our phenotypic variation is in part a reflection of this. So the "evolution on the savannah" narrative is one I will forgo. But, as I alluded to in my reference to The Number Sense, it seems to me that abstract mathematics is a cultural innovation which is contingent upon a wide range of congitive faculties. When I state abstract mathematics, I am being precise because analog numeracy is a capacity which rats and pigeons exhibit. We do possess a gestalt ability to "count" a set of objects arrayed before us. But, this ability reaches its limit somewhere around 10 objects for the vast majority of humans. Though we can assess rough proportions and have a general sense of "amount," if you threw 58 marbles at front of any normal person they couldn't spit back a number with any level of accuracy. They could tell you about how many marbles they were seeing with rough consistency, but they would have to count sequentially in their mind's eye or verbally. Counting precisely (as opposed to comprehending the rough gist of proportions and amounts) requires extending our learning facilities, and coopts language and meta-representational capacities. I offered the example of individuals who suffer brain damange which imobilizes both their fingers and their ability to meta-represent sequences of numbers to suggest that even this cultural innovation is in part rooted in a pre-existent cognitive architecture.
This brings me back to variation: though humans are a cultural animal with an innate aptitude toward plasticity of behavior and thought, we exhibit universal and intraspecies cognitive biases. The French cognitive anthropologist Dan Sperber has been developing a theory over the past 25 years which posits that cultural ideas inhabit a cognitive fitness landscape where particular motifs and tendencies are strong favored and spread as mental epidemics. Sperber takes the basic thrust of memes and adds a large dosage of cognitive science and hypothesizes that the probability of a particular path of cultural development is conditional upon the weights of various initial mental parameters. This is premised upon the common evolutionary psychological idea that humans share a unified cognitive substrate, so universal preferences will result in particularly favored cultural conformations. To use a chemical analogy, the various conformations of cyclohexane are characterized by different levels of strain. Though the chair conformation is energetically favored, that does not imply that the other forms are not ever extent or possible, rather, the steric parameter is simply a strong biasing factor. Similarly, an evolutionary adaptive landscape might have a primary peak, but temporal and spatial vicissitudes may result in stabilization of a population's gene frequencies at a local fitness peak which is "sub-optimal."
What does this have to do with algebra? We need to move behind the common universal set of cognitive biases towards an appreciation of human variation in said biases. I suspect that abstract mathematics draws upon a wide range of cognitive faculties. I also suspect that virtuosity in this phenotype is contingent upon a particular combination of aptitudes, as well as the necessary developmental and social nurturing (ie, aptitude is irrelevant outside of a cultural matrix which allows the expression of a given trait). The various cognitive faculties and developmental and social factors can be thought of as random variables with particular states, and these states together contribute to a distribution of mathematical aptitude. In contrast, I believe that language is a relatively tightly integrated module (if not necessarily localized to one region of the brain), and so humans without basic language fluency are pathological. I believe that arithmetic is "close enough the code" to our innate numeracy, (which is just one of the various faculties that contribute to mathematical aptitude) that the number of random variables are limited, and that they aren't so random (so they skew toward a distribution which leaves almost all humans within the range of teachability). As mathematics becomes more "decoupled" from our cognitive substrate and necessarily coopts a far wider range of our hardware and necessitates more and more software input, the higher the probability becomes that you are shifting up the distribution of mathematical aptitude so that a greater and greater proprotion of the population falls below the threshold needed for educability in a particular domain.
