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A Math Paradox: The Widening Gap Between High School and College Math

36 points by gnosis 17y ago

tokenadult 17y ago
The book mentioned in this blog post is a pretty good book. I like another book http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773 http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773 for slightly more advanced math learners, including those I teach in the more experienced section of the math classes I teach in my town. Below is a set of quotations and links about math education I send out to parents of newly enrolled students in my classes: LINKS ABOUT LEARNING MATH AND OTHER SUBJECTS I was first introduced to a mathematician writing about how to teach math when a parent told me a decade ago about the article "Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education," http://www.aft.org/pubs-reports/american_educator/fall99/wu.pdf http://www.aft.org/pubs-reports/american_educator/fall99/wu.... by Professor Hung-hsi Wu. His writings have been very influential on my thinking about math education. A link that furthered my process of pondering how to teach mathematics better was Richard Askey's review of the book Knowing and Teaching Elementary Mathematics by Liping Ma. http://www.aft.org/pubs-reports/american_educator/fall99/amed1.pdf http://www.aft.org/pubs-reports/american_educator/fall99/ame... Another review of that excellent book http://www.ams.org/notices/199908/rev-howe.pdf http://www.ams.org/notices/199908/rev-howe.pdf is also food for thought. Professor John Stillwell writes, in the preface to his book Numbers and Geometry (New York: Springer-Verlag, 1998): "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/ http://www.amazon.com/gp/product/0387982892/ Richard Rusczyk, a champion math competitor in high school and now a publisher of math textbooks, among other ventures, has written an interesting article "The Calculus Trap": http://www.artofproblemsolving.com/Resources/AoPS_R_A_Calculus.php http://www.artofproblemsolving.com/Resources/AoPS_R_A_Calcul... I particularly like this article's statement, "If ever you are by far the best, or the most interested, student in a classroom, then you should find another classroom. Students of like interest and ability feed off of each other. They learn from each other; they challenge and inspire each other." which is one reason to encourage able math learners to learn together. Another good article about a broader rather than narrower mathematics education is http://www.math.sunysb.edu/~mustopa/thurston_edu.pdf http://www.math.sunysb.edu/~mustopa/thurston_edu.pdf by William Thurston, a Fields medalist. "Another problem is that precocious students get the idea that the reward is in being ‘ahead’ of others in the same age group, rather than in the quality of learning and thinking. With a lifetime to learn, this is a shortsighted attitude. By the time they are 25 or 30, they are judged not by precociousness but on the quality of work." I recently read an issue of the MAA Focus newsletter of the Mathematical Association of America, and in the newsletter I saw a link to an article by a math professor http://www.ams.org/notices/200502/fea-kenschaft.pdf http://www.ams.org/notices/200502/fea-kenschaft.pdf who is very concerned about the quality of elementary mathematics education in the United States. She provides many interesting examples of ways elementary teachers can think more mathematically about elementary mathematics and thus teach better. A very interesting article about overcoming learning challenges and thriving from them http://www.stanfordalumni.org/news/magazine/2007/marapr/features/dweck.html http://www.stanfordalumni.org/news/magazine/2007/marapr/feat... reports the work of psychologist Carol Dweck, who shows how "growth mindset" can make learners smarter.
gchpaco 17y ago
Something that burned a friend of mine is that college math is nothing like high school math even conceptually. She liked calculus and algebra and became a math major to discover that the math classes she was taking contained very little algebra and no calculus but required really abstract thinking that she had trouble with. I came at this from the other side, where I dislike mathematics and calculus but love abstract thinking.
- RiderOfGiraffes 17y ago
  The education system is like a low pass filter follwed by a high pass filter. The people who really enjoy school math, doing sums, applying equestions, get to university and find it's really hard and abstract and switch courses or drop out. The ones who might be really good at university style math get bored, disengaged, and don't do math at university, never suspecting that it's completely different. The problem is that school math is trying to accomplish two things at once, and they're at odds with each other. On the one hand, regular people who go on to regular jobs need numeracy and some facility to understand graphs, sums, and to follow a line of reasoning. On the other hand, anyone going on to the sciences needs a good background in (school-type) algebra, some geometry, a little calculus, and some statistics. Way more than most people need in everyday life. The latter group also needs a glimpse of some cool stuff to keep them interested. That's why in the UK there are math masterclasses, "Maths Inspiration", Further Maths, and other groups that try to get kids interested.
yurisagalov 17y ago
I have a younger sister who is currently going through the High School system in Canada, and as the son of a Teacher and an Engineer (both of Russian upbringing), education is a constant hot topic at family gatherings. While we are not experts on these topics, I'll share some of the opinions that we've reached. The education in North America (not necessarily the US alone, I live in Canada) is significantly lagging behind the rest of the world when it comes to mathematics. This should come to no surprise to anyone who has studied with anyone outside of this country. The Asian and Israeli kids (I am singling them out as I lived in a predominantly immigrant heavy Jewish/Asian neighborhood of Toronto, so I only have them to compare against) in my High School classes consistently scored higher than those born in Canada in all math courses, and this continued throughout my undergrad. This widening gap between high school and college math is most prominent here, as compared to Europe, the Middle East, and Asia. In my humble opinion, this is largely due to our policy of "no child left behind" when it comes to Education and, as a secondary cause, due to Teachers teaching the "Fear of Math" (I'll get to that later). As a society, we are constantly attempting to make sure that all our kids have an equal opportunity at success in life. Pursuing this endeavor, we have decided that rather than recognizing