Unlike many members of the educational commentariat, Andrew Hacker, writing the the New York Times yesterday, is more concerned with finding a solution to mathematical under-performance than wringing his hands about the problem. In my book, this automatically makes him about five times more worth listening to than your ordinary educrat. Dr. Hacker suggests that the formal math standard in American schools is largely not useful either to most workers or to the man as citizen. He proposes that math courses should instead of the discipline of formal mathematics focus on logical and quantitative reasoning and application.
While he has a point - "useful" mathematical knowledge is far from as common as it should be - I think he misunderstands the real problem. He begins by citing the number of students failing and doing poorly across the United States in these traditional classes. He observes that most teachers are dedicated and competent. He concludes that the problem must therefore be the material. "Algebra is a stumbling block..."
As a teacher myself, I believe Hacker has conflated two problems. Many students, I think, find algebra difficult because they have not mastered arithmetic. I inherit seventh graders, half of whom do not know how many feet in a yard, and the other half are too unsure to volunteer the answer. Tenth grade students new to me have forgotten or never learned their perfect squares. Students at all levels are incapable of or unwilling to do unit conversions.
Algebra I would define loosely as the art of manipulating and finding the values of unknown numbers. Two critical ideas in carrying out algebraic operations (by which I mean mathematical manipulation performed on expressions - "clauses", if you will - with variables), can be taught simply and directly from basic arithmetic techniques.
The first is the concept of the variable itself. A student familiar with measurements and conversions can be introduced to the idea easily: he is used to answering questions like, "How many meters are in 2400 millimeters?" Arithmetically, we set the problem up in stages: find the starting amount, and then from the final units set up the conversion factors - which should be memorized by 4th or 5th grade but in any case are easy to look up. Algebraically, we introduce the idea of equivalence, and mathematical symbols as a language. The student already knows that, for instance, meters in "math" are m. Now he learns that "question words" are represented by a symbol: a box, a question mark, or an x. He learns that "are" and "equals" are (in basic algebra) equivalent. So we reach the algebraic statement "x m = 2400 mm". The conversion factor still needs to be reintroduced in its algebraic place, of course, and enough practice done to learn the methods.
The second technique which is an easy extrapolation from basic arithmetic is that of variable operations. Polynomials, which are (simple version) expressions with multiple variable powers, can be added, subtracted, and multiplied (and therefore exponentiated) almost exactly like ordinary numbers, but with the different powers indicating "place value". (Division, while also analogous, is slightly different and therefore harder.) So a student who knows why his arithmetic works the way it does, who has been taught the decimal system, will recognize "place value" and make the connection in most cases without significant difficulty.
In short, I believe the root cause of high school failure is far more likely to be inadequate elementary education than difficulty with the material.
While I think Hacker's reasoning as to cause is flawed, his main concern - is this worth teaching anyway? - is a question worth asking. I think the answer is still "yes", but with caveats. I have blogged before about the state of mathematics textbooks in the modern United States: torn between traditional American mathematics and the "unified" approach of much of the rest of the world, having to teach or re-teach concepts not taught successfully before high school, fascinated with fiddly properties at the expense of overview, and so forth.
Algebra or Geometry as disciplines, though, are branches of a formal mathematics which exists - as people realized thousands of years ago - not only for practical purposes but as a training tool for the mind and as an art. Algebra is valuable for much the same reason that a foreign language is useful, or music, or memory: to cultivate the human spirit in all its facilities. Certainly most people will not use hyperbolic equations or non-Euclidean axioms in their day-to-day life, any more than most people will use French or German. Certainly you can enjoy the original Hugo novels - you can also dabble in Newton.
The practical mathematics which Dr. Hacker champions is of course still valuable. It is found in applied form in physics, chemistry, economics - all things which we say should be normally studied as useful. But its basic tools can and should be provided before high school.