Now, I will tack to Chad's example of new techniques in literary presentation. I don't know much about literature, though I am trying to read Jane Austen, so I am a righteous philistine. But, to go back to a mathematical analogy, I would hold that literature is of a different magnitude of a rather universal vector, our love of story telling. This is a strong cognitive bias which draws upon both our linguistic and social intelligence, both of which we are generally endowed with in spades. While I can not do subpar algebraic topology, I can write down a subpar story. In other words, the seed for this aptitude is there, I can conceive of the lands I have not seen. In contrast, I have a really difficult time even conceiving of the wilds of algebraic topology, "algebraic topology" are two words whose general outlines I can sea, but whose face is hidden. In short, mathematics is a vector that is exploring an entirely new space. I do not believe this space is always intuitive to people who attain fluency, even virtuosity, in mathematics. When I took introductory linear algebra one of the most bizarre moments was when I made a joke about "6-space being confused for 5-space." I will spare you the details because they escape me, but I don't know too many people who can conceive of dimensionalities beyond 3 (a physicist friend tells me that some physicists claim they can imagine higher dimensionalities). To go into the strange lands of abstract mathematics we need to depend on axioms and the systems we derive from them, but we are taking a cultural boat into uncharted waters, a vessel whose planks are constructed from a wide range of materials.
So finally, let me respond to the most hurtful charge Chad has cast my way:
I'm not really comfortable with the claim that the relatively late development of an idea indicates that it's somehow counter to our brain chemistry, and thus ok for people to not understand it.
First, to marginal issues, I think my prose above elucidates my position that it is not brain chemistry alone which I am speaking to. Of course, there are issues relating to necessary and sufficient conditions, and a particular brain chemistry, roughly interpreted, might be necessary to algebraic fluency. But the primary issue is that I am not engaging in the naturalistic fallacy, anymore than I would assert that because the chair conformation is energenetically favored it is the form of cyclohexane we must utilize. Rather, I am suggesting that to actually get everyone in our society to understand algebra, or be educated in a modern sense, will be as feasible as trapping noble gases in bucky balls.
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(Emphasis added)
I'm not sure what "virtuosity" or "aptitude" means in this context, since we're talking about basic skills in symbolic algebra. Does it take a virtuoso to understand that (a+b)/b is not the same as a? What kind of aptitude is required to grasp the notion of graphing a function?
Of course there are people who are (for whatever reason) better at mathematical reasoning than most, just as there are better writers, better orators, better jumpers, better swimmers, etc. It would not be reasonable to time a runner who has an aptitude for the 100 meter and then ask all people to meet that standard. Just as it would be unreasonable to ask people to even do a modicum of algebraic topology (though, interestingly, they kinda do when they distinguish between a basketball and a donut).
But I've not seen why it is that the concepts of algebra need fall into the category of "only a minority can do this". Surely most people are able to abstract from examples to patterns, and surely most people are able to apply rules (and if someone can't, he is an example of a pathological case, similar to someone who cannot learn a language). And high school algebra is, in large part, simply that: applying rules to abstracted patterns.
Do you have in mind particular algebraic skills or abilities expected of high school students that are things you would not expect that most people could do?
(Note that this question is distinct from asking about what graduation requirements ought to be. I agree that getting everyone to become proficient at high school algebra is asking too much, just as it would be asking too much for everyone to learn language.)
kipli, i used 'virtuosity' in a sloppy fashion. but, note: If algebra and geometry are the standard for being educated, I suspect a large minority are not educable. by implication i am offering that most ppl can grasp basic algebra. i would offer that 10-20% of the population can not just as a guess, basically those with an IQ below 85.
I'm not sure language is even a good analogy. Virtually every person can learn language, since virtually every person does. Not every person learns a langage other than his native language, of course, and learning a new language after childhood is more difficult than doing it, for example, as a toddler. How does learning algebra compare to this? Is there some limited set that every person learns and is therefore obviously capable of learning? Should we compare being a poet to knowing algebra? Is there some window after which learning algebra is more difficult? Or is algebra so different from spoken languages that making a comparison is close to meaningless?
Or is algebra so different from spoken languages that making a comparison is close to meaningless?
i don't think it is meaningless, but let me make clear that i used to the language analogy as a contrast, not to show a correspondence. while a 'language module' might be a plausible contention, a 'mathematics module' is not. i think that learning a foreign language as an adult is closer to learning abstract mathematics.
...learning a foreign language as an adult is closer to learning abstract mathematics.
I'm not quite sure about that, but I suppose you could argue that learning a new language as an adult requires a more conscious, rational process than learning a language as a child. I agree that a mathematics module is less plausible than a language module.