that everyone has different strengths and weaknesses, we should level the playing field. But how do we level this field? We simply adjust to the lowest common denominator. We want our children to score higher on the standardized tests, and when the scores are low, we simply make the tests simpler. This lowers the bar so that the weakest links in a particular subject area can keep up (it also allows politicians to get reelected and show how "great" of a job they're doing). I think this is inherently wrong. First, not everyone needs to be educated with the same level of intensity on a particular subject area. Second, you are preventing "super stars" in particular areas from ever developing since they will never get to grow to their full potential. Integration is too hard for some Grade 12's? Let's remove it altogether. My sister, who is a decade my junior, will never learn even derivatives in high school. Algebra and geometry? Gone. My dad's high school ended in grade 10, and by the end of Grade 10 they have learnt derivatives, integration, and the importance of the natural logarithm. How many of us in our 20's can say the same thing? Yes, the drop out rate was higher, and if you didn't push yourself the system sure as hell did not care, but this is also the system that sent a man who shares my first name into space. As a society, we have gone soft. We are babying our kids into believing that everything is roses. It made me furious when even as early as Grade 4, my sister would come home with a poor mark and say that it's ok because she "did her best". Really? What do kids really know about doing their best? of course if she truly did her best it would be ok and often when I see someone truly struggling to grasp a concept that they can't I feel both sympathy and have great respect for them, even if they fail. But at such a young age, there is no way that anyone knows what "their best" is, and this constant statement from her teachers is certainly not helping. In fact, I've never heard of "doing your best" until I immigrated to Canada. And then, then there's the teachers. Oh the teachers. Few people are as important to a society's success as the teachers of children. They are in a direct position to influence and train our minds at a very important stage. And yet, they are underpaid and disrespected, such that only "those who can't do, teach" has become an unfortunate reality in many school systems. This is of course (and luckily) not always true, as some of my closest friends have dreamt of being teachers for as long as I can remember prior to ever beginning their Bachelor's degrees. In a high school environment the situation is also not as terrible as in the elementary schools as the teachers have at least specialized in the subjects they are now teaching. However, in the elementary school, the situation is in fact worrisome. Class coordinators (those that you spend most of your day with), teach practically everything from Math, to English, to History, at least until grade 5, and sometimes even through middle school. As such, they are in a particularly strong position to put the fear of _____ into you. Generally, this happens to be the "Fear of Math". The teachers may in fact be great educators of English, and may have a very strong foundation in early childhood education, but quite often, these same teachers are the ones who were not great at math growing up, and so they subconsciously teach their own fear and lack of understanding to the kids. I've seen this first hand from kids who are afraid of math without having even tried, and it used to baffle me, until I talked to some of my Math teachers in high school and they explained this concept to me (yeah, I was a nerd growing up and still keep in touch with some of my teachers ;) I realize that I place a lot of blame and offer few solutions. The truth is, I don't really have many solutions, if I did I would be trying to implement them. I think the position of teacher should be highly respected -- as much as a Doctor and/or a Lawyer or even an Engineer. I think this can only be achieved if teachers are not as heavily underpaid as they are today... I think that the policy of "no child left behind", is fantastic, but it needs to incorporate the realization that not everyone is the same. Finally, I think that Arthur Benjamin's formula for changing math is something everyone should spend 3 minutes, watch, and think about: http://www.ted.com/talks/lang/eng/arthur_benjamin_s_formula_for_changing_math_education.html http://www.ted.com/talks/lang/eng/arthur_benjamin_s_formula_... I'd like to apologize to my sister for singling her out in this post, she is a great kid with a good head on her shoulder's. I'd also like to apologize to any teachers that I have inadvertently offended -- I have nothing but the greatest respect for you and your profession. I have been BLESSED by having teachers who did not tell me to just "do my best", but rather challenged me at every turn.
jey 17y ago
I think the "Lockhart's Lament" explains a good part of the source of this apparent paradox: http://www.maa.org/devlin/devlin_03_08.html http://www.maa.org/devlin/devlin_03_08.html For some reason they don't teach math in high school, but instead "math" class seems to focus on developing your memorization and robotic homework-completion skills.
- unalone 17y ago
  Anybody who hasn't read this paper half a dozen times should give it another read. It's a fascinating article that will change your outlook on education a little bit.
christopherolah 17y ago
I'm a high school student presently. The Math is trivial. The only way to learn anything is to read by myself and audit university courses. I'm actually trying to write a math textbook that will actually teach math. If you're interested email me at christopherolah.co@gmail.com and I'll email you a copy. Please include `math textbook' in you subject (I get lots of emails).
jamesbressi 17y ago
What is most clever about this article (besides the intense and long dialogue created by the subject matter which has resulted in some of the longest comments and replies I have seen for any article) is how his wife purchased that book for their child. Think about the reactions many parents would fall to by default: Lecturing their child that if they want to be "x" they have to do "x" in school. Or, not taking seriously what their child wants to do in the future or finding it too much effort to debate why they should be doing their school work because "kids, teenagers, heck even adults, change their minds frequently about what they want to do in the future and look at how many college students change their majors." That was an impressive solution to the issue his wife pursued and an example that will stay with me a very long time. +1 to the author of this post and his wife for parenting ingenuity.