Showing posts with label education. Show all posts
Showing posts with label education. Show all posts
2012/07/29
2012/03/27
Some Thoughts on Mathematics
It is a well-rehearsed complaint that the United States is "falling behind" the rest of the world in technical studies - meaning mathematics and science. It is paradoxical, then, that the general United States curriculum features specific study of advanced mathematical topics - Algebra and Geometry. Sometimes requirements have even advanced up to basic calculus or at least its underlying principles (indignified with the moniker "pre-calculus"). Most other countries teach Mathematics, perhaps glorified with a grade number, throughout middle and high school.
Speculating on the explanation for this oddity, I would suspect something like the following happened in American education: Years ago when "high school" would have been considered fairly advanced education, algebra and trigonometry and other advanced math would reasonably have been studied, possibly as electives; essential math would have been learned earlier. Mathematics instruction likely suffered from the overall decline in American education over the (later part of the?) 20th century. The language problems, I suppose, are documented better because writers felt them more critically. Either the courses were never dropped in name, or people noticing the problems codified the "Algebra" and "Geometry" to be studied; however the rigor of the courses was slackened and the material diluted with the addition of other elements which were no longer being learned earlier - or which were part of newly expanding fields and felt to be "necessary".
As evidence, I have mainly the (I am tempted to say absurd) amount of review contained from year to year in the average modern US mathematics textbook series. Admittedly mathematics is a subject I have some affinity for, but in my estimation the material generally spread over the three years from pre-algebra to second-year algebra either could be condensed into at most two years; or should be taught earlier; or is (at least from a conceptual standpoint) superfluous to the subject at hand. An "Algebra II" class - I am speaking here from experience - by the end of the first month of classes can still be reviewing material theoretically learned up to three years before in "Pre-Algebra" (possibly even the first semester of that course) - linear equations. True, the more advanced course has more detail and harder problems; I am not convinced that excuses the state of affairs.
I do not think - at this point in my career, at least - that the problem is the (nominal) focus of American courses. The integrated approach adopted most famously by most Asian schools does have its advantages, mainly in maintaining a unity in the discipline; the focused study of a particular branch of a subject has off-setting advantages, mainly in ability to explain details rather than teaching by rote. In disadvantages, the "subject-based" approach of American textbooks does tend to create artificial distinctions of one thing from another that ought to be known as related; however, an integrated approach obscures the focus achieved by distinguishing between the parts of a field of study. Beyond that I am not qualified to comment - except to mention that more American textbooks seem to be actually using an essentially integrated approach, while maintaining their supposed subject matter in name only.
A more pressing problem is the study habits expected of students. When a conscientious American parent wants the student to master a subject, and is willing to sacrifice grades if necessary, all is well and good. When a lazy parent (to say nothing of the student or teachers) with "high expectations" wants the student to have an "A" but could care less about the subject, there is a problem - and the Asian parent determined that his child will have an A and prepared to expect that and make him work for it creates an advantage. (I dislike relying on overplayed stereotypes, and can say from personal experience that - which should be obvious to any observer of human nature - not every Asian parent is in this regard an "Asian parent". Some of them are quite prepared to look the other way on instance of, for instance, cheating; or even to berate teachers who attempt to discipline that behavior. On a general comparison, however, the stereotype does hold true for fairly clear cultural reasons. The comparison is therefore useful. Also, I know next to nothing of European education, which is the other reasonable comparison point.)
So much for the problem. What about a solution? One obvious comment is that the best method of improvement must be increased expectations, even demands, on the part of parents and schools. The effect of this simple change can be most clearly seen in modern America within the home- and classical-schooling community, where concerned parents created a demand - and have often been part of creating a supply - of improved, or at least diverse, curricula for language study both in English and the Classics. Contrast this with mathematics education, especially in the public schools, where the number of available curricula in print has been steadily decreasing. I believe there is a total of two courses remaining put out by large publishers, and one has much greater presence as best as I can ascertain. My knee-jerk reaction is to blame Federal government meddling with standards, together with a human inclination to follow the Next Greatest Thing, but I could be wrong - the point is that with no serious competition and a largely captive market, innovation and diversity seems to be quickly becoming a matter of who has prettier pictures.
