Does math have big scary teeth or something?(ask.metafilter.com)
ask.metafilter.com
Does math have big scary teeth or something?
http://ask.metafilter.com/135720/Does-math-have-big-scary-teeth-or-something#1939302
48 comments
The comment that the link leads directly to is devastatingly good. If you don't want to do a lot of reading, just skip the main article and read that comment.
No it's not, it's narcissism. Look at me, I'm a computer engineer, I solve math problems all day, so I must be much smarter than everyone else.
You think consultants don't have to think about the needs of their client? You think salesmen don't have to think about how to make the deal go through? You think lawyers don't have to think about how to make their case? You think doctors don't have to think about their patients problems? You think wall street traders don't have to think about how to beat the market? You think managers don't have to think about the thoughts and emotions of their employees and customers? (ok, here lies a real problem) You think real estate agents don't need to be creative to stay afloat?
I've never seen these stupid people everyone keeps talking about, but I have a huge suspicion it's a case of projection.
You think consultants don't have to think about the needs of their client? You think salesmen don't have to think about how to make the deal go through? You think lawyers don't have to think about how to make their case? You think doctors don't have to think about their patients problems? You think wall street traders don't have to think about how to beat the market? You think managers don't have to think about the thoughts and emotions of their employees and customers? (ok, here lies a real problem) You think real estate agents don't need to be creative to stay afloat?
I've never seen these stupid people everyone keeps talking about, but I have a huge suspicion it's a case of projection.
He's probably right in that the American culture / schooling seems to favor memorization, but the distinction between "problem solving" and doing things by recall is fuzzier than he implies. A lot of problem solving can be cast as examining details to find a match and looking up how to proceed. (That's not a terrible description of Prolog, come to think of it.) The reverse is true, as well.
Is a sysadmin troubleshooting an erratic network and starting with checking DNS doing what he calls problem solving ("my experience says checking this will narrow down several potential causes") or working off memorization ("many cross-cutting issues are caused by DNS, check there first")?
Is a sysadmin troubleshooting an erratic network and starting with checking DNS doing what he calls problem solving ("my experience says checking this will narrow down several potential causes") or working off memorization ("many cross-cutting issues are caused by DNS, check there first")?
Devastatingly good? Meh. It's okay, until it runs off the tracks and starts making massive over-generalizations, and ends up mired in populist propaganda (i.e. "carpenters" being problem-solvers, whereas "academics" are not). By that point, it's rubbish -- a victim of the same sort of lazy, pattern-based thinking that it's trying to critique.
I think the hypothesis proposed might have some merit: most people can (and do) get by without the need for strong analytical skills. Beyond that, I think it's probably a stretch to try to match those skills to certain professions, when it's far more reasonable to expect that any random sample of people (taken from any profession) will fall on a continuum of analytical prowess (just as with any other metric). For all we know, "problem solving" is the trait that distinguishes the top performers in any field from the rest of the crowd.
I think the hypothesis proposed might have some merit: most people can (and do) get by without the need for strong analytical skills. Beyond that, I think it's probably a stretch to try to match those skills to certain professions, when it's far more reasonable to expect that any random sample of people (taken from any profession) will fall on a continuum of analytical prowess (just as with any other metric). For all we know, "problem solving" is the trait that distinguishes the top performers in any field from the rest of the crowd.
> a victim of the same sort of lazy, pattern-based thinking that it's trying to critique.
Over generalization based on incomplete data or biased vision of the world has nothing to do with the problem solving/memorization dichotomy the author is speaking about. In fact, (and this is a big generalization) problem solvers have a tendency to reduce every problem to its simplest forms in order to find new ways to analyze it. I'd agree that when it is applied to human/social world, it often leads to partially or even totally wrong hypotheses though.
I'd also like people to stop just dissing over generalizations where in fact we just use these all the time. This comment is not meant to be a "end of discussion by scientific proof" about anything. But it's doing an extremely good job opening a new interresting way to see the world we have around us as intellectuals. And that's what generalizations are for.
