It's time for a favorite quotation about mathematics again:
"What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics.
". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the whole idea that mathematics is a rigorous, cumulative discipline in which each mathematician stands on the shoulders of giants.
"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety."
Stillwell demonstrates what he means about the interconnectedness and depth of "elementary" topics in the rest of his book, which is a delight to read and full of thought-provoking problems.
Wasn't there an attempt to do this in the 1960s? They tried to give students a better mathematical foundation for more advanced maths that ended up backfiring politically? Namely they were teaching rudimentary set and number theory to K-5 kids. This came at the expense of kids ability to multiply and divide, and when the press caught wind of this, the program was quickly shut down.
It seems like any attempt to restructure the math curriculum will be met with massive resistance from parents who were eminently satisfied with their own (likely poor quality) math education, and want their children to have the same.
which describes improved university courses for students who plan to become elementary teachers. That is a big emphasis in the United States now--international comparisons have shown that mathematics education of elementary pupils in the United States is lousy largely because the mathematical education of elementary teachers (at all levels) is lousy,
so United States mathematicians are trying to do something about that that is more effective than the 1960s attempt at "new math."
Yes, Stillwell's book, mostly aimed at mathematics students who will go on to be mathematicians rather than schoolteachers, is also an outcome of thinking about curriculum reform. He describes his motivation for writing his excellent book as attempting to understanding concepts of mathematics he still didn't understand after he earned his Ph.D. at MIT.
"mathematics education of elementary pupils in the United States is lousy largely because the mathematical education of elementary teachers (at all levels) is lousy"
This is something I've noticed in the time I spent working with teachers. It amazes me how many 4th-6th grade teachers don't understand fractions, but are trying to teach them to kids!
I mean, I have watched groups of elementary school teachers work on the same sort of problems they assign (fractions being one example) and struggle mightily. They understand the basic concept of what a fraction is, but many of them get bogged down in the algorithms because they don't really understand what the algorithms represent.
What is a "common denominator" beyond "the thing you put fractions over to be able to add them"? Many of the teachers I've worked with would struggle to explain this to students.
Comments
It's time for a favorite quotation about mathematics again:
"What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics.
". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the whole idea that mathematics is a rigorous, cumulative discipline in which each mathematician stands on the shoulders of giants.
"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety."
Stillwell demonstrates what he means about the interconnectedness and depth of "elementary" topics in the rest of his book, which is a delight to read and full of thought-provoking problems.
http://www.amazon.com/gp/product/0387982892/
Wasn't there an attempt to do this in the 1960s? They tried to give students a better mathematical foundation for more advanced maths that ended up backfiring politically? Namely they were teaching rudimentary set and number theory to K-5 kids. This came at the expense of kids ability to multiply and divide, and when the press caught wind of this, the program was quickly shut down.
It seems like any attempt to restructure the math curriculum will be met with massive resistance from parents who were eminently satisfied with their own (likely poor quality) math education, and want their children to have the same.
You are referring to "new math," which is the kind of math instruction I had when I was a child.
I am reading just now a very interesting book The Mathematics Pre-Service Teachers Need to Know
ftp://math.stanford.edu/pub/papers/milgram/FIE-book.pdf
(posted on
http://hub.mspnet.org/index.cfm/13083
which warns that it may be a slow download)
which describes improved university courses for students who plan to become elementary teachers. That is a big emphasis in the United States now--international comparisons have shown that mathematics education of elementary pupils in the United States is lousy largely because the mathematical education of elementary teachers (at all levels) is lousy,
http://www.ams.org/notices/200502/fea-kenschaft.pdf
so United States mathematicians are trying to do something about that that is more effective than the 1960s attempt at "new math."
Yes, Stillwell's book, mostly aimed at mathematics students who will go on to be mathematicians rather than schoolteachers, is also an outcome of thinking about curriculum reform. He describes his motivation for writing his excellent book as attempting to understanding concepts of mathematics he still didn't understand after he earned his Ph.D. at MIT.
"mathematics education of elementary pupils in the United States is lousy largely because the mathematical education of elementary teachers (at all levels) is lousy"
This is something I've noticed in the time I spent working with teachers. It amazes me how many 4th-6th grade teachers don't understand fractions, but are trying to teach them to kids!
What do you mean when you say they don't understand fractions?
I mean, I have watched groups of elementary school teachers work on the same sort of problems they assign (fractions being one example) and struggle mightily. They understand the basic concept of what a fraction is, but many of them get bogged down in the algorithms because they don't really understand what the algorithms represent.
What is a "common denominator" beyond "the thing you put fractions over to be able to add them"? Many of the teachers I've worked with would struggle to explain this to students.