My own suggestions are limited by unfamiliarity with lower elementary curricula. A makeshift solution I would propose as an upper school teacher would be to spend seventh (and maybe eighth?) grade focusing on practical math, starting wherever necessary and working up to whatever difficulty level is needed for scraping by (and hopefully more than scraping by) in life but without particular emphasis on unifying principles of "subjects"; the unifying principle of mathematics generally is (in my opinion - this would require an entire other essay) description and problem-solving. With that foundation in place, you have a fourteen-year-old student who is capable of doing things like working out a simple budget (say, not overspending his allowance), working out basic problems in compound interest, calculating the price of a room's carpet, or not drawing to an inside straight (for reasons other than every book ever written with a card game in it saying so, not that that is a bad reason).
A high school can then teach those subjects which will be either necessary to a future career (as an engineer, architect, or the like) as electives, and require those useful to the formation of a thoughtful citizen ("Let no one ignorant of geometry..."). The question "how much math?" is a fascinating one I am not prepared to answer in the general case. I am prepared to say that it would be easier to teach algebra thoroughly if a text did not interrupt - and perhaps need to interrupt - the fairly logical progression through the various variable functions with silliness about translations which should have been learned earlier, and bits of a trigonometry (for instance) which could either be learned later or cheerfully ignored. Every calculus textbook I ever remember seeing not only confined itself politely to a strictly logical presentation of the calculus, but made assumptions of what the student knows. If we imagine a calculus textbook written on the same principles as the average "Algebra 2", it would start by presenting methods to solve equations in one variable and would reteach the quadratic formula; in between types of derivatives there would be a discussion on graphing complex numbers. An actual calculus text teaches a subject; I am not sure what the algebra book is trying to do.
Speculating on the explanation for this oddity, I would suspect something like the following happened in American education: Years ago when "high school" would have been considered fairly advanced education, algebra and trigonometry and other advanced math would reasonably have been studied, possibly as electives; essential math would have been learned earlier. Mathematics instruction likely suffered from the overall decline in American education over the (later part of the?) 20th century. The language problems, I suppose, are documented better because writers felt them more critically. Either the courses were never dropped in name, or people noticing the problems codified the "Algebra" and "Geometry" to be studied; however the rigor of the courses was slackened and the material diluted with the addition of other elements which were no longer being learned earlier - or which were part of newly expanding fields and felt to be "necessary".
As evidence, I have mainly the (I am tempted to say absurd) amount of review contained from year to year in the average modern US mathematics textbook series. Admittedly mathematics is a subject I have some affinity for, but in my estimation the material generally spread over the three years from pre-algebra to second-year algebra either could be condensed into at most two years; or should be taught earlier; or is (at least from a conceptual standpoint) superfluous to the subject at hand. An "Algebra II" class - I am speaking here from experience - by the end of the first month of classes can still be reviewing material theoretically learned up to three years before in "Pre-Algebra" (possibly even the first semester of that course) - linear equations. True, the more advanced course has more detail and harder problems; I am not convinced that excuses the state of affairs.
I do not think - at this point in my career, at least - that the problem is the (nominal) focus of American courses. The integrated approach adopted most famously by most Asian schools does have its advantages, mainly in maintaining a unity in the discipline; the focused study of a particular branch of a subject has off-setting advantages, mainly in ability to explain details rather than teaching by rote. In disadvantages, the "subject-based" approach of American textbooks does tend to create artificial distinctions of one thing from another that ought to be known as related; however, an integrated approach obscures the focus achieved by distinguishing between the parts of a field of study. Beyond that I am not qualified to comment - except to mention that more American textbooks seem to be actually using an essentially integrated approach, while maintaining their supposed subject matter in name only.