36 points by gnosis 17y ago

The book mentioned in this blog post is a pretty good book. I like another book http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773 http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773 for slightly more advanced math learners, including those I teach in the more experienced section of the math classes I teach in my town. Below is a set of quotations and links about math education I send out to parents of newly enrolled students in my classes: LINKS ABOUT LEARNING MATH AND OTHER SUBJECTS I was first introduced to a mathematician writing about how to teach math when a parent told me a decade ago about the article "Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education," http://www.aft.org/pubs-reports/american_educator/fall99/wu.pdf http://www.aft.org/pubs-reports/american_educator/fall99/wu.... by Professor Hung-hsi Wu. His writings have been very influential on my thinking about math education. A link that furthered my process of pondering how to teach mathematics better was Richard Askey's review of the book Knowing and Teaching Elementary Mathematics by Liping Ma. http://www.aft.org/pubs-reports/american_educator/fall99/amed1.pdf http://www.aft.org/pubs-reports/american_educator/fall99/ame... Another review of that excellent book http://www.ams.org/notices/199908/rev-howe.pdf http://www.ams.org/notices/199908/rev-howe.pdf is also food for thought. Professor John Stillwell writes, in the preface to his book Numbers and Geometry (New York: Springer-Verlag, 1998): "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/ http://www.amazon.com/gp/product/0387982892/ Richard Rusczyk, a champion math competitor in high school and now a publisher of math textbooks, among other ventures, has written an interesting article "The Calculus Trap": http://www.artofproblemsolving.com/Resources/AoPS_R_A_Calculus.php http://www.artofproblemsolving.com/Resources/AoPS_R_A_Calcul... I particularly like this article's statement, "If ever you are by far the best, or the most interested, student in a classroom, then you should find another classroom. Students of like interest and ability feed off of each other. They learn from each other; they challenge and inspire each other." which is one reason to encourage able math learners to learn together. Another good article about a broader rather than narrower mathematics education is http://www.math.sunysb.edu/~mustopa/thurston_edu.pdf http://www.math.sunysb.edu/~mustopa/thurston_edu.pdf by William Thurston, a Fields medalist. "Another problem is that precocious students get the idea that the reward is in being ‘ahead’ of others in the same age group, rather than in the quality of learning and thinking. With a lifetime to learn, this is a shortsighted attitude. By the time they are 25 or 30, they are judged not by precociousness but on the quality of work." I recently read an issue of the MAA Focus newsletter of the Mathematical Association of America, and in the newsletter I saw a link to an article by a math professor http://www.ams.org/notices/200502/fea-kenschaft.pdf http://www.ams.org/notices/200502/fea-kenschaft.pdf who is very concerned about the quality of elementary mathematics education in the United States. She provides many interesting examples of ways elementary teachers can think more mathematically about elementary mathematics and thus teach better. A very interesting article about overcoming learning challenges and thriving from them http://www.stanfordalumni.org/news/magazine/2007/marapr/features/dweck.html http://www.stanfordalumni.org/news/magazine/2007/marapr/feat... reports the work of psychologist Carol Dweck, who shows how "growth mindset" can make learners smarter.

