[math-fun] NYTimes: How to Fix Our Math Education
FYI -- There may be something to this. Abstraction too early leaves many (most?) students in the dust. Humans are pretty good at generalizing from the specific to the abstract on their own, but they aren't very good about learning in the abstract without any concrete examples. http://www.nytimes.com/2011/08/25/opinion/how-to-fix-our-math-education.html August 24, 2011 How to Fix Our Math Education By SOL GARFUNKEL and DAVID MUMFORD THERE is widespread alarm in the United States about the state of our math education. The anxiety can be traced to the poor performance of American students on various international tests, and it is now embodied in George W. BushÂs No Child Left Behind law, which requires public school students to pass standardized math tests by the year 2014 and punishes their schools or their teachers if they do not. All this worry, however, is based on the assumption that there is a single established body of mathematical skills that everyone needs to know to be prepared for 21st-century careers. This assumption is wrong. The truth is that different sets of math skills are useful for different careers, and our math education should be changed to reflect this fact. Today, American high schools offer a sequence of algebra, geometry, more algebra, pre-calculus and calculus (or a Âreform version in which these topics are interwoven). This has been codified by the Common Core State Standards, recently adopted by more than 40 states. This highly abstract curriculum is simply not the best way to prepare a vast majority of high school students for life. For instance, how often do most adults encounter a situation in which they need to solve a quadratic equation? Do they need to know what constitutes a Âgroup of transformations or a Âcomplex numberÂ? Of course professional mathematicians, physicists and engineers need to know all this, but most citizens would be better served by studying how mortgages are priced, how computers are programmed and how the statistical results of a medical trial are to be understood. A math curriculum that focused on real-life problems would still expose students to the abstract tools of mathematics, especially the manipulation of unknown quantities. But there is a world of difference between teaching Âpure math, with no context, and teaching relevant problems that will lead students to appreciate how a mathematical formula models and clarifies real-world situations. The former is how algebra courses currently proceed  introducing the mysterious variable x, which many students struggle to understand. By contrast, a contextual approach, in the style of all working scientists, would introduce formulas using abbreviations for simple quantities  for instance, EinsteinÂs famous equation E=mc2, where E stands for energy, m for mass and c for the speed of light. Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering. In the finance course, students would learn the exponential function, use formulas in spreadsheets and study the budgets of people, companies and governments. In the data course, students would gather their own data sets and learn how, in fields as diverse as sports and medicine, larger samples give better estimates of averages. In the basic engineering course, students would learn the workings of engines, sound waves, TV signals and computers. Science and math were originally discovered together, and they are best learned together now. Traditionalists will object that the standard curriculum teaches valuable abstract reasoning, even if the specific skills acquired are not immediately useful in later life. A generation ago, traditionalists were also arguing that studying Latin, though it had no practical application, helped students develop unique linguistic skills. We believe that studying applied math, like learning living languages, provides both useable knowledge and abstract skills. In math, what we need is Âquantitative literacy, the ability to make quantitative connections whenever life requires (as when we are confronted with conflicting medical test results but need to decide whether to undergo a further procedure) and Âmathematical modeling, the ability to move practically between everyday problems and mathematical formulations (as when we decide whether it is better to buy or lease a new car). Parents, state education boards and colleges have a real choice. The traditional high school math sequence is not the only road to mathematical competence. It is true that our students proficiency, measured by traditional standards, has fallen behind that of other countries students, but we believe that the best way for the United States to compete globally is to strive for universal quantitative literacy: teaching topics that make sense to all students and can be used by them throughout their lives. It is through real-life applications that mathematics emerged in the past, has flourished for centuries and connects to our culture now. Sol Garfunkel is the executive director of the Consortium for Mathematics and Its Applications. David Mumford is an emeritus professor of mathematics at Brown.