Over generalization based on incomplete data or biased vision of the world has nothing to do with the problem solving/memorization dichotomy the author is speaking about. In fact, (and this is a big generalization) problem solvers have a tendency to reduce every problem to its simplest forms in order to find new ways to analyze it. I'd agree that when it is applied to human/social world, it often leads to partially or even totally wrong hypotheses though.
I'd also like people to stop just dissing over generalizations where in fact we just use these all the time. This comment is not meant to be a "end of discussion by scientific proof" about anything. But it's doing an extremely good job opening a new interresting way to see the world we have around us as intellectuals. And that's what generalizations are for.
It's an extremely insightful post! But even he is still stuck in "memorization" as shown by this little footnote:
> *true, in America shocking number of people can't tell you the name of their congressman or the capital of North Carolina.
Both of those things are utterly useless things to know. Someone once asked Einstein how many feet in a mile, and he replied that that's what almanacs are for.
> *true, in America shocking number of people can't tell you the name of their congressman or the capital of North Carolina.
Both of those things are utterly useless things to know. Someone once asked Einstein how many feet in a mile, and he replied that that's what almanacs are for.
I think the reason he points this out is to illustrate the idea that the memorization isn't even necessarily directed towards information which has an direct impact on that person's life and decision making ability
Almanacs: the internet of the early 20th century.
He does have some good points, but they're drowned in nonsense. For example, he suggests that memorization is taught in lieu of problem solving because this is what society thinks is a smart person. Memorization is taught in lieu of problem solving, but it's because we haven't yet figured out a good way to teach problem solving. It's not some vast conspiracy; the outcome we actually want is just a really hard one.
I don't think that the author was implying the existence of a vast conspiracy to keep people dumb; rather, the author was suggesting that society is operating in a manner which he believes is suboptimal.
Let's say memorisation is taught in lieu of problem solving for the reasons you stated, then consequently the people who are good at memorisation become thought of as smart, since they do well academically. This, in turn, influences the education system to focus more on memorisation (though, of course, this is all hypothetical).
Let's say memorisation is taught in lieu of problem solving for the reasons you stated, then consequently the people who are good at memorisation become thought of as smart, since they do well academically. This, in turn, influences the education system to focus more on memorisation (though, of course, this is all hypothetical).
Interestingly, "smart" is only equated with memorization in artificial situations (like academics and trivia shows). Case in point: if you ask people who's presently the smartest person in America, they might say the jeopardy guy (I actually suspect a lot of them would say Hawking). If you ask who's the smartest person in American history, the overwhelming response will probably be Einstein (except a few who will say Hawking, or -- very rarely -- Tesla). Einstein was a notorious problem-solver. Certainly, if you asked someone if they'd prefer their child to grow up to be like Einstein or like the Jeopardy guy, they'd probably pick Einstein.
Likewise, I don't think the author was implying any sort of conspiracy. I probably used the wrong word. I should have said that there's no real effort on anyone's part to promote memorization over problem-solving. Memorization is more prominent in these artificial situations because it's easier to train and test, rather than because people think it's superior.
Likewise, I don't think the author was implying any sort of conspiracy. I probably used the wrong word. I should have said that there's no real effort on anyone's part to promote memorization over problem-solving. Memorization is more prominent in these artificial situations because it's easier to train and test, rather than because people think it's superior.
I agree to some extent, though I interpret "haven't yet figured out a good way to teach problem solving" as "haven't yet figured out how to quantify the unquantifiable." In other words, problem solving -- in the sense of creatively overcoming new challenges and not just plugging values into memorized formulas -- is probably impossible to measure and standardize.
I believe you really can teach things like analytical thinking, lateral thinking, creativity, etc... (I have a number of books on these subjects) but how do you measure them via standardized tests? You can't. And so our educational system measures what can be measured: recall, application of rote formulas, etc...
Even the ostensibly freer essay portion of the SAT suffers from this problem: http://www.theatlantic.com/doc/200403/katzman
I believe you really can teach things like analytical thinking, lateral thinking, creativity, etc... (I have a number of books on these subjects) but how do you measure them via standardized tests? You can't. And so our educational system measures what can be measured: recall, application of rote formulas, etc...