If algebra and geometry are the standard for being educated, I suspect a large minority are not educable.
It also depends on the country.
I'm not quite sure about that, but I suppose you could argue that learning a new language as an adult requires a more conscious, rational process than learning a language as a child.
well, i had a friend who was a classicist, and she posited that learning latin and greek was a lot like learning mathematics. i was skeptical, but, i think the general idea that it requires a lot of conscious processing is what i was trying to get across. to give a trivial example, bayesian probability "makes sense" if you are taught it, but the vast majority of humans (including scientists) really just don't intuitively expect the results it gives them.
I'm not sure that your suggestion that if differential calculus would exclude most people. When I went to highschool some ten years ago, we had to take differential calculus for the "advanced diploma" which was required for university. Furthermore, from how I remember things most people I knew took it. I think whether people succeed or fail (at least in secondary school classes) in most classes is a matter of preparation rather than a matter of some sort of innate ability. If I remember correctly, highschool courses required more learning by rote and any true deep understanding of the subject matter.
As an aside, as for the requirement for a second language, here in Canada you have to learn French to graduate. Indeed, I think you have to take French starting in primary school. That's not to say that everyone gets fluent in French but everyone is required to study it to some extent.
ss, most people socialize with individuals who around their own intelligence. in any case, 30 seconds of google: Males, by contrast, were more likely than females to have taken physics. The percentage of male graduates who took calculus increased from 6 to 12 percent and the percentage of female graduates who took calculus increased from 4 to 11 percent between 1982 and 2000. (high school i believe)
Didn't Dehaene say the French tried to teach axiomatic math to grade schoolers? And they say we're too literal-minded! There's an easy way to make math more accessible -- just lie to them using graspable analogies, and let them explore the nuances of the truth if they major in math. In high school physics, you're taught little to no relativity, and that's fine.
Ex.: functions -- holy hell, the standard method must be an import from France. "A function is a set of ordered pairs (x,y) s.t. ..." and they're already lost. Our mind has an intuitive grasp of what a "tool" is, so just use that instead. A function is a box that takes in something, typically changes it, and turns each thing into only one other thing. Things that would break the box aren't allowed in. Ta-da! Let them learn fine-grained stuff later.
As you said, even when the method is clear, sub-85 kids still won't get it. As an aside, the most g-loaded IQ test is Ravens Progressive Matrices, which basically tests pattern recognition. Only trouble? It's hard. So the assumption that "most people can abstract from examples to patterns" is true only when the patterns are obvious to, e.g., McDonald's cashiers.
If I gave 100 Harvard humanities majors the Hasse diagrams for the "is a subset of" relation defined on the power set of a set of cardinality N, w/ N = 0, 1, 2, 3 already provided to them -- could they give me the next diagram, that of a hypercube? Even assuming I gave them toothpicks & marshmellows to play w/ rather than draw in 2-D? Impossible for all but a few. Now, you're thinking this example is unfair, but it was posed for Harvard humanities majors, who probably have an avg IQ of 135 or 140.
The point is, any math problem that involves pattern abstraction is nearly unsolvable for even the straight-A students. That's why they only show up at the end of SAT Math sections (a de facto IQ test). And that's why we teach kids algorithms like long division and reverse-FOIL: so the bad pattern-recognizers can still traverse rough terrain.
razib,
Is that an artifact of what kids are capable of or of an educational system that doesn't put much emphasis on analytical thinking? I admit I'm not an expert in cognitive psychology or even educational theory but it strikes me that what and how kids are taught in primary and middle school most likely plays some role in how they perform in highschool. If such is the case then how do you separate the influence of innate ability from the influence of educational history on whether most kids could possibly succeed in some area of study. I mean, granted I'm sure that there are some limits and that at some point it'll be the case that the subject matter will be too difficult for some people whatever their educational history. However, I would be hesitant to hastily decide that material should be excluded from curricula based upon a belief that most people wouldn't be able to grasp it.
