The points brought up here, especially by Thoma but less so by the quoted passage by Woolley, are very valid in my experience. Often times the shortest way surpass a difficult abstract challenge is to find a practical instance of it and then slough through the mechanics. You abuse the natural pattern searching of the human brain very productively that way.
Simultaneously, these arguments tend to be reactionary and ignore the benefits of calculators. A calculator, use effectively, allows for a far larger number of examples to be considered together than the brain is often able to achieve. This is was most apparent for me in pre-calculus classes where students should be gaining abstract intuition about the behaviors of functions. Here, graphing large numbers of functions varying their parameters allows one to quickly get a sense of a parametric family.
There's an implicit statement here that the "gap" is one such that the lower generation is worse off than their elders --- which is a pretty common human narrative, really. I think instead that this difference is less well-ordered than assumed. Technology is definitely capable of improving human cognition and learning by providing new capacity, and curricula need to explicitly study and take advantage of these capacities.
The argument though is that it is simply impossible to use a calculator effectively without first understanding the basics.
If you don't understand that 50*80 should give you something starting with 40, you don't even understand that getting 3745 as an answer on your calculator because you mistyped is horribly incorrect.
You can chalk it up to "kids these days" all you want, but if you do some basic math problems with a 60 year old the odds are they will simply leave you in the dust while you go looking for a calculator.
"Conrad Wolfram says the part of math we teach -- calculation by hand -- isn't just tedious, it's mostly irrelevant to real mathematics and the real world. He presents his radical idea: teaching kids math through computer programming."
Y'know, maybe what we should really do is to completely decouple "mathematics" from "arithmetic". Arithmetic is useful but boring and you mostly learn it from age 5 to 12. Mathematics is harder, more interesting, and either useful or useless depending on what you eventually do with your life (no doubt 60% of people could go through life without understanding any mathematics at all -- all they need is arithmetic).
Seems to me there was an article a few months back on how the use of GPS in cars was handicapping drivers -- damaging their ability to form and manipulate abstract mental maps.
I'm of the opinion that you should always learn the thing at about two levels down in automation from where you normally use it. So, for programming, I'm all for assembler, compiler, and C programming skills, even though those might never be used in the real world. By the same principle, if you're learning navigation in an airplane, you should learn dead reckoning and a wet compass. If you're learning to driver and shoot a tank you should have pretty good concepts of how rifle combat works, etc.
In math and economics, however, I'm not sure what "2 levels down" means. Is math the rote memorization and repetition of stuff? Most definitely not. But does it depend on it? Maybe. Is it the application of pre-existing patterns in any fashion -- such as punching numbers into a calculator? I don't think so. I think it's much more about the ability to teach yourself to find and exploit patterns through trial and error. That's one of the reasons I've always thought so highly of High School Geometry classes -- when done well, they begin to teach how to think, not just what to think.
In my mind, math is a language like any other. Whereas you wouldn't be considered fluent in English until you have memorized a significant number of words and phrases, the same holds for mathematics.
Al that rote memorization that everyone hates is the same as understanding a basic sentence.
Back in the 90s there was an episode of the Outer Limits that touched on this subject. Effectively there was a nexus of information that everyone connected to through their minds. Anytime they wanted to know something, they just looked it up by thought and presto, instant expert. One guy couldn't connect, so he had to learn things the old fashioned way by research and practice. Naturally, the nexus goes down and people can barely function. He goes from being the village idiot to the village savior.
Tangentially, up until grade 6 or 7 (in the late 80's), my math books also included BASIC code related to the concepts --- usually at the end of the chapter. It's how I became curious about programming and more so about math (though in my case none of the teachers ever did anything with the material, I either just typed the code into my family's TI-99A or tried to mentally work out what the program was doing).
The trend has been away from this as programming has gotten more difficult and the general decline in resources for developing and updating school curricula.