Even the ostensibly freer essay portion of the SAT suffers from this problem: http://www.theatlantic.com/doc/200403/katzman
We most certainly do know how to teach problem solving. I do it all the time in my Boy Scout troop. Of course, exactly none of the methods I use would work in the context of a school classroom, but that's as much a critique of our schooling methods as it is of my teaching methods.
Umm, yeah. Just to save people some time, he sums up by saying that doctors don't solve problems but football players do. So, you know. Bullshit.
I mean, his central thesis is: MY job requires problem solving, whereas everybody else's job requires more memorization than football does (which, you know, is a sport that consists largely of continual drilling.)
I mean, his central thesis is: MY job requires problem solving, whereas everybody else's job requires more memorization than football does (which, you know, is a sport that consists largely of continual drilling.)
He didn't say doctors don't solve problems. He said they "don't do much problem solving". He wasn't talking about the result at all. He was talking about the method: mere recalling, or (re)discovery?
Like ahpeeyem said below, this is a lot like mappers and packers. Here is the link he provided: http://the-programmers-stone.com/the-original-talks/day-1-th...
Like ahpeeyem said below, this is a lot like mappers and packers. Here is the link he provided: http://the-programmers-stone.com/the-original-talks/day-1-th...
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point in fact, I was intentionally linking to that particular comment. It does a wonderful job of spelling out a lot of ideas which are not ordinarily articulated very well, if at all!
A handshake and a medal to anybody who can define "devastatingly good."
Counter to the linked-comment: maths is not pure problem solving.
There are a lot of facts to be learned in mathematics; it is more like a language, with a huge vocabulary. And, like language, very little of mathematics is actively justified as to whether it is or is not the best way to go about solving problems. People just build on the existing mathematics as a basis... in a similar way to how people write novels in a language.
The attitude of mathematicians is as an imperial, aristocratic attitude to courtly protocol. The true problem-solving mathematician is more like the wild barbarians, who break all the protocols to achieve actual outcomes. One might say: empirical not imperial.
But the aristicratic attitude makes a lot of sense, because facts and protocols are things that one can learn and advance with, whereas a problem solver is only as good as their last solution. It's a bit magical. While one can practice problem solving, the return on that investment is no where near as reliable as the return on investing in learning established, conventional, standard facts. It's a good investment; it gives you a sustainable competitive advantage.
However, there is something massively cool about being one of the people on the frontier, who germinate and disseminate facts, rather than passively receive them.
There are a lot of facts to be learned in mathematics; it is more like a language, with a huge vocabulary. And, like language, very little of mathematics is actively justified as to whether it is or is not the best way to go about solving problems. People just build on the existing mathematics as a basis... in a similar way to how people write novels in a language.
The attitude of mathematicians is as an imperial, aristocratic attitude to courtly protocol. The true problem-solving mathematician is more like the wild barbarians, who break all the protocols to achieve actual outcomes. One might say: empirical not imperial.
But the aristicratic attitude makes a lot of sense, because facts and protocols are things that one can learn and advance with, whereas a problem solver is only as good as their last solution. It's a bit magical. While one can practice problem solving, the return on that investment is no where near as reliable as the return on investing in learning established, conventional, standard facts. It's a good investment; it gives you a sustainable competitive advantage.
However, there is something massively cool about being one of the people on the frontier, who germinate and disseminate facts, rather than passively receive them.
Math requires a lot of memorized knowledge to solve anything interesting.
Without knowing the right theorems to use in the proof you are trying to compose ... all your efforts might be futile despite having excellent problem solving skills.
Math uses mountains of knowledge derived over many years by hundreds of the smartest people in history.
You just can't physically repeat their work for the purpose of making your proof. You just have to know what is known so far.
For me one of the most fun parts of educations was non-organic chemistry in primary school. So little to learn and you could attempt to solve problems that twisted your brain like a pretzel (and succeed). I could solve problems my teacher couldn't because I was probably smarter then him and we both knew the same about the problem domain because it was all that was to know.
Without knowing the right theorems to use in the proof you are trying to compose ... all your efforts might be futile despite having excellent problem solving skills.
Math uses mountains of knowledge derived over many years by hundreds of the smartest people in history.
You just can't physically repeat their work for the purpose of making your proof. You just have to know what is known so far.