If such is the case then how do you separate the influence of innate ability from the influence of educational history on whether most kids could possibly succeed in some area of study.
you can't. you can offer conditional suppositions. unfortunately there is only one natural science like physics, and that is physics :)
However, I would be hesitant to hastily decide that material should be excluded from curricula based upon a belief that most people wouldn't be able to grasp it.
the issue i am pointing to is not exclusion of material, rather, it is the assumption that said material serves as a minimum base. this problem is not that big of a problem on the grand scale of things, but, i have younger siblings and friends who are educators, and some school districts are having issues with extremely high standards blind-siding students who are unprepared or never capable of the tasks asked of them. the overall point is that a diverse cirriculum is necessary and though we should encourage all to do their best and follow their interests, not everyone will be a doctor, engineer or scientist. i think the perceptions of human capacities can be biased by who you spend your time with. almost everyone i know rather well has some familiarity with calculus. the numbers above suggest that this is exceptional. the particular mix of genotype, development and environment, and their interactional components and correlations, are up for debate. the point is that these factors to come into the mix and a black-white dichotomy between all-children-can-learn and there-is-no-need-to-learn is not really realistic in my opinion.
A question that really interests me: How can we take advantage of our truly amazing childhood language ability (you have to have a few to see just how amazing it is!) to provide a foundation on which we can later build math ability. For example, 1st graders could easily learn addition and multiplication tables if taught as songs - maybe that could give them some mathematical intuition that they can build on later - as opposed to rational understanding that comes from, e.g., counting fingers? Maybe there's a way to harness language ability to teach some kind of pre-algebra, so that equations "speak" to us like sentences? I do feel, to some extant, that equations speak to me - but I was taught algebra in 7th grade, just when my childhood language ability was beginning to fade. Maybe it would be stronger if I had been taught earlier?
In other words, it seems to me that instead of teaching early childhood math as a rational faculty, maybe we should be focusing on teaching math as an intuitive faculty? Our rational-learning ability gets better as we get older, but our intuitive-learning ability gets worse. We should hone our intuitions as much as possible before time runs out.
Algebra is all about proportion. And as we know, it comes from the Arabic word al-jabr. Western Europe was quite lucky to acquire this cognitive technology developed in the Middle East. And I agree that only some people employ this essential element of proportion and balancing in their thinking.
Question: Did the Danish newspaper editor that intentionally solicited inflammatory cartoons directed at an ethnic minority employ this algebraic sense of proportion?
This, I'd like to know.
"A diverse curriculum ..." Maybe what we need is a rational education system. Maybe the debate should be backed up a little. Questions like whether algebra is needed by everyone are not questions that should be asked when laying out a curriculum at a high school. They are questions that should be asked when designing an educational system. We ought to decide what the minimum level of education should be for every citizen, because that decision is a civic one. That's what every student should have when he leaves high school. They we ought to decide how many different paths to life's work should exist and make curriculums for those. Some might involve algegra and some might not. Some might involve running a personal business and some might not.
Of course, in this country, a rational educational system is not very likely.
I guess I'm still unclear just what it is about high school algebra that makes it cognitively difficult, even when taught the 'best' way. I agree that the use of minimal standards for graduation is problematic, but that is really a secondary issue. The initial questions to ask center around what elements of mathematical thought students are able to grasp with appropriate teaching; next, we can consider what elements they should grasp.
Given that it is impossible to ask that everyone know something (unfortunately there are severely cognitively disabled people in the world), let's just worry about reaching 90% of the population.
Can 90% of the population be taught when multiplication is the appropriate operation? Is the concept of a variable beyond the abilities of 90% of the population? What about the concept of a function? Can 90% understand the connection between the algebraic and the graphical representations of a function? Can 90% recognize polynomial and exponential functions and distinguish their behaviors? Can 90% solve linear equations? Quadratic equations? Solve inequalities or work with the absolute value function?
What is it, beyond the broad category of 'algebra', that more than 10% of the population simply cannot seem to grasp?