gchpaco 17y ago

Something that burned a friend of mine is that college math is nothing like high school math even conceptually. She liked calculus and algebra and became a math major to discover that the math classes she was taking contained very little algebra and no calculus but required really abstract thinking that she had trouble with. I came at this from the other side, where I dislike mathematics and calculus but love abstract thinking.

RiderOfGiraffes 17y ago
The education system is like a low pass filter follwed by a high pass filter. The people who really enjoy school math, doing sums, applying equestions, get to university and find it's really hard and abstract and switch courses or drop out. The ones who might be really good at university style math get bored, disengaged, and don't do math at university, never suspecting that it's completely different. The problem is that school math is trying to accomplish two things at once, and they're at odds with each other. On the one hand, regular people who go on to regular jobs need numeracy and some facility to understand graphs, sums, and to follow a line of reasoning. On the other hand, anyone going on to the sciences needs a good background in (school-type) algebra, some geometry, a little calculus, and some statistics. Way more than most people need in everyday life. The latter group also needs a glimpse of some cool stuff to keep them interested. That's why in the UK there are math masterclasses, "Maths Inspiration", Further Maths, and other groups that try to get kids interested.

yurisagalov 17y ago

I have a younger sister who is currently going through the High School system in Canada, and as the son of a Teacher and an Engineer (both of Russian upbringing), education is a constant hot topic at family gatherings. While we are not experts on these topics, I'll share some of the opinions that we've reached. The education in North America (not necessarily the US alone, I live in Canada) is significantly lagging behind the rest of the world when it comes to mathematics. This should come to no surprise to anyone who has studied with anyone outside of this country. The Asian and Israeli kids (I am singling them out as I lived in a predominantly immigrant heavy Jewish/Asian neighborhood of Toronto, so I only have them to compare against) in my High School classes consistently scored higher than those born in Canada in all math courses, and this continued throughout my undergrad. This widening gap between high school and college math is most prominent here, as compared to Europe, the Middle East, and Asia. In my humble opinion, this is largely due to our policy of "no child left behind" when it comes to Education and, as a secondary cause, due to Teachers teaching the "Fear of Math" (I'll get to that later). As a society, we are constantly attempting to make sure that all our kids have an equal opportunity at success in life. Pursuing this endeavor, we have decided that rather than recognizing that everyone has different strengths and weaknesses, we should level the playing field. But how do we level this field? We simply adjust to the lowest common denominator. We want our children to score higher on the standardized tests, and when the scores are low, we simply make the tests simpler. This lowers the bar so that the weakest links in a particular subject area can keep up (it also allows politicians to get reelected and show how "great" of a job they're doing). I think this is inherently wrong. First, not everyone needs to be educated with the same level of intensity on a particular subject area. Second, you are preventing "super stars" in particular areas from ever developing since they will never get to grow to their full potential. Integration is too hard for some Grade 12's? Let's remove it altogether. My sister, who is a decade my junior, will never learn even derivatives in high school. Algebra and geometry? Gone. My dad's high school ended in grade 10, and by the end of Grade 10 they have learnt derivatives, integration, and the importance of the natural logarithm. How many of us in our 20's can say the same thing? Yes, the drop out rate was higher, and if you didn't push yourself the system sure as hell did not care, but this is also the system that sent a man who shares my first name into space. As a society, we have gone soft. We are babying our kids into believing that everything is roses. It made me furious when even as early as Grade 4, my sister would come home with a poor mark and say that it's ok because she "did her best". Really? What do kids really know about doing their best? of course if she truly did her best it would be ok and often when I see someone truly struggling to grasp a concept that they can't I feel both sympathy and have great respect for them, even if they fail. But at such a young age, there is no way that anyone knows what "their best" is, and this constant