A more pressing problem is the study habits expected of students. When a conscientious American parent wants the student to master a subject, and is willing to sacrifice grades if necessary, all is well and good. When a lazy parent (to say nothing of the student or teachers) with "high expectations" wants the student to have an "A" but could care less about the subject, there is a problem - and the Asian parent determined that his child will have an A and prepared to expect that and make him work for it creates an advantage. (I dislike relying on overplayed stereotypes, and can say from personal experience that - which should be obvious to any observer of human nature - not every Asian parent is in this regard an "Asian parent". Some of them are quite prepared to look the other way on instance of, for instance, cheating; or even to berate teachers who attempt to discipline that behavior. On a general comparison, however, the stereotype does hold true for fairly clear cultural reasons. The comparison is therefore useful. Also, I know next to nothing of European education, which is the other reasonable comparison point.)
So much for the problem. What about a solution? One obvious comment is that the best method of improvement must be increased expectations, even demands, on the part of parents and schools. The effect of this simple change can be most clearly seen in modern America within the home- and classical-schooling community, where concerned parents created a demand - and have often been part of creating a supply - of improved, or at least diverse, curricula for language study both in English and the Classics. Contrast this with mathematics education, especially in the public schools, where the number of available curricula in print has been steadily decreasing. I believe there is a total of two courses remaining put out by large publishers, and one has much greater presence as best as I can ascertain. My knee-jerk reaction is to blame Federal government meddling with standards, together with a human inclination to follow the Next Greatest Thing, but I could be wrong - the point is that with no serious competition and a largely captive market, innovation and diversity seems to be quickly becoming a matter of who has prettier pictures.
My own suggestions are limited by unfamiliarity with lower elementary curricula. A makeshift solution I would propose as an upper school teacher would be to spend seventh (and maybe eighth?) grade focusing on practical math, starting wherever necessary and working up to whatever difficulty level is needed for scraping by (and hopefully more than scraping by) in life but without particular emphasis on unifying principles of "subjects"; the unifying principle of mathematics generally is (in my opinion - this would require an entire other essay) description and problem-solving. With that foundation in place, you have a fourteen-year-old student who is capable of doing things like working out a simple budget (say, not overspending his allowance), working out basic problems in compound interest, calculating the price of a room's carpet, or not drawing to an inside straight (for reasons other than every book ever written with a card game in it saying so, not that that is a bad reason).
A high school can then teach those subjects which will be either necessary to a future career (as an engineer, architect, or the like) as electives, and require those useful to the formation of a thoughtful citizen ("Let no one ignorant of geometry..."). The question "how much math?" is a fascinating one I am not prepared to answer in the general case. I am prepared to say that it would be easier to teach algebra thoroughly if a text did not interrupt - and perhaps need to interrupt - the fairly logical progression through the various variable functions with silliness about translations which should have been learned earlier, and bits of a trigonometry (for instance) which could either be learned later or cheerfully ignored. Every calculus textbook I ever remember seeing not only confined itself politely to a strictly logical presentation of the calculus, but made assumptions of what the student knows. If we imagine a calculus textbook written on the same principles as the average "Algebra 2", it would start by presenting methods to solve equations in one variable and would reteach the quadratic formula; in between types of derivatives there would be a discussion on graphing complex numbers. An actual calculus text teaches a subject; I am not sure what the algebra book is trying to do.
2012/02/25
Comments on Something Directly Relevant to Me
Dick Cavett, who is an editorialist on the New York Times website, has written an article lambasting Santorum, for, of all things, homeschooling his children. There are many reasonable causes for objecting to Santorum (though I think not as many as there are to object to almost any other candidate), but a blanket dismissal of homeschooling is not one of them. I say this as a homeschooled student who is now a teacher.