I think they're exactly right. The US math curriculum is great for its originally-designed purpose: being the first to send a man to the moon using a slide rule. Time to recognize that this is no longer the relevant objective function. --Michael On Thu, Aug 25, 2011 at 9:39 AM, Henry Baker <hbaker1@pipeline.com> wrote:
FYI -- There may be something to this. Abstraction too early leaves many (most?) students in the dust. Humans are pretty good at generalizing from the specific to the abstract on their own, but they aren't very good about learning in the abstract without any concrete examples.
http://www.nytimes.com/2011/08/25/opinion/how-to-fix-our-math-education.html
August 24, 2011
How to Fix Our Math Education
By SOL GARFUNKEL and DAVID MUMFORD
THERE is widespread alarm in the United States about the state of our math education. The anxiety can be traced to the poor performance of American students on various international tests, and it is now embodied in George W. Bush’s No Child Left Behind law, which requires public school students to pass standardized math tests by the year 2014 and punishes their schools or their teachers if they do not.
All this worry, however, is based on the assumption that there is a single established body of mathematical skills that everyone needs to know to be prepared for 21st-century careers. This assumption is wrong. The truth is that different sets of math skills are useful for different careers, and our math education should be changed to reflect this fact.
Today, American high schools offer a sequence of algebra, geometry, more algebra, pre-calculus and calculus (or a “reform” version in which these topics are interwoven). This has been codified by the Common Core State Standards, recently adopted by more than 40 states. This highly abstract curriculum is simply not the best way to prepare a vast majority of high school students for life.
For instance, how often do most adults encounter a situation in which they need to solve a quadratic equation? Do they need to know what constitutes a “group of transformations” or a “complex number”? Of course professional mathematicians, physicists and engineers need to know all this, but most citizens would be better served by studying how mortgages are priced, how computers are programmed and how the statistical results of a medical trial are to be understood.
A math curriculum that focused on real-life problems would still expose students to the abstract tools of mathematics, especially the manipulation of unknown quantities. But there is a world of difference between teaching “pure” math, with no context, and teaching relevant problems that will lead students to appreciate how a mathematical formula models and clarifies real-world situations. The former is how algebra courses currently proceed introducing the mysterious variable x, which many students struggle to understand. By contrast, a contextual approach, in the style of all working scientists, would introduce formulas using abbreviations for simple quantities for instance, Einstein’s famous equation E=mc2, where E stands for energy, m for mass and c for the speed of light.
Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering. In the finance course, students would learn the exponential function, use formulas in spreadsheets and study the budgets of people, companies and governments. In the data course, students would gather their own data sets and learn how, in fields as diverse as sports and medicine, larger samples give better estimates of averages. In the basic engineering course, students would learn the workings of engines, sound waves, TV signals and computers. Science and math were originally discovered together, and they are best learned together now.
Traditionalists will object that the standard curriculum teaches valuable abstract reasoning, even if the specific skills acquired are not immediately useful in later life. A generation ago, traditionalists were also arguing that studying Latin, though it had no practical application, helped students develop unique linguistic skills. We believe that studying applied math, like learning living languages, provides both useable knowledge and abstract skills.
In math, what we need is “quantitative literacy,” the ability to make quantitative connections whenever life requires (as when we are confronted with conflicting medical test results but need to decide whether to undergo a further procedure) and “mathematical modeling,” the ability to move practically between everyday problems and mathematical formulations (as when we decide whether it is better to buy or lease a new car).
Parents, state education boards and colleges have a real choice. The traditional high school math sequence is not the only road to mathematical competence. It is true that our students’ proficiency, measured by traditional standards, has fallen behind that of other countries’ students, but we believe that the best way for the United States to compete globally is to strive for universal quantitative literacy: teaching topics that make sense to all students and can be used by them throughout their lives.
It is through real-life applications that mathematics emerged in the past, has flourished for centuries and connects to our culture now.
Sol Garfunkel is the executive director of the Consortium for Mathematics and Its Applications. David Mumford is an emeritus professor of mathematics at Brown.
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-- Forewarned is worth an octopus in the bush.