That's amazing, I wish any of my math books had that; we would probably see a lot more young programmers these days. I can seriously say that the only reason I even got into programming was discovering TI-Basic in 7th grade (although the habits it taught me had to be slowly unlearned as I started using a real language ;)
It seems from many of the other comments is that people recognize that performing exact mental-arithmetic calculations is rarely necessary; however, the more intuitive understanding of how numbers relate in magnitude etc. is critically important. Estimation is something that I think many of us take for granted, but that has some significant mental pre-requisites.
Interestingly, a growing branch of mathematics education has been working to explore whether the traditional rote memorization is the most effective way of instilling this more hollistic understanding of numbers. If people were interested, I could ask some educator friends for more up-to-date links/citations on this topic.
I think the problem with Economics in this case (I'm an undergraduate) is that it has become overly mathematical, often for its own sake.
A great article to read is here: http://www.jstor.org/stable/30042661 - much of UG level Economics can be taught and explained with the use of diagrams and graphs.
Overall, I think the generation gap argument is fairly sound - we aren't taught in the same way that our parents and their parents were, and for better or worse, this is how it is.
I like the way Sowell did it (recounted in his A Personal Odyssey); when teaching economics to engineers he concentrated on a "literary" conceptual teaching style, since he already knew they were competent in the needed math.
Presumably, though he didn't discuss this directly, he used more math with those who needed it to understand the mathematical underpinnings of the theories.
His idea was to present the coursework in a way that would make the students think about what they were doing. As he put it about the engineering students, if he presented it in the regular mathematical way, they could have just plugged the numbers into the formulas without necessarily understanding what he wanted them to learn.
I remember doing "Mad Minutes" in Grade 4 (1986). We'd have to do 30 multiplication problems (up to 12 * 12) in 60 seconds. It was a game to see how many we could complete.
I think I might need to revive these for my technical-college math students.
About that Mad Minute.... Sure it may be great for for kids that are already good at it. The fact that you thought it was a "game" says a lot.
But for kids that haven't gotten the fluency with the arithmetic tables yet it just raises their anxiety level, even to the point of near-panic. Which of course shuts down exactly those parts of the brain that you need to be running well to be good at math. The anxiety association with math becomes the lesson learned and as a result students can end up absolutely hating math because they feel sure that "I'm just not good at it".
Of course, it could be a fun experiment to try on some college students.
From the original article 'Students today might be taught by a teacher who is himself unable to work out 37+16 without help' I take it that this is extreme exaggeration to make the point. Such a person would clearly not be fit to teach any subject let alone one requiring numeracy skills at its core.
I think most people who seem unable are just unwilling. I can do that in my head, but it's a lot of effort without practice. Someone less patient might put "37" and "16" in their mind, start adding, and interpret the block they hit when trying to carry the 1 as "can't do it."
It's hard to imagine not being able to do it if you're practiced, but it's as hard to do as anything else you never have to do when you never do it.
You're kidding right? I know many teachers that can't utilize the correct "your|you're" instance with any degree of consistency; I don't see how they'd be able to work out 37+16 without at least writing it down.
Conservative generation gap nonsense. The prevelance of Google Maps means people can't plan directions and hence can't read some graphs?
Ban television, radio, electric lights, the horseless carriage, automated looms, steam engines, the printing press, and writing while you're at it.
This sort of "the basics are really important" nonsense is sometimes heard from the old directed at the young because they can no longer argue against calculators themselves, so they make a proxy.
The big issue here, since most reading this are probably of the generation in question, is found in the first comment:
As a result they have no quantitative intuition, which means they have no idea if their arithmetic results (achieved by simply hitting buttons on a keypad) even make sense.
That's scary. It'll be quite difficult to teach my kids math if their own schools don't see it as a requirement.
I think we should aim for "quantitative intuition" by teaching how to do rough calculations instead of spending time teaching precise calculations.
There is a lot of "cheats" possible when calculating approximations that makes it much easier to learn/perform.
In real life there is often so many uncertainties in the source numbers that a 'precise' answer is not meaningful anyway. Many people think that all the digits displayed in the calculator are significant/meaningful.
If you are a painter that estimates an offer price to the customer, you don't need to be able to work out in your head that 10.5 * 21.5 is 225.75, the estimate 'approx 215-230' is almost always good enough.