For me one of the most fun parts of educations was non-organic chemistry in primary school. So little to learn and you could attempt to solve problems that twisted your brain like a pretzel (and succeed). I could solve problems my teacher couldn't because I was probably smarter then him and we both knew the same about the problem domain because it was all that was to know.
Taking math classes in school, even at the graduate level, feels like pattern matching to me. On all tests and most homeworks you're expected to stare at a problem long enough until its structure becomes isomorphic to something about which you memorized a theorem. Whether you understand what makes the theorem tick or not is a separate question, but I found a workable test taking strategy in my classes consists of memorizing the theorems that the prof made a big deal about in class, and then just running down the list for each problem.
Research math is more open-ended but the idea is the same. The quantity of genius mathematicians who sit down, a la Newton or Gauss, and invent their own mathematics out of thin air is exceedingly small. Most research in math today, even among the "big" papers, is pretty incremental.
Research math is more open-ended but the idea is the same. The quantity of genius mathematicians who sit down, a la Newton or Gauss, and invent their own mathematics out of thin air is exceedingly small. Most research in math today, even among the "big" papers, is pretty incremental.
APPLYING math is problem solving. And that is the hard part of math too. Or doesn't anyone else remember all the complaints in school about word problems.
I memorized my way through all kinds of math in high school. I was in my thirties before I knew some of those formulas had practical applications. I knew my oldest son couldn't memorize his way through math. So I taught a conceptual approach. It was the math concepts I retained anyway. When I went back to school, knowing the concepts stood me in good stead. Looking up what all the little squiggles meant and refamiliarizing myself with some of the formulas quickly got me up to speed. I was waivered into my statistics class based on my 17 year old SAT scores because I had never taken college algebra. I ended up with the highest grade in the class and explained a lot to my classmates. So I made sure my son knew the concepts, even though he is terrible at dealing with numbers. It worked beautifully.
How it's taught can make a big difference.
How it's taught can make a big difference.
After reading a lot of comments in a lot of threads like this one on a lot of sites, I've noticed a particular factor which sticks out: people who direct this sort of commentary at education in the US almost all seem to have attended very large universities.
I did not attend such a university, and my experience of "academics" was rather different from what's described in the linked comment; there were some things that I know were unusual about the way my college education was structured, and that not even all of the students there were getting the same level of or approach to education, but I suspect that there's a fundamental difference between large and small schools. The larger the institution, the more "efficient" it has to be with the resources at hand; this seems to lead naturally to the sort of bulk-processed "memorize this and regurgitate it on an exam" approach being complained about here. A smaller institution, meanwhile, isn't under the same pressure to move large volumes of students through its curriculum, and can afford to do things differently. I'm well aware that mine did, and thankful every day for that fact.
I did not attend such a university, and my experience of "academics" was rather different from what's described in the linked comment; there were some things that I know were unusual about the way my college education was structured, and that not even all of the students there were getting the same level of or approach to education, but I suspect that there's a fundamental difference between large and small schools. The larger the institution, the more "efficient" it has to be with the resources at hand; this seems to lead naturally to the sort of bulk-processed "memorize this and regurgitate it on an exam" approach being complained about here. A smaller institution, meanwhile, isn't under the same pressure to move large volumes of students through its curriculum, and can afford to do things differently. I'm well aware that mine did, and thankful every day for that fact.
The comment is quite similar to the idea of "mappers and packers:"
http://the-programmers-stone.com/the-original-talks/day-1-th...
This theory goes that some people (mappers) create a mental map of how the world works, and any new knowledge has to be integrated into the map - or if it doesn't fit, the entire map has to be rearranged. Packers just store little packets of knowledge for each situation, without connecting them together.
The mappers therefore can do problem solving by connecting the different parts of their map when they come across something they haven't seen before, whereas packers are will try and apply one of their existing knowledge packets to a new situation, and if none of them fit they are lost.
I think the analogy works, certainly some people I've helped with using their computer don't seem to be able to apply knowledge learned in one program to another one, and need to learn everything by rote instead of figuring it out as they go. So either they don't have enough experience yet to build a meta map/model that would let them figure it out on their own, or they don't even try to.