Did Fermat develop his Little Theorem, and Euler extend it, in the knowledge that it would be used nowadays in RSA cryptography? Sincerely, Adam P. Goucher
I'm sure I irritate many people by a capacity for sweeping generalisation, but it does seem to me that this phenomenon needs to be viewed in a much wider context. AFAK, Ivan Ilyitch first enunciated the relevant principle [though I shouldn't be at all surprised to learn that it goes back to some ancient Greek]: like a river building dams and meanders, every organisation finally frustrates its original purpose. Fallible memory suggests this was formulated in the context of XIII-th century monastic communities: but plenty of examples surround us today, and nowhere better than in education. Formal maths teaching put me off the subject for years, as it has discouraged so many others. In this respect nothing has changed throughout my lifetime, despite many well-intentioned attempts to improve the situation. No solution to the problem can be more than temporary (at best) which does not recognise the fundamental administrative contradiction involved. Fred Lunnon
The dismal state of math and science public education shows that the current process is a failure, and its deterioration over several decadesindicates that the system is incapable of self-repair. The tragedy is that students who might have become mathematicians and scientists are frustrated and discouraged by a system that is geared towards the dumbest of the dumb. There is no "one size fits all" solution. Only by breaking the public education monopoly, and opening up education to competition will we discover the best processes. This sounds like a political rant; and it is. But math education is more about politics than math. -- Gene
The tragedy is that students who might have become mathematicians and scientists are frustrated and discouraged by a system that is geared towards the dumbest of the dumb. There is no "one size fits all" solution.
I have to agree with you there. Indeed, most maths tests are so routine that anyone capable of typing expressions into Wolfram Alpha could attain very high grades. I advocate the approach in Gerry Leversha's "Crossing the Bridge", which promotes a more articulate, thoughtful way of solving problems.
Only by breaking the public education monopoly, and opening up education to competition will we discover the best processes.
Hmm, that sounds like Darwinian evolution. Unsuccessful curricula die out, and the schools originally teaching them adopt successful curricula and alter them slightly... Sincerely, Adam P. Goucher
When a government spends around 8* more on "defence" than education is it any wonder that knives and guns are more highly valued than learning ? The UK education budget is considerably higher than the US one, in fact it's actually around double our defence budget. Of course that means that per capita we spend vastly more on education than the US plus we also have many pupils, in particular aged from 11-16, whose parent's choose "public schools" i.e. private education - though IMO these should be gradually phased out along with any educational institutions having religious bias of any variety - except for those specifically for training of religion/theology aimed at having such a vocation - and only then at 16+. On 25 Aug 2011, at 19:48, Adam P. Goucher wrote:
The tragedy is that students who might have become mathematicians and scientists are frustrated and discouraged by a system that is geared towards the dumbest of the dumb. There is no "one size fits all" solution.
I have to agree with you there. Indeed, most maths tests are so routine that anyone capable of typing expressions into Wolfram Alpha could attain very high grades. I advocate the approach in Gerry Leversha's "Crossing the Bridge", which promotes a more articulate, thoughtful way of solving problems.
Only by breaking the public education monopoly, and opening up education to competition will we discover the best processes.
Hmm, that sounds like Darwinian evolution. Unsuccessful curricula die out, and the schools originally teaching them adopt successful curricula and alter them slightly...
Sincerely,
Adam P. Goucher
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="David Makin" <makinmagic@tiscali.co.uk> When a government spends around 8* more on "defence" than education is it any wonder
Right, that's why, for example, Lesotho, which apparently spends 3rd highest per pupil in % of GDP on education (10.4%) but ranks only 138th in per capita military expenditures (2.8%)--a factor of about 3.7 in the other direction--leads the world in mathematical contributions.
these should be gradually phased out along with any educational institutions having religious bias of any variety
Right, that's why, for example, Utah (presumably more religious) apparently gets .21 SAT points for every dollar they spend, while Washington DC (presumably more secular) apparently only gets .07 SAT points per dollar. Along similar lines, I recall an article in the AMS Notices a few years back indicating that Chinese teacher performance was many times that of American, despite carrying roughly twice as dense class sizes. Personally, I'm not able to discern any positive correlation between throwing resources at this and achieving better results. It appears something else appears to be a more dominant influence. And finally there's Garfunkel and Mumford's position "All this worry, however, is based on the assumption that there is a single established body of mathematical skills that everyone needs to know[...]. This assumption is wrong." Surely then, spending more on something that's not broken won't fix anything, right? I do agree we're drifting further afield from any fun factor in this thread, so I'll shut up now.