Cavett's argument is really two-fold. In the first place, he obviously regards Santorum as deranged or at least deviant. Eight kids? Professing faithfulness to his church's teaching? Not looking like a president? Wearing sweater-vests? These are all things he cites as problems with Santorum, and by the time he is done with all of that Cavett ought to be out of a job, because - well, he admits himself that we all know making fun of a person's appearance is rude. But having the opinions he does, for Cavett homeschooling is clearly just a stick to beat Santorum with, or maybe Santorum is a stick to beat homeschooling with, because to Cavett the only conceivably legitimate teaching is that done by a professional. After all, he says,
There is a basic flaw here in his chain of reasoning. Certainly if teaching were the demanding, specialized thing he makes it out to be, then he might have a point. Unfortunately for his case, it is not. To take his claim back to front, rather than asking how parents can figure out how to teach it would be more accurate to ask how many parents do not teach their children anything. Ignoring school for the moment, who taught you to read, to rake leaves, to change a tire, to clean your room, to share with others, to use the right fork? Who coached your first soccer game, went out driving with you for your permit, showed you how to do laundry, cook pasta, take care of a dog (or the lawnmower), manage a checking account? (I have looked: that last definitely was not your math book these days.) Of course mom and dad do not usually have the expertise to teach all the nuanced multiplicity of disciplines the modern education wants to demand, but the old standbys: reading, writing, and arithmetic? They know those, and they can teach them. Teaching is a human thing.
Of course I am not saying they always do teach them. I have dealt with plenty of homeschooled students who cannot figure out a simple proportion and get lost in decimals. On the other hand, there are a fair number of students from a "regular" school who do no better. I would say that the craze for online classes seems to have produced some troubling results - but no more troubling than, say, the numbers of students who have trouble reading because they were faddishly protected from phonics. (And I would note, while I am at it, that online classes, like most things, can be done well or badly.)
The obvious retort to my above description is to ask, "What about dealing with all kinds of kids?" or, "What about the tricky subjects?" On the first point, I would have been inclined to admit that classroom management (incidentally not something pertinent to homeschooling) might be a carefully learned professional skill, were it not for the curious case that I find myself a teacher, and I believe a reasonably good one, having taken exactly zero classes or training sessions in any sort of educationism. I had what we call a liberal education, which is to say some of everything with a lot of books in different languages; I then took my bachelor's degree in a subject - mathematics, for the curious - and after some of the usual post-collegiate adventures, I have settled down to teach that subject to highschool students, with what seems to me very little difficulty fitting in. For those keeping score at home, let me sum up: I was a homeschooled student who have fit comfortably into the school environment in the profession Cavett considers so exclusive with absolutely no training. For those more inclined to the economic argument, there is a reason teachers' salaries are low (at least out on the market): almost anybody can do it.
As to the second: see above what I said about doing things well or badly. The conscientious homeschooler - and even Cavett would have to admit that whatever else Santorum is, he is conscientious - finds a tutor, or a class to share, or works cooperatively with others. And even if they did not - if the homeschooled child's education was limited to reading, writing, basic arithmetic, and nothing else at all, I posit that while limited it would still be sufficient to get along. How many of us really use a foreign language even monthly, or need to know the phylogenetic origin of the potatoes we eat?
The funny thing is, Cavett is forced in the end to admit that there are times homeschooling is a responsible choice, thus making this blogpost mostly irrelevant beyond pointing out that it might be responsible more often than he thinks. But he is absolutely appalled at homeschooling, in general - the truth comes out! - not really because of academic or even social concerns so much as such as for worldview reasons. Santorum's (or other) homeschooled children might not learn the definite truth that the universe has been around forever and life accidentally happened somehow, or that we should respect all other cultures (but not our own), or that the woman over there is "him" because she, I mean he, said so. These people have to learn that the logic stuff they are all so fond of only extends so far, and definitely not to reality!