Can 90% of the population be taught when multiplication is the appropriate operation? Is the concept of a variable beyond the abilities of 90% of the population? What about the concept of a function? Can 90% understand the connection between the algebraic and the graphical representations of a function? Can 90% recognize polynomial and exponential functions and distinguish their behaviors? Can 90% solve linear equations? Quadratic equations? Solve inequalities or work with the absolute value function?
i think most of these can be solved algorithmically, but is that true understanding? (whatever that means) i think 90% can understand multiplication. i think you are getting on thin ground beyond that point (frankly, many college educated people have problems with this, though is more ignorance than incomprehension)
razib,
the point is that these factors to come into the mix and a black-white dichotomy between all-children-can-learn and there-is-no-need-to-learn is not really realistic in my opinion.
I would agree with that argument. I do think that there should be some degree of flexibility regarding high school curricula. However, I think my concern is that one has to find a balance between flexibility addressing variation within the population and an education system that is too lax and allows people to finish school without a solid educationaly grounding.
I think my concern is that one has to find a balance between flexibility addressing variation within the population and an education system that is too lax and allows people to finish school without a solid educationaly grounding.
of course, this is about good intentions, so we can all agree. the reality is that many schools are allowing people to finish without solid educational grounding. i will give you an explicit example: a friend of mine employed a young woman who graduated summa with a degree in french and history who did not know her multiplication tables. my friend described her in terms of math as a 'pre-modern tribesperson.'
i will simply reiterate a tendency i believe i have seen on SB for the highly educated posters and bloggers to assume that everyone is like them, gifted with the same aptitudes and possessing the same interests and priorities. that ain't so. just because richard cohen is dumb doesn't mean that we need all our prescriptions to be inverted. responding to kipli, i really don't know what could be hard about basic algebra! but that's the point, just because i can't understand what is difficult about it doesn't mean that others don't have difficulty. some of this is due to poor didactic technique (or no didactic technique), but not all.
Here's another wrinkle: since blacks are on average 3 or 4 years behind whites, how do we work that in re: requirements? If the average white kid is expected to do algebra in 9th grade, he's got plenty of time to get it before graduation, whereas the average black kid can't even start trying to get it until time's almost up, which is unfair.
A naive solution would just be to track w/in the school, and the kids stay there until they meet requirements, whether that's at age 17 or 20. But imagine the tension that'd create -- "Why am I here at age 20 when they are free at 17? That's bullshit!" A better solution would be creating separate schools based on ability: lower-level for not-so-brights, mid-level for regulars, and upper-level for gifted. Lower students would start and end later (say, 9 to 21), and they wouldn't have to suffer walking constant reminders of their lower ability level.
This assumes no PC backlash once someone notices disproportionate representation of some groups in different levels, but PC is easier to get rid of than the instinctual hatred lower-level people have of higher-level people, and vice versa, that plagues high school existence.
my understanding is that berkeley high school (for example) engages in extensive 'tracking.'
razib,
i believe i have seen on SB for the highly educated posters and bloggers to assume that everyone is like them, gifted with the same aptitudes and possessing the same interests and priorities. that ain't so.
I agree, personally I'm terrible at math, despite having passable ability in formal logic. I'm quite aware that people have differing aptitudes. I'm just concerned that we're overestimating how stupid (for lack of a better word) the non-gifted are.
I'm just concerned that we're overestimating how stupid (for lack of a better word) the non-gifted are.
cost vs. benefit, do you think it is better that policy is designed which overestimates human aptitudes or underestimates them?
David Boxenhorn says:
In other words, it seems to me that instead of teaching early childhood math as a rational faculty, maybe we should be focusing on teaching math as an intuitive faculty? Our rational-learning ability gets better as we get older, but our intuitive-learning ability gets worse. We should hone our intuitions as much as possible before time runs out.
I can remember in Grade 3 when we all learned parts of our times tables by chanting them out. This was in about 1964. I am also told that in Hong Kong kids are expected to learn this quite early on.
So, I suspect that we can teach much of this stuff at an earlier age.
I'm a bit of a strange duck: I really like math, but I'm not very good at it. I found I have to master the basics hardcore to learn it. I think that people who are good at math can teach other people who are good at math, but for those of us who aren't, they seem to gloss over the basics in order to get to the more interesting things.