David Makin: When a government spends around 8* more on "defence" than education is it any wonder that knives and guns are more highly valued than learning ? The UK education budget is considerably higher than the US one, in fact it's actually around double our defence budget. Of course that means that per capita we spend vastly more on education than the US plus we also have many pupils, in particular aged from 11-16, whose parent's choose "public schools" i.e. private education - though IMO these should be gradually phased out along with any educational institutions having religious bias of any variety - except for those specifically for training of religion/theology aimed at having such a vocation - and only then at 16+. -------------------------------------------------------- I don't quite understand what you mean by ""public schools" i.e. private education". Do you mean publicly funded, privately operated schooling? In the UK, is this an option available only to children at least 11 year old? The most extreme case of enforced educational indoctrination that I'm aware of in a not overtly communist country is in Germany. Home schooling is illegal there. A German home schooling family was granted political asylum in the U.S. because they faced persecution (e.g. having their children seized). [ http://www.nytimes.com/2010/03/01/us/01homeschool.html ] -- Gene
Gene Salamin wrote:
I don't quite understand what you mean by ""public schools" i.e. private education". Do you mean publicly funded, privately operated schooling?
No. The UK has some rather bizarre terminology. US "public school" : UK "state school" US "private school": UK "private school" *and* "public school". The exact meaning of "public school" isn't perfectly clear. Sometimes it seems to mean "old, expensive, socially prestigious private school". Sometimes it means any non-state-funded school. (The term "independent school" is also used and may in fact now be the usual UK term for non-state-funded schools.) According to Wikipedia, the terminology dates to 1868 and derives from these schools' openness to any pupils whose parents are willing to pay the fees. The OED, however, has citations going back to the 16th century and suggests that the distinction was that "private schools" are run for the personal benefit of their owners, whereas "public schools" are charitable institutions run for the benefit of the public (despite receiving fees). Neither distinction has much to do with how the terms are used today. -- g
Quoting "Adam P. Goucher" <apgoucher@gmx.com>:
The tragedy is that students who might have become mathematicians and scientists are frustrated and discouraged by a system that is geared towards the dumbest of the dumb. There is no "one size fits all" solution.
I have to agree with you there. Indeed, most maths tests are so routine that anyone capable of typing expressions into Wolfram Alpha could attain very high grades. I advocate the approach in Gerry Leversha's "Crossing the Bridge", which promotes a more articulate, thoughtful way of solving problems. I will hypothesize that mathematical ability is the least understood aspect of human psychology, and perhaps it is the most complicated part. I would guess there is huge variation in which parts of the brain are used for the operations. It is amazing that it can be taught at all.
I'd like to see more research on how people actually learn and do math in all its varieties. I have no confidence in any particular curriculum or methods. I don't even believe that the evaluation methods make any sense. Hilarie
Only by breaking the public education monopoly, and opening up education to competition will we discover the best processes.
Hmm, that sounds like Darwinian evolution. Unsuccessful curricula die out, and the schools originally teaching them adopt successful curricula and alter them slightly...
Sincerely,
Adam P. Goucher _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
It has been known, although apparently not well known, since antiquity that _spatial memory_ is incredibly powerful. By utilizing the _visual cortex_, the human brain can memorize a lot more than most people realize. In retrospect, this should be obvious to anyone who has ever been hiking in the woods very much. Due to evolutionary pressure (presumably), humans (and many animals) can relatively effortlessly store & remember an enormous number of bits about their location. http://en.wikipedia.org/wiki/Method_of_loci It is for reasons like this that it is often easier to remember _very specific facts_ and the rule to generalize from them, instead of a more general rule. I know it sounds crazy, but if I listen to a program on the radio while driving & later recall the information discussed on the program, I often recall where I was on the road at the precise time that information was discussed. At 12:47 PM 8/25/2011, hilarie@xmission.com wrote:
I will hypothesize that mathematical ability is the least understood aspect of human psychology, and perhaps it is the most complicated part. I would guess there is huge variation in which parts of the brain are used for the operations. It is amazing that it can be taught at all.