Math is one of the huge things leading me to home school rather than sending the kids to school. It's bad enough that schools teach "calculation" rather "math", but now they're not even teaching "calculation"‽
It's a lot of work but it has gotten to the point that I feel like I'd be screwing my kids over to send them to school. Changes are in play that may make me comfortable with them sending their children to a 21st century school, but I can't stomach the thought of sending them to these 19th century monstrosities in their final days.
I have little quantitative intuition, which typically leads me to have to work through everything rather than give an off-the-cuff answer and checking my results multiple times.
This generation gap doesn't exist in the sciences and engineering, IMO.
For example, even the crappiest students in my class (in Physics) knew how to do a taylor expansion to approximate the sine of something, and they knew power series and all sorts of other stuff. You have to know how in order to simplify algebra in many cases.
If there is a generation gap, it's because professors aren't strict enough and unwavering on their decisions to not use calculators, or the courses aren't doing hard enough algebra and calculus to really merit not using a calculator. When you get into the really hard stuff, not even Maple can help you half the time.
I see the lack of mental arithmetic ability all the time when scuba diving. There are many experienced divers who are totally dependent on wrist computers or pre-calculated tables, and can't do simple things like figuring out decompression schedules or gas consumption rates in their heads. If you have a good sense of numbers then it's really easy to do these things in your head after you memorize a few simple rules. Yet so many divers think it's some kind of black magic, or even that it's somehow "dangerous" to make your own calculations.
It's a good idea to keep a table and a wrist calculator when scuba diving anyway, in case you're subject to nitrogen narcosis, which makes you really stupid.
It is a difficult issue. I do a mathematics and computer science degree hence there's obviously no real lack of mental arithmetic maths skills for this degree (well, at least the basics are consistent across the maths students).
Anywhoo, this is - IMO - something that the author's college should look at collectively. If there are some students and professors with massively different ideas of the required level of mental maths skills, perhaps the college should look at introducing a mandatory first year 'mental maths' crash course/module?
This would help to make things a little more consistent. If there's genuine confusion/disagreement between the students and Professors, this should - IMO - be addresses by a course-wide decision being made.
Regarding programmable calculators - they can be reset in about 2 seconds total (it's usually Menu -> Settings -> Memory -> Reset All).
In our University, the exam invigilators ensure that all programmable calculators are reset (with them watching them being reset, of course) before the start of the exam.
So I'm not sure why this (to me) fairly obvious idea seems to be overlooked in the article? As I say, it takes 2 seconds total.
An interesting article though; even though I think the author/prof is approaching things in a slightly muddled (for want of a better - non insulting- term!) way. The college should (IMO) decide on how they want to approach things, and then be consistent across all modules and all Professors.
Skills become outdated over time. The skill of mental calculations, I think, is not really that valuable on the average nor is it unlearnable later on if you sit at a cashier's desk at some point ... but the skill of doing quantitative calculations mentally greatly helps me make connections that I suspect I won't make otherwise. I think this is because I'm morphing pictures in my head instead of crunching numbers.
Next step - lets allow all kids to use google search during their tests.
There has been a profound change in mathematics education in the years indicated, and the author of the submitted article is on to something. One of my favorite authors on mathematics, 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.
I have a collection of analytic geometry and calculus books, accumulated as used books from various readers, that includes the books used by my late father in his higher education as a chemistry major during the Truman administration, followed by books from other previous owners reflecting "new math," "back to basics," and "reform" approaches to mathematics education. Plainly today's secondary and tertiary students of mathematics need to take advantage of current technology so that they can devote more time to THINKING about the mathematics they learn and less time to what even any mathematician would call "tedious calculation." But too few students have ever been guided to through the kind of insight-producing problems in which the tedious steps themselves and the false starts while struggling with the problem produce deep understanding. Stillwell gives examples of such problems in his books, and the minority of students who participate in math contexts or who voluntarily work the "challenge" problems not assigned in their textbooks may gain such insight, but most school textbook problems of all eras are mere exercises, and too few students do enough of those thoughtfully to have hope of learning mathematical concepts.