This theory goes that some people (mappers) create a mental map of how the world works, and any new knowledge has to be integrated into the map - or if it doesn't fit, the entire map has to be rearranged. Packers just store little packets of knowledge for each situation, without connecting them together.
The mappers therefore can do problem solving by connecting the different parts of their map when they come across something they haven't seen before, whereas packers are will try and apply one of their existing knowledge packets to a new situation, and if none of them fit they are lost.
I think the analogy works, certainly some people I've helped with using their computer don't seem to be able to apply knowledge learned in one program to another one, and need to learn everything by rote instead of figuring it out as they go. So either they don't have enough experience yet to build a meta map/model that would let them figure it out on their own, or they don't even try to.
I would think this is more like what happens in each individual on a given topic - you start off with little packets and eventually you integrate them into a map. How well you are at doing this depends on how good you are at linking the packets together, how you are taught or learn them, and whether you have constructed a similar enough map before to construct the new one more easily.
The highlighted response was bullseye hit of what I noticed going through college. The couple classes I remember it being most evident in were the very low level programming classes: Intro to Java/Javascript and a PHP class. These were the classes that pretty much everyone in the business school had to take.
People would work through problems assigned out of the book, but if they ever ran into an issue not specifically outlined for them, they were usually stuck. I spent many nights in the computer lab helping classmates through problems which just required stepping back and taking it one small step at a time. And even if they couldn't work through it on their own, they had no idea where to start on researching a solution -- they didn't know how to ask questions, etc. (Disclaimer: I nearly failed every damn accounting class I had to take, 5 in total I believe.)
I find myself being the exact opposite of what the linked reply states as being the 'norm'. I can't remember simple facts to save my life. I love playing guitar, but I can't memorize notes/chords. To this day I can't honestly tell you what a noun/pronoun/verb/adverb, etc are. Forget about people's names, it ain't happening. I find no real joy in reading fiction, I forget it all anyways. I couldn't ever remember the bajillion accounting terms I had to deal with in business school, but I loves me some calculus.
Also, he seems to point out "America" quite a bit in his response. I have a hard time believing this trend only applies to Americans. I'd like to hear either a non-American or someone with a little international time chime in on this.
People would work through problems assigned out of the book, but if they ever ran into an issue not specifically outlined for them, they were usually stuck. I spent many nights in the computer lab helping classmates through problems which just required stepping back and taking it one small step at a time. And even if they couldn't work through it on their own, they had no idea where to start on researching a solution -- they didn't know how to ask questions, etc. (Disclaimer: I nearly failed every damn accounting class I had to take, 5 in total I believe.)
I find myself being the exact opposite of what the linked reply states as being the 'norm'. I can't remember simple facts to save my life. I love playing guitar, but I can't memorize notes/chords. To this day I can't honestly tell you what a noun/pronoun/verb/adverb, etc are. Forget about people's names, it ain't happening. I find no real joy in reading fiction, I forget it all anyways. I couldn't ever remember the bajillion accounting terms I had to deal with in business school, but I loves me some calculus.
Also, he seems to point out "America" quite a bit in his response. I have a hard time believing this trend only applies to Americans. I'd like to hear either a non-American or someone with a little international time chime in on this.
"I find no real joy in reading fiction, I forget it all anyways"
But that is good: you can read lord of the rings over and over again as if it was the first time.
But that is good: you can read lord of the rings over and over again as if it was the first time.
I was taught programming at a (quite big) university at Toulouse, France. First years were taught Ocaml (Caml-light, actually). The first 2 hours took place in a lecture hall and were about the lexicon of the language: authorized characters, possible identifiers, keywords… A typical "by rote" course.
On my second year, I took a more specialized route. Among other thing, we were again taught Ocaml, as if we learned programming for the first time —which was the case for many of us. This time we were no more than 30. The first 2 hours directly jumped into expressions and functions. Much less to learn, much more to understand. Overall, this second course was faster, by a factor of at least 3. And the class understood it all.
Anyway, one aspect of rote learning we always had was the downgrading of our program in the case of minor syntax errors.