I'd like to see more research on how people actually learn and do math in all its varieties. I have no confidence in any particular curriculum or methods. I don't even believe that the evaluation methods make any sense.
Hilarie
Perhaps this is a good argument for introducing study of the abacus into the maths curriculum. Calculation with the abacus uses the "visiospatial" hemisphere of the brain, unlike calculation with pen and paper, which uses the "linguistic" hemisphere. Given the links between memory and calculation, if spatial memory is an untapped resource maybe spatial calculation is too. When I was researching this for my book, I failed to find anyone doing serious research into the educational benefits of the abacus. Recently, Michael Frank at Stanford has been looking into it however: http://news.stanford.edu/news/2011/august/abacus-mental-math-080311.html On 26 Aug 2011, at 15:19, Henry Baker wrote:
It has been known, although apparently not well known, since antiquity that _spatial memory_ is incredibly powerful. By utilizing the _visual cortex_, the human brain can memorize a lot more than most people realize.
In retrospect, this should be obvious to anyone who has ever been hiking in the woods very much. Due to evolutionary pressure (presumably), humans (and many animals) can relatively effortlessly store & remember an enormous number of bits about their location.
http://en.wikipedia.org/wiki/Method_of_loci
It is for reasons like this that it is often easier to remember _very specific facts_ and the rule to generalize from them, instead of a more general rule.
I know it sounds crazy, but if I listen to a program on the radio while driving & later recall the information discussed on the program, I often recall where I was on the road at the precise time that information was discussed.
At 12:47 PM 8/25/2011, hilarie@xmission.com wrote:
I will hypothesize that mathematical ability is the least understood aspect of human psychology, and perhaps it is the most complicated part. I would guess there is huge variation in which parts of the brain are used for the operations. It is amazing that it can be taught at all.
I'd like to see more research on how people actually learn and do math in all its varieties. I have no confidence in any particular curriculum or methods. I don't even believe that the evaluation methods make any sense.
Hilarie
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="Eugene Salamin" <gene_salamin@yahoo.com> The dismal state of math and science public education shows that the current process is a failure,
Not to make too light of a serious topic, but I can't resist noting that we've recently offered (comments welcome!) a tool to address "the dismal state of math", namely "dismal arithmetic": http://arxiv.org/abs/1107.1130 (which BTW cites contributions our proprietor RCS made on this very list). -------- And, a very small suggestion: why don't we introduce kids to partitions right after they learn how to add?
Did really none of the replies mention FUN? Math eduction (school level) suffers brutally from the style of teaching that implicitly suggests that nothing could be farther away from math than fun. People are horrified by math because in school they learned that it is something utterly dreadful. Huge surprise they are neither very good at it nor much inclined to learn more? Big fail for the teachers, they _are_ to blame. This is from my (German) perspective where teachers do have a decent salary and working conditions. If you have to mention politics make sure it's similar for the place you are referring to (regarding cash, conditions, and perspective) before anything else. Cheers, jj, now a teacher (post school, students of engineering)
Joerg, Yes Indeed. By contrast millions of people voluntarily (and with relish) spend hours each week doing Sudoku's. We would see this as being an example of mathematical reasoning. Invariably when I meet someone who's not in a scientific field, and I tell them that I'm a mathematician, I see a look of panic in their face, and I hear "Math was my worst subject". It's clear that the experience of being taught mathematics is traumatic for most people. It would be better if we structured mathematics as a general problem solving experience -- for example look at the introduction of Polya's "How to solve it". The general public needs to learn logic and reasoning skills. Historically almost all mathematical skills have evolved as part of *natural sciences* -- necessary skills that one needs to use in the context of general problem solving. Most word problems in algebra and calculus are worse than stupid. An imperfect analogy is physical education. At least when I was in school (and I hope that things have gotten better) -- we spent most of the time doing calisthenics (which nobody really liked -- but we were assured were necessary. Most math skills are taught like that). The rest of the time was spent in *real* physical activities -- like playing various sports -- which would be enjoyable, but only after acquiring a certain level of skill. But the real problem that I saw here, is that the instructors (who were invariably ex star athletes themselves, or in my case in high school an ex-Marine drill sargeant) catered toward the students who were already skilled in the sports, and never tried to motivate or help the others (sound familiar). Victor On Sat, Aug 27, 2011 at 1:41 PM, Joerg Arndt <arndt@jjj.de> wrote:
Did really none of the replies mention FUN?
