I was linked today to this article by a Dr. Frank Quinn employed by Virginia Polytechnic University. Dr. Quinn outlines a philosophical split among mathematicians about a century ago and discusses its outcomes. He then writes briefly on how the advances made might be introduced into mathematics education with profit.
Dr. Quinn's major contention is that mathematics, understood in a modern sense, is entirely rule-based, with no demand that the rules match any physical results exactly. When applied educationally - as it commonly is in Geometry - this results in a system which while difficult to learn leaves room for entirely logical conclusions and is therefore in a sense "easy to master" if unlikely to ever be anything remotely close to fully explored.
The primary drawback to such an understanding is of course easy to spot: if it does not actually describe anything, what is the use of mathematics anyway? It becomes little more than a convoluted and peculiarly abstract art form, at least to the common understanding. I happen to be attracted to logical puzzles and purity of logic, but in teaching I have been made aware - sometimes more forcefully than others - that many students have no real interest in such things, and more importantly little use for them.
Now Dr. Quinn makes the fair observation that elementary education is still dominated by an earlier view of mathematics, using the teaching of fractions as an example. That being noted, though, I want to say that there remain at least two possibilities I can come up with:
First, he might be right. Perhaps at least one part of the reason for, say, falling educational results is that early education goes blithely on with outdated methods while the higher mathematics - starting in, say, high school, where the teachers teaching algebra and calculus have probably at least studied mathematics - is relying on the new framework. In which case, it would clearly be beneficial to unify the system and use the new and improved math throughout.
On the other hand, the older methods may persist (outside academia) because they are superior in a general sense. Mathematical proof may frown on analogy, but experience does not. In a related field, the fact is that elementary science remains mostly Newtonian even though it will yield errors of fact to varying degrees. Why? Accounting for relative or quantum effects is simply too difficult. When we are attempting to train nuclear physicists, then by all means we try get the science correct. But even an engineer rarely has to worry about such complications, far less a plumber or an accountant. Similarly, professional (which is at least largely to say, academic) mathematics is certainly very useful as its own thing, but it is hardly a field which everyone needs to be prepared for. If it turns out that Johnny and his friends can be taught to balance a checkbook more easily by considering half of an apple than by learning about the properties of the ratio of the integers 1 and 2, so much the worse for the integers.
In fact, I suspect - both because I am a cynical person, and from experience - that the old methods have been partly discarded while the new ones have not been fully adopted; or perhaps worse, both methods are attempted simultaneously and the result is confusion. Let me take a concrete example.
Is a half, after all, a part of a thing, or an arithmetically constructed ratio of two integers whose idea is (almost) entirely man-made? Even the latter definition is hardly rigorous and results, for instance, in mass confusion when fractional exponents - also known as roots - are introduced. It now becomes apparent that for these "higher" relations, it would be best if we could have gone back and made sure that the part stressed in the introduction to fractions was their reciprocal relationships - which is not even ratio, per se, but an application of ratio and multiplication. Meanwhile, three students divide two candy bars equally, and the complete abstraction is suddenly of limited value. "Three parts of a candy bar" is a simple idea. "A part of a candy bar which if you could have three candy bars each split into similar parts would make a whole candy bar" is a bit unwieldy, and attempting to ignore the actual candy in order to keep the definition manageable and the numbers pure is not helpful either.
Now, I confess I was not particularly aware there had been any radical changes in the understanding of what mathematics is and how it should be conducted before reading this article. I would have said that there has been a gradually progressing tightening of definitions, standards, and proofs over the centuries, and certainly "modern" - twentieth century - mathematics is more enamored of these closed system, definitionally-based, completely logical approaches than most, while also yielding some impressive constructions, many of which have even proved useful.
So, maybe for that reason, I fail to see how this "crisis" is to be resolved, or even that it is much of a crisis. I suspect the problem is not that Johnny and Susan cannot define fractions in a "mathematically correct" fashion, but that they are in ninth or tenth grade and cannot add them because a calculator does their thinking for them, or possibly for worse reasons.
In short I am hesitant about prescribing "modern mathematics" as a solution, for the very reason Dr. Quinn finds it so appealing: its lack of connection to the actual world. If we are going to set up "science" and "core mathematics" as rivals, I am more inclined to the scientific approach in anything serious. But even more importantly, Dr. Quinn's worries strike me as those of someone who would complain that each and every driver is not able to design an engine: of course it would be fantastic if it could be managed, but it is also a truly impossible request.
Showing posts with label mathematics. Show all posts
Showing posts with label mathematics. Show all posts
2012/07/29
"Practical Math"
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.
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.
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.
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