On my second year, I took a more specialized route. Among other thing, we were again taught Ocaml, as if we learned programming for the first time —which was the case for many of us. This time we were no more than 30. The first 2 hours directly jumped into expressions and functions. Much less to learn, much more to understand. Overall, this second course was faster, by a factor of at least 3. And the class understood it all.
Anyway, one aspect of rote learning we always had was the downgrading of our program in the case of minor syntax errors.
Rote memorization is a problem in programming, too. Not to rag on PHP or Visual Basic, but people could get by with just memorizing certain keywords - or not solve hard problems by using a third-party plugin or googling for code snippets.
I hated my Automata and Functional Programming Language classes because I was forced to write out formal proofs to prove really obvious programs to be correct, or to demonstrate something arcane as the Turing machine. But it forced me to think about programming in a different light and find my own way to find the solution.
This reminds me to always to look under the hood of all of the web frameworks, data ORM's that I'm using and to improve my 3rd party libraries, in programming. Also in music, not to blindly play the scales or the tabs of popular music, but learn the different patterns, what makes a song tick, and composition. Also in sports, not just play or practice according to drills, but to analyze post-game what went right, what went wrong, and to apply it in future matches.
I hated my Automata and Functional Programming Language classes because I was forced to write out formal proofs to prove really obvious programs to be correct, or to demonstrate something arcane as the Turing machine. But it forced me to think about programming in a different light and find my own way to find the solution.
This reminds me to always to look under the hood of all of the web frameworks, data ORM's that I'm using and to improve my 3rd party libraries, in programming. Also in music, not to blindly play the scales or the tabs of popular music, but learn the different patterns, what makes a song tick, and composition. Also in sports, not just play or practice according to drills, but to analyze post-game what went right, what went wrong, and to apply it in future matches.
"Not to rag on PHP or Visual Basic, but people could get by with just memorizing certain keywords"
That's bullshit. Yes, PHP/VB programming may be easier, and yes, they may not be tackling hard problems, but "problem solving" approach revolves around analysis and how to address different situations.
That is best expressed by the "if.. then.." construct.
Look - I'm not going to claim the ability to program in VB makes you a good problem solver. However, I am claiming that it is a difference of skill level, not skill set. Memorization is a different skill to problem solving.
That's bullshit. Yes, PHP/VB programming may be easier, and yes, they may not be tackling hard problems, but "problem solving" approach revolves around analysis and how to address different situations.
That is best expressed by the "if.. then.." construct.
Look - I'm not going to claim the ability to program in VB makes you a good problem solver. However, I am claiming that it is a difference of skill level, not skill set. Memorization is a different skill to problem solving.
Memorization, at least in its meaning of remembering facts, is a necessary prerequisite of understanding, which is needed for effective problem solving. http://williambswift.blogspot.com/2009/03/learning-journal-a...
It's pretty sad how strongly memorization, as opposed to critical thinking, is encouraged and rewarded. This seems (based on my University education) to be increasingly true even in disciplines like Math and CS.
Beginning in grade school, the "smart kids" are those who memorize things, not those who solve problems. I'm better at rote memorization, but I pride myself on solving problems.
Beginning in grade school, the "smart kids" are those who memorize things, not those who solve problems. I'm better at rote memorization, but I pride myself on solving problems.
I run screaming from courses that cover mathematical background I don't have (e.g., machine learning because of linear algebra) because it's impossible to keep up without having terminology and definitions memorized.
Recently got annoyed with the overuse of math in http://www.eecs.umich.edu/courses/eecs461/lecture/Lecture8.p... . The lecture covers numerical integration in an embedded system. If you know dx/dt = f(x,u) (where u is the input), the completely obvious approximation by computer is to execute x += dt * f(x, u) at intervals of dt seconds. Said lecture takes ages to get to this point, unnecessarily puts things in matrices, and never puts the point so concisely. IMO, it's easy to use linear algebra to belabor simple points.
Recently got annoyed with the overuse of math in http://www.eecs.umich.edu/courses/eecs461/lecture/Lecture8.p... . The lecture covers numerical integration in an embedded system. If you know dx/dt = f(x,u) (where u is the input), the completely obvious approximation by computer is to execute x += dt * f(x, u) at intervals of dt seconds. Said lecture takes ages to get to this point, unnecessarily puts things in matrices, and never puts the point so concisely. IMO, it's easy to use linear algebra to belabor simple points.