See "Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education," American Educator, Fall 1999, Vol. 23, No. 3, pp. 14-19, 50-52 for additional commentary on mathematics education,
I side much more Conrad Wolfram on this than Stillwell [1]. Computation simply is a lot less important now as we have tools that do them with far greater speed and accuracy. And I'm from the generation that did tons of computation, but I honestly never had a great grasp of it. I just knew there were steps to be taken (leave an extra space to right as you do long multiplication) -- and I was one of the top math students in the district.
In fact I'd argue that I never really understood much of any math until grad school. I was computational sophisticated, but lacked understanding.
And oddly, I seem to find quite the opposite problem from what the blog author describes. I find students who know 3x5. But struggle to understand when the Fourier Transform is appropriate. Sure, if they're looking at problem sets at the back of the chapter about Fourier Transforms then they'll start with it, but in the real world they lack the conceptual understanding of it. I've met students who can compute the SVD, they can tell you the text book definition, but don't actually intuitively know what it means. They don't know when it should be applied, or when it is applied, what it means.
"I plan to remain hard-headed about this until I am convinced that abandoning the rote sorts of exercises done in, say, a linear algebra class (which can also be done on a calculator) does not hinder our ability to form intuition about how to do proofs, etc"
It is positive statements that require proof, not negative ones. If you believe that the introduction of calculators, google maps, etc., has negatively impacted number sense and human spacial reasoning, it is on you to prove it.
what i gained from learnign math using trig and log tables, pencil and paper to calculate fractions, multiplying them, more numbers, more calculations, => feel for a solution. yes, it is subjective, by looking at a problem there are no magic that pops out, just a feel for what would be a correct answer. objectively, the experience how to simplify is a great gain as well. ... then i studied logics. today i can calcualte areas, circumferences in my head, faster then using my phones calculator. the precesion is acceptable for my daily use.
[+] [-] tel|15 years ago|reply
Simultaneously, these arguments tend to be reactionary and ignore the benefits of calculators. A calculator, use effectively, allows for a far larger number of examples to be considered together than the brain is often able to achieve. This is was most apparent for me in pre-calculus classes where students should be gaining abstract intuition about the behaviors of functions. Here, graphing large numbers of functions varying their parameters allows one to quickly get a sense of a parametric family.
There's an implicit statement here that the "gap" is one such that the lower generation is worse off than their elders --- which is a pretty common human narrative, really. I think instead that this difference is less well-ordered than assumed. Technology is definitely capable of improving human cognition and learning by providing new capacity, and curricula need to explicitly study and take advantage of these capacities.
[+] [-] run4yourlives|15 years ago|reply
If you don't understand that 50*80 should give you something starting with 40, you don't even understand that getting 3745 as an answer on your calculator because you mistyped is horribly incorrect.
You can chalk it up to "kids these days" all you want, but if you do some basic math problems with a 60 year old the odds are they will simply leave you in the dust while you go looking for a calculator.
[+] [-] torstein|15 years ago|reply
http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_m...
[+] [-] hugh3|15 years ago|reply
[+] [-] DanielBMarkham|15 years ago|reply
I'm of the opinion that you should always learn the thing at about two levels down in automation from where you normally use it. So, for programming, I'm all for assembler, compiler, and C programming skills, even though those might never be used in the real world. By the same principle, if you're learning navigation in an airplane, you should learn dead reckoning and a wet compass. If you're learning to driver and shoot a tank you should have pretty good concepts of how rifle combat works, etc.
In math and economics, however, I'm not sure what "2 levels down" means. Is math the rote memorization and repetition of stuff? Most definitely not. But does it depend on it? Maybe. Is it the application of pre-existing patterns in any fashion -- such as punching numbers into a calculator? I don't think so. I think it's much more about the ability to teach yourself to find and exploit patterns through trial and error. That's one of the reasons I've always thought so highly of High School Geometry classes -- when done well, they begin to teach how to think, not just what to think.
Food for thought.
[+] [-] run4yourlives|15 years ago|reply
Al that rote memorization that everyone hates is the same as understanding a basic sentence.