Math eduction (school level) suffers brutally from the style of teaching that implicitly suggests that nothing could be farther away from math than fun.
People are horrified by math because in school they learned that it is something utterly dreadful.
Huge surprise they are neither very good at it nor much inclined to learn more?
Big fail for the teachers, they _are_ to blame.
This is from my (German) perspective where teachers do have a decent salary and working conditions. If you have to mention politics make sure it's similar for the place you are referring to (regarding cash, conditions, and perspective) before anything else.
Cheers, jj, now a teacher (post school, students of engineering)
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Wow, that brought back memories! My high school gym teacher was an ex-Israeli gymnast. (But that was only part of why I hated gym; being my first class, at 8? in the morning after an hour commute to school, didn't help.) I am also reminded of a math problem set my daughter got in elementary school. One of the problems started "19 people compete in a 3-legged race..." The "correct" solution was to mindlessly multiply the number of people by the length of the race. (And that wasn't the worst of the problems.) --ms On 8/27/2011 2:01 PM, Victor Miller wrote:
Joerg, Yes Indeed. By contrast millions of people voluntarily (and with relish) spend hours each week doing Sudoku's. We would see this as being an example of mathematical reasoning. Invariably when I meet someone who's not in a scientific field, and I tell them that I'm a mathematician, I see a look of panic in their face, and I hear "Math was my worst subject". It's clear that the experience of being taught mathematics is traumatic for most people. It would be better if we structured mathematics as a general problem solving experience -- for example look at the introduction of Polya's "How to solve it". The general public needs to learn logic and reasoning skills. Historically almost all mathematical skills have evolved as part of *natural sciences* -- necessary skills that one needs to use in the context of general problem solving. Most word problems in algebra and calculus are worse than stupid.
An imperfect analogy is physical education. At least when I was in school (and I hope that things have gotten better) -- we spent most of the time doing calisthenics (which nobody really liked -- but we were assured were necessary. Most math skills are taught like that). The rest of the time was spent in *real* physical activities -- like playing various sports -- which would be enjoyable, but only after acquiring a certain level of skill. But the real problem that I saw here, is that the instructors (who were invariably ex star athletes themselves, or in my case in high school an ex-Marine drill sargeant) catered toward the students who were already skilled in the sports, and never tried to motivate or help the others (sound familiar).
Victor
On Sat, Aug 27, 2011 at 1:41 PM, Joerg Arndt<arndt@jjj.de> wrote:
Did really none of the replies mention FUN?
Math eduction (school level) suffers brutally from the style of teaching that implicitly suggests that nothing could be farther away from math than fun.
People are horrified by math because in school they learned that it is something utterly dreadful.
Huge surprise they are neither very good at it nor much inclined to learn more?
Big fail for the teachers, they _are_ to blame.
This is from my (German) perspective where teachers do have a decent salary and working conditions. If you have to mention politics make sure it's similar for the place you are referring to (regarding cash, conditions, and perspective) before anything else.