You know the missing mathematical background isn't actually that hard to learn. And once you learn it, you may notice important details you're currently missing.
For instance the lecture you got annoyed with was covering how to correctly model a second degree differential equation. So you know that d^2x/dx^2 = f(x, u). This is a different and more complicated problem than a first order differential equation. They handled it by turning it into a system of first order differential equations. And then demonstrated how the obvious way to do it lead to numerical instability, creating a physically incorrect result. And then they had to do more complex stuff to avoid the numerical instability.
The underlying concepts share a lot with a simple first order equation. But the generality is letting you tackle and deal with issues that a simple model can't. Plus the technique described can be expanded in a straightforward way to model a combination of interacting things.
But you froze up at the idea of linear algebra and didn't even realize that they were dealing with a much more complicated problem than you thought.
For instance the lecture you got annoyed with was covering how to correctly model a second degree differential equation. So you know that d^2x/dx^2 = f(x, u). This is a different and more complicated problem than a first order differential equation. They handled it by turning it into a system of first order differential equations. And then demonstrated how the obvious way to do it lead to numerical instability, creating a physically incorrect result. And then they had to do more complex stuff to avoid the numerical instability.
The underlying concepts share a lot with a simple first order equation. But the generality is letting you tackle and deal with issues that a simple model can't. Plus the technique described can be expanded in a straightforward way to model a combination of interacting things.
But you froze up at the idea of linear algebra and didn't even realize that they were dealing with a much more complicated problem than you thought.
It could be easier to understand if instead using z-Transform they shown that you could do better numerical second order integration if you take few previous points into account instead of just one as in naive method of integrating twice with forward Euler integration.
Of course purpose of this lecture was probably not to make you understand what you are actually doing while numerically integrating, just to teach you how to put this stuff into simulink that has z-Transform as one of it's primitive building blocks.
Of course purpose of this lecture was probably not to make you understand what you are actually doing while numerically integrating, just to teach you how to put this stuff into simulink that has z-Transform as one of it's primitive building blocks.
I did just fine, thank you. Matrix form is not necessary to tell me that we need to approximate both x and dx/dt in order to use our knowledge of d^2x/dt^2. It is not particularly difficult to remember that the second derivative is the first derivative of the first derivative.
I run screaming from courses that cover mathematical background I don't have
I had the opposite attitude when it came prerequisites in college. For example, I had no problem taking a course that required linear algebra, or signal analysis, or data structures (none of which I had taken; eg, I took Data Structures II before I). You pick up the missing pieces as you go. I actually had more fun that way - "Signal Analysis" was much more interesting when I already knew the applications!
BTW, I didn't do this on purpose. I was trying to get these classes in to fit my schedule.
I had the opposite attitude when it came prerequisites in college. For example, I had no problem taking a course that required linear algebra, or signal analysis, or data structures (none of which I had taken; eg, I took Data Structures II before I). You pick up the missing pieces as you go. I actually had more fun that way - "Signal Analysis" was much more interesting when I already knew the applications!
BTW, I didn't do this on purpose. I was trying to get these classes in to fit my schedule.
I was always annoyed during college how math notations occluded what was actually happening.
I always had to translate math to common sense. Translate derivatives to speeds of gain, gradients to slopes, vectors to movements and so on. Only then I could understand why given theorem actually might be true (and useful). I could often spot some error in lecture during this process. Unfortunately my peers were most of the times not able to do the same (at least not in real-time).
I think much more general understanding could be achieved if my teachers could stand back from math notation and do some hand-waving more often.
I always had to translate math to common sense. Translate derivatives to speeds of gain, gradients to slopes, vectors to movements and so on. Only then I could understand why given theorem actually might be true (and useful). I could often spot some error in lecture during this process. Unfortunately my peers were most of the times not able to do the same (at least not in real-time).
I think much more general understanding could be achieved if my teachers could stand back from math notation and do some hand-waving more often.
It's weird - I find it a lot easier to understand theory when I'm presented with an opportunity to use said theory. Linar algebra isn't terribly hard to get started on, and seeing it used will reinforce your understanding.