[+] [-] droz|15 years ago|reply
[+] [-] mkr-hn|15 years ago|reply
[+] [-] run4yourlives|15 years ago|reply
[+] [-] noblethrasher|15 years ago|reply
[+] [-] sedachv|15 years ago|reply
The trend has been away from this as programming has gotten more difficult and the general decline in resources for developing and updating school curricula.
[+] [-] fferen|15 years ago|reply
[+] [-] akikuchi|15 years ago|reply
It seems from many of the other comments is that people recognize that performing exact mental-arithmetic calculations is rarely necessary; however, the more intuitive understanding of how numbers relate in magnitude etc. is critically important. Estimation is something that I think many of us take for granted, but that has some significant mental pre-requisites.
Interestingly, a growing branch of mathematics education has been working to explore whether the traditional rote memorization is the most effective way of instilling this more hollistic understanding of numbers. If people were interested, I could ask some educator friends for more up-to-date links/citations on this topic.
[+] [-] 44Aman|15 years ago|reply
A great article to read is here: http://www.jstor.org/stable/30042661 - much of UG level Economics can be taught and explained with the use of diagrams and graphs.
Overall, I think the generation gap argument is fairly sound - we aren't taught in the same way that our parents and their parents were, and for better or worse, this is how it is.
[+] [-] billswift|15 years ago|reply
Presumably, though he didn't discuss this directly, he used more math with those who needed it to understand the mathematical underpinnings of the theories.
His idea was to present the coursework in a way that would make the students think about what they were doing. As he put it about the engineering students, if he presented it in the regular mathematical way, they could have just plugged the numbers into the formulas without necessarily understanding what he wanted them to learn.
[+] [-] JabavuAdams|15 years ago|reply
I think I might need to revive these for my technical-college math students.
[+] [-] marshray|15 years ago|reply
But for kids that haven't gotten the fluency with the arithmetic tables yet it just raises their anxiety level, even to the point of near-panic. Which of course shuts down exactly those parts of the brain that you need to be running well to be good at math. The anxiety association with math becomes the lesson learned and as a result students can end up absolutely hating math because they feel sure that "I'm just not good at it".
Of course, it could be a fun experiment to try on some college students.
[+] [-] gsmaverick|15 years ago|reply
[+] [-] vixen99|15 years ago|reply
[+] [-] mkr-hn|15 years ago|reply
It's hard to imagine not being able to do it if you're practiced, but it's as hard to do as anything else you never have to do when you never do it.
[+] [-] run4yourlives|15 years ago|reply
[+] [-] rmc|15 years ago|reply
Ban television, radio, electric lights, the horseless carriage, automated looms, steam engines, the printing press, and writing while you're at it.
This sort of "the basics are really important" nonsense is sometimes heard from the old directed at the young because they can no longer argue against calculators themselves, so they make a proxy.
[+] [-] run4yourlives|15 years ago|reply
As a result they have no quantitative intuition, which means they have no idea if their arithmetic results (achieved by simply hitting buttons on a keypad) even make sense.
That's scary. It'll be quite difficult to teach my kids math if their own schools don't see it as a requirement.
[+] [-] silvestrov|15 years ago|reply
There is a lot of "cheats" possible when calculating approximations that makes it much easier to learn/perform.
In real life there is often so many uncertainties in the source numbers that a 'precise' answer is not meaningful anyway. Many people think that all the digits displayed in the calculator are significant/meaningful.
If you are a painter that estimates an offer price to the customer, you don't need to be able to work out in your head that 10.5 * 21.5 is 225.75, the estimate 'approx 215-230' is almost always good enough.
[+] [-] jerf|15 years ago|reply
It's a lot of work but it has gotten to the point that I feel like I'd be screwing my kids over to send them to school. Changes are in play that may make me comfortable with them sending their children to a 21st century school, but I can't stomach the thought of sending them to these 19th century monstrosities in their final days.
[+] [-] sn|15 years ago|reply
[+] [-] chopsueyar|15 years ago|reply
It is the curriculum.