Cheers, jj, now a teacher (post school, students of engineering)
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I'm going to jump in with a defense of my teachers (1960's San Jose public schools). My 7th grade teacher, Mr. Juel, showed us how you could prove an infinite number of facts with a finite amount of work. His favorite example was the sum of consecutive cubes -- it's always a perfect square! That got me hooked. Infinity was a hot topic, and we would argue endlessly about what it "meant". In 8th grade, Mrs. Coffee gave me one of Martin Gardner's books. It was a present she just decided to give me. I don't think I had done anything special to deserve it. It's true that later on (high school) I had a stream of less inspired math teachers*. But by then it didn't matter anymore. I believe those random mechanisms are still at work today, even if we don't immediately recognize them. Consider Vi Hart's doodles. Veit *Some were actual drill sergeants. On Aug 27, 2011, at 3:33 PM, Mike Speciner wrote:
Wow, that brought back memories! My high school gym teacher was an ex-Israeli gymnast. (But that was only part of why I hated gym; being my first class, at 8? in the morning after an hour commute to school, didn't help.)
I am also reminded of a math problem set my daughter got in elementary school. One of the problems started "19 people compete in a 3-legged race..." The "correct" solution was to mindlessly multiply the number of people by the length of the race. (And that wasn't the worst of the problems.)
--ms
On 8/27/2011 2:01 PM, Victor Miller wrote:
Joerg, Yes Indeed. By contrast millions of people voluntarily (and with relish) spend hours each week doing Sudoku's. We would see this as being an example of mathematical reasoning. Invariably when I meet someone who's not in a scientific field, and I tell them that I'm a mathematician, I see a look of panic in their face, and I hear "Math was my worst subject". It's clear that the experience of being taught mathematics is traumatic for most people. It would be better if we structured mathematics as a general problem solving experience -- for example look at the introduction of Polya's "How to solve it". The general public needs to learn logic and reasoning skills. Historically almost all mathematical skills have evolved as part of *natural sciences* -- necessary skills that one needs to use in the context of general problem solving. Most word problems in algebra and calculus are worse than stupid.
An imperfect analogy is physical education. At least when I was in school (and I hope that things have gotten better) -- we spent most of the time doing calisthenics (which nobody really liked -- but we were assured were necessary. Most math skills are taught like that). The rest of the time was spent in *real* physical activities -- like playing various sports -- which would be enjoyable, but only after acquiring a certain level of skill. But the real problem that I saw here, is that the instructors (who were invariably ex star athletes themselves, or in my case in high school an ex-Marine drill sargeant) catered toward the students who were already skilled in the sports, and never tried to motivate or help the others (sound familiar).
Victor
On Sat, Aug 27, 2011 at 1:41 PM, Joerg Arndt<arndt@jjj.de> wrote:
Did really none of the replies mention FUN?
Math eduction (school level) suffers brutally from the style of teaching that implicitly suggests that nothing could be farther away from math than fun.
People are horrified by math because in school they learned that it is something utterly dreadful.
Huge surprise they are neither very good at it nor much inclined to learn more?
Big fail for the teachers, they _are_ to blame.
This is from my (German) perspective where teachers do have a decent salary and working conditions. If you have to mention politics make sure it's similar for the place you are referring to (regarding cash, conditions, and perspective) before anything else.
Cheers, jj, now a teacher (post school, students of engineering)
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An inspirational, committed, capable teacher. Now why didn't someone think of that before? WFL On 8/27/11, Veit Elser <ve10@cornell.edu> wrote:
I'm going to jump in with a defense of my teachers (1960's San Jose public schools).
My 7th grade teacher, Mr. Juel, showed us how you could prove an infinite number of facts with a finite amount of work. His favorite example was the sum of consecutive cubes -- it's always a perfect square! That got me hooked. Infinity was a hot topic, and we would argue endlessly about what it "meant".
In 8th grade, Mrs. Coffee gave me one of Martin Gardner's books. It was a present she just decided to give me. I don't think I had done anything special to deserve it.
It's true that later on (high school) I had a stream of less inspired math teachers*. But by then it didn't matter anymore. I believe those random mechanisms are still at work today, even if we don't immediately recognize them. Consider Vi Hart's doodles.
Veit
*Some were actual drill sergeants.
participants (14)
-
Adam P. Goucher -
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