Hi. I'm the author of that post. Wow. I wrote my opinion on a message board and now it's everywhere. On some sites they are referring to it as an "article." Ah, the risks you take when you post stuff on the web...
In any case, I think some of you make good points below about weaknesses in my argument (which, if I WAS writing an article, would have been more rigorously researched and reasoned), but I stand behind the majority of what I wrote.
Feel free to ask questions, challenge me or call me a narcissist. It's all good.
In any case, I think some of you make good points below about weaknesses in my argument (which, if I WAS writing an article, would have been more rigorously researched and reasoned), but I stand behind the majority of what I wrote.
Feel free to ask questions, challenge me or call me a narcissist. It's all good.
I think all jobs require problem solving to some degree just that some jobs are easier than others and others while not necessarily easy are sufficiently well defined to enable a worker to develop expertise in that area. And with expertise in a field problems are often solved intuitively without going through all the "problem solving mechanics".
Getting to this point where problems can be solved intuitively can sometimes take years of study and practice. We can look at this as an investment. A person invests time and energy in wiring his or her brain to solve a particular set of problems and after achieving this goal the person is left with little desire to repeat the process for another problem domain. A doctor may apply problem solving skills to diagnose and treat a rare disease but may not be able muster the effort required to learn photoshop. It's not due to a lack of problem solving abilities but more of a desire to step not out of their problem solving comfort zone.
Getting to this point where problems can be solved intuitively can sometimes take years of study and practice. We can look at this as an investment. A person invests time and energy in wiring his or her brain to solve a particular set of problems and after achieving this goal the person is left with little desire to repeat the process for another problem domain. A doctor may apply problem solving skills to diagnose and treat a rare disease but may not be able muster the effort required to learn photoshop. It's not due to a lack of problem solving abilities but more of a desire to step not out of their problem solving comfort zone.
The main purpose of the educational system, up to and including graduate school, is to produce obedient workers. We're fortunate to work in a field too young to have developed an orthodoxy. That's how we can get away with thinking so much. Most people are not so fortunate.
See also http://www.metafilter.com/70699 (and the recently published printed version, although IMO the material added beyond the original article is not worth getting the book for, which is a shame).
I was trying to explain the difference between hackers, if you will, and "programmers" just going through the motions. I likened it to the difference between someone who spoke a language frequently and someone who was working out of a conversational phrasebook. The thing is one cannot help but learn a language as a human being, but you can sure go through life without problem solving skills!
Well all my school life I have had some great English/Hindi literature teachers. I may be safe in saying that I have learned problem solving from them. They were not of the types which tell you to learn the poem or prose by heart.Give prepared notes on certain lines and then tell you to rote learn them.They told us how to get to heart of a story.On of my most experienced teacher(our batch was her last,she retired at the age of 60 I think),told us that just reading a story is a disgrace in itself to the author [when you are in a Literature class].Every story was a part of his/her life when he/she wrote it.Every poem was a real emotion when it was laid out on paper.She used to throw random books at us (on me sometime literally), told us to "understand" them. Then we had this huge discussion over the plot and how this can be interpreted or how that is misinterpreted by most people.Those were one of the best days of my school life.
Now this may seem not a particular type of "problem solving" to some people. But it is when you take it seriously.You get the same sensation when you make a perfect connection between an event in the authors life and how he interpreted it in his story,that you get when you solve a mathematical problem.Because these two have essentially the same procedure.1)Finding the roots of the problem ,2)Understanding the roots of the problem,3)Interpreting the roots of the problem,4)Using those to solve the problem. Perhaps that's why one of my Maths professor used to say "Look at the problem,its screaming its solution to you".
Now this may seem not a particular type of "problem solving" to some people. But it is when you take it seriously.You get the same sensation when you make a perfect connection between an event in the authors life and how he interpreted it in his story,that you get when you solve a mathematical problem.Because these two have essentially the same procedure.1)Finding the roots of the problem ,2)Understanding the roots of the problem,3)Interpreting the roots of the problem,4)Using those to solve the problem. Perhaps that's why one of my Maths professor used to say "Look at the problem,its screaming its solution to you".