statement from her teachers is certainly not helping. In fact, I've never heard of "doing your best" until I immigrated to Canada. And then, then there's the teachers. Oh the teachers. Few people are as important to a society's success as the teachers of children. They are in a direct position to influence and train our minds at a very important stage. And yet, they are underpaid and disrespected, such that only "those who can't do, teach" has become an unfortunate reality in many school systems. This is of course (and luckily) not always true, as some of my closest friends have dreamt of being teachers for as long as I can remember prior to ever beginning their Bachelor's degrees. In a high school environment the situation is also not as terrible as in the elementary schools as the teachers have at least specialized in the subjects they are now teaching. However, in the elementary school, the situation is in fact worrisome. Class coordinators (those that you spend most of your day with), teach practically everything from Math, to English, to History, at least until grade 5, and sometimes even through middle school. As such, they are in a particularly strong position to put the fear of _____ into you. Generally, this happens to be the "Fear of Math". The teachers may in fact be great educators of English, and may have a very strong foundation in early childhood education, but quite often, these same teachers are the ones who were not great at math growing up, and so they subconsciously teach their own fear and lack of understanding to the kids. I've seen this first hand from kids who are afraid of math without having even tried, and it used to baffle me, until I talked to some of my Math teachers in high school and they explained this concept to me (yeah, I was a nerd growing up and still keep in touch with some of my teachers ;) I realize that I place a lot of blame and offer few solutions. The truth is, I don't really have many solutions, if I did I would be trying to implement them. I think the position of teacher should be highly respected -- as much as a Doctor and/or a Lawyer or even an Engineer. I think this can only be achieved if teachers are not as heavily underpaid as they are today... I think that the policy of "no child left behind", is fantastic, but it needs to incorporate the realization that not everyone is the same. Finally, I think that Arthur Benjamin's formula for changing math is something everyone should spend 3 minutes, watch, and think about: http://www.ted.com/talks/lang/eng/arthur_benjamin_s_formula_for_changing_math_education.html http://www.ted.com/talks/lang/eng/arthur_benjamin_s_formula_... I'd like to apologize to my sister for singling her out in this post, she is a great kid with a good head on her shoulder's. I'd also like to apologize to any teachers that I have inadvertently offended -- I have nothing but the greatest respect for you and your profession. I have been BLESSED by having teachers who did not tell me to just "do my best", but rather challenged me at every turn.

jey 17y ago

I think the "Lockhart's Lament" explains a good part of the source of this apparent paradox: http://www.maa.org/devlin/devlin_03_08.html http://www.maa.org/devlin/devlin_03_08.html For some reason they don't teach math in high school, but instead "math" class seems to focus on developing your memorization and robotic homework-completion skills.

unalone 17y ago
Anybody who hasn't read this paper half a dozen times should give it another read. It's a fascinating article that will change your outlook on education a little bit.

christopherolah 17y ago

I'm a high school student presently. The Math is trivial. The only way to learn anything is to read by myself and audit university courses. I'm actually trying to write a math textbook that will actually teach math. If you're interested email me at christopherolah.co@gmail.com and I'll email you a copy. Please include `math textbook' in you subject (I get lots of emails).

jamesbressi 17y ago

What is most clever about this article (besides the intense and long dialogue created by the subject matter which has resulted in some of the longest comments and replies I have seen for any article) is how his wife purchased that book for their child. Think about the reactions many parents would fall to by default: Lecturing their child that if they want to be "x" they have to do "x" in school. Or, not taking seriously what their child wants to do in the future or finding it too much effort to debate why they should be doing their school work because "kids, teenagers, heck even adults, change their minds frequently about what they want to do in the future and look at how many college students change their majors." That was an impressive solution to the issue his wife pursued and an example that will stay with me a very long time. +1 to the author of this post and his wife for parenting ingenuity.