[+] [-] juiceandjuice|15 years ago|reply
For example, even the crappiest students in my class (in Physics) knew how to do a taylor expansion to approximate the sine of something, and they knew power series and all sorts of other stuff. You have to know how in order to simplify algebra in many cases.
If there is a generation gap, it's because professors aren't strict enough and unwavering on their decisions to not use calculators, or the courses aren't doing hard enough algebra and calculus to really merit not using a calculator. When you get into the really hard stuff, not even Maple can help you half the time.
[+] [-] arethuza|15 years ago|reply
[+] [-] nradov|15 years ago|reply
[+] [-] wazoox|15 years ago|reply
[+] [-] tristanperry|15 years ago|reply
Anywhoo, this is - IMO - something that the author's college should look at collectively. If there are some students and professors with massively different ideas of the required level of mental maths skills, perhaps the college should look at introducing a mandatory first year 'mental maths' crash course/module?
This would help to make things a little more consistent. If there's genuine confusion/disagreement between the students and Professors, this should - IMO - be addresses by a course-wide decision being made.
Regarding programmable calculators - they can be reset in about 2 seconds total (it's usually Menu -> Settings -> Memory -> Reset All).
In our University, the exam invigilators ensure that all programmable calculators are reset (with them watching them being reset, of course) before the start of the exam.
So I'm not sure why this (to me) fairly obvious idea seems to be overlooked in the article? As I say, it takes 2 seconds total.
An interesting article though; even though I think the author/prof is approaching things in a slightly muddled (for want of a better - non insulting- term!) way. The college should (IMO) decide on how they want to approach things, and then be consistent across all modules and all Professors.
[+] [-] aufreak3|15 years ago|reply
Next step - lets allow all kids to use google search during their tests.
[+] [-] tokenadult|15 years ago|reply
"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/
I have a collection of analytic geometry and calculus books, accumulated as used books from various readers, that includes the books used by my late father in his higher education as a chemistry major during the Truman administration, followed by books from other previous owners reflecting "new math," "back to basics," and "reform" approaches to mathematics education. Plainly today's secondary and tertiary students of mathematics need to take advantage of current technology so that they can devote more time to THINKING about the mathematics they learn and less time to what even any mathematician would call "tedious calculation." But too few students have ever been guided to through the kind of insight-producing problems in which the tedious steps themselves and the false starts while struggling with the problem produce deep understanding. Stillwell gives examples of such problems in his books, and the minority of students who participate in math contexts or who voluntarily work the "challenge" problems not assigned in their textbooks may gain such insight, but most school textbook problems of all eras are mere exercises, and too few students do enough of those thoughtfully to have hope of learning mathematical concepts.
See "Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education," American Educator, Fall 1999, Vol. 23, No. 3, pp. 14-19, 50-52 for additional commentary on mathematics education,
http://www.aft.org/newspubs/periodicals/ae/fall1999/index.cf...
and see an earlier HN comment
http://news.ycombinator.com/item?id=2515796
for a FAQ on the distinction between problems and exercises in mathematics education.
[+] [-] kenjackson|15 years ago|reply
In fact I'd argue that I never really understood much of any math until grad school. I was computational sophisticated, but lacked understanding.
And oddly, I seem to find quite the opposite problem from what the blog author describes. I find students who know 3x5. But struggle to understand when the Fourier Transform is appropriate. Sure, if they're looking at problem sets at the back of the chapter about Fourier Transforms then they'll start with it, but in the real world they lack the conceptual understanding of it. I've met students who can compute the SVD, they can tell you the text book definition, but don't actually intuitively know what it means. They don't know when it should be applied, or when it is applied, what it means.
[1] http://blog.wolfram.com/2010/11/23/conrad-wolframs-ted-talk-...
[+] [-] reso|15 years ago|reply
It is positive statements that require proof, not negative ones. If you believe that the introduction of calculators, google maps, etc., has negatively impacted number sense and human spacial reasoning, it is on you to prove it.
[+] [-] wmobit|15 years ago|reply
[+] [-] unknown|15 years ago|reply
[deleted]
[+] [-] peterbotond|15 years ago|reply