Cavett's argument is really two-fold. In the first place, he obviously regards Santorum as deranged or at least deviant. Eight kids? Professing faithfulness to his church's teaching? Not looking like a president? Wearing sweater-vests? These are all things he cites as problems with Santorum, and by the time he is done with all of that Cavett ought to be out of a job, because - well, he admits himself that we all know making fun of a person's appearance is rude. But having the opinions he does, for Cavett homeschooling is clearly just a stick to beat Santorum with, or maybe Santorum is a stick to beat homeschooling with, because to Cavett the only conceivably legitimate teaching is that done by a professional. After all, he says,
Teaching is an art and a profession requiring years of training. Where did the idea come from that anybody can do it? How many parents can intuit how to do it?
There is a basic flaw here in his chain of reasoning. Certainly if teaching were the demanding, specialized thing he makes it out to be, then he might have a point. Unfortunately for his case, it is not. To take his claim back to front, rather than asking how parents can figure out how to teach it would be more accurate to ask how many parents do not teach their children anything. Ignoring school for the moment, who taught you to read, to rake leaves, to change a tire, to clean your room, to share with others, to use the right fork? Who coached your first soccer game, went out driving with you for your permit, showed you how to do laundry, cook pasta, take care of a dog (or the lawnmower), manage a checking account? (I have looked: that last definitely was not your math book these days.) Of course mom and dad do not usually have the expertise to teach all the nuanced multiplicity of disciplines the modern education wants to demand, but the old standbys: reading, writing, and arithmetic? They know those, and they can teach them. Teaching is a human thing.
Of course I am not saying they always do teach them. I have dealt with plenty of homeschooled students who cannot figure out a simple proportion and get lost in decimals. On the other hand, there are a fair number of students from a "regular" school who do no better. I would say that the craze for online classes seems to have produced some troubling results - but no more troubling than, say, the numbers of students who have trouble reading because they were faddishly protected from phonics. (And I would note, while I am at it, that online classes, like most things, can be done well or badly.)
The obvious retort to my above description is to ask, "What about dealing with all kinds of kids?" or, "What about the tricky subjects?" On the first point, I would have been inclined to admit that classroom management (incidentally not something pertinent to homeschooling) might be a carefully learned professional skill, were it not for the curious case that I find myself a teacher, and I believe a reasonably good one, having taken exactly zero classes or training sessions in any sort of educationism. I had what we call a liberal education, which is to say some of everything with a lot of books in different languages; I then took my bachelor's degree in a subject - mathematics, for the curious - and after some of the usual post-collegiate adventures, I have settled down to teach that subject to highschool students, with what seems to me very little difficulty fitting in. For those keeping score at home, let me sum up: I was a homeschooled student who have fit comfortably into the school environment in the profession Cavett considers so exclusive with absolutely no training. For those more inclined to the economic argument, there is a reason teachers' salaries are low (at least out on the market): almost anybody can do it.
As to the second: see above what I said about doing things well or badly. The conscientious homeschooler - and even Cavett would have to admit that whatever else Santorum is, he is conscientious - finds a tutor, or a class to share, or works cooperatively with others. And even if they did not - if the homeschooled child's education was limited to reading, writing, basic arithmetic, and nothing else at all, I posit that while limited it would still be sufficient to get along. How many of us really use a foreign language even monthly, or need to know the phylogenetic origin of the potatoes we eat?
The funny thing is, Cavett is forced in the end to admit that there are times homeschooling is a responsible choice, thus making this blogpost mostly irrelevant beyond pointing out that it might be responsible more often than he thinks. But he is absolutely appalled at homeschooling, in general - the truth comes out! - not really because of academic or even social concerns so much as such as for worldview reasons. Santorum's (or other) homeschooled children might not learn the definite truth that the universe has been around forever and life accidentally happened somehow, or that we should respect all other cultures (but not our own), or that the woman over there is "him" because she, I mean he, said so. These people have to learn that the logic stuff they are all so fond of only extends so far, and definitely not to reality!
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