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I don't remember "move" being a valid mathematical operation. I'll have to go look that one up.
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It's a division or multiplication by ten, or anyway by the base of the number system you are using. Duh.Posts: 10645 | Registered: Jul 2004
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quote:Originally posted by King of Men: You might as well doubt the existence of science fiction because you cannot define it satisfactorily. We can see that intelligence exists.
We can also see that it is not one-dimensional and therefore not identifiable by a single integer.
I do not disagree. I also do not think that this contradicts anything I have been saying.
quote:One reason I think many people have difficulty understanding math is because teachers rarely explain how the math concept he/she is teaching applies to the real world.
I think you have the causality backwards. People who understand calculus can see it working all around them. If you don't get it, though, it's just abstract symbols to you, and how many abstract symbols do you deal with outside of math class? You have to be able to think on two levels at once: One level is (to take an example I encountered in real life recently) the manipulation of lambda, f, and c to get f alone on one side of the equation; the other is the equivalence between f and frequency, lambda and wavelength, c and the speed of light. It is a bit like pointers in C++: Either you understand what they are and how they work, or you just combine the *s and &s until you get something that compiles.
Now, it is true that this ability to think in two levels at once can be improved by practice. But I don't think that explaining things in terms of wavelengths and frequencies is necessarily helpful; what is wanted is a map between concepts and symbols, plus a skill set in manipulating the symbols, plus perhaps the meta-skill of making your own symbol map when you encounter a new concept. Thinking about concepts is not going to help on its own.
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To take another example: Nobody complains about being unable to learn multiplication on the grounds that they don't see the relevance. But really, how often do you actually need to do hardcore multiplication and not have a calculator handy? But pretty much everyone understands what multiplication is about and how it works (provided they haven't been taught New Math, of course...), even though they may sometimes make mistakes in the actual operation. So if you asked someone to give an example of multiplication in the real world, they would think for a moment and say, perhaps, "Tax rate times income", or "price of gas times gallons bought", or something. But it has never been explained to them in those terms! They just learned quite abstract lookup tables - three times two is six, seven times two is fourteen - and then did some problems with it.
If you know the math, then you can see applications easily. But knowing the applications doesn't make learning the symbols any easier; to see this, consider a student with some degree of mental retardation, who struggles with multiplication. Is he going to have an easier time when you explain that "This is used for calculating how much you have to pay for gas"? I think not. Or similarly Synesthesia with her dyscalculia. Certainly she understands how arithmetic applies in real life! She's just missing the relevant circuit in her brain, so she can't do it.
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I wonder if the way I do math in my head is similar to anyone else's?
Say, for example, I wanted to figure out a 15% tip on a meal that cost $43.26. What I'll usually do is just round to the nearest 10, in this case 40, and figure that for every $10, I pay $1.5 in tips. So I take 1.5 and multiply by 4, which, in my brain is 1.5x2=3 and 3x2=$6.
Then it's $.15 for every dollar, or $.45; $.02 for every 10 cents, or .04. So I end up with a tip of about 6.49, which I'll usually just round up till I get an even dollar amount on the final charge.
This I figure by just taking the cents of the charge and adding enough so I get an even cent amount, in this case I add $.04 so that I'd get 43.30, then it's easy to see that I just need $.70 more to get to 44.00. So I leave a tip of 6.74 which leaves a total charge of $50 even.
Now to check my math on a calculator: 43.26x.15=6.489, or 6.49 rounded up. $50-43.26=6.74.
Sure, I could do long multiplication or division to figure this out, but that's hard to do in my head. By breaking it down, I can do the whole thing in my head and it doesn't require much effort.
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quote:Originally posted by katdog42: I don't remember "move" being a valid mathematical operation. I'll have to go look that one up.
It's more commonly called "Shift Left" and "Shift Right." I think my teacher used the word "move" because he probably also wanted to include moving from one memory location to another. And as King of Men said, it's just multiplication or division by the base. And what do mean by "valid" mathematical operation, anyway? I would have thought just about any function could be called a mathematical operation.
quote:Originally posted by King of Men: The 'move' operation implicitly assumes that you understand division and fractions, though.
Not necessarily. "move" contains only a subset of all division, and is a simpler operation. One doesn't really have to know anything about division, or what a decimal point means in order to move it.
But you are right in that I'm just giving an algorithm. Meaning and understanding is what we really want, but an algorithm alone will not provide it. Blindly following a series of steps won't help. But, like I said before: Just using the higher operations help make life easier.
quote:I think you have the causality backwards. People who understand calculus can see it working all around them.
Yeah, you're probably right, that does make a lot of sense. I guess I said that because the biggest complaint I hear from people who don't understand math is "I don't get this. Why am I learning this? When am I ever going to use this in real life?" So maybe they only ask that because they haven't been seeing it working all around them, so why should something new?
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They did a study looking at how math was taught and how well students did and it turned out understanding why it was useful, or having it taught as fun did not improve students understanding. Simply being taught to do it because you have to was the best method.
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Well, passing the test was the measure they used for determining understanding. I don't see a good way to test understanding other then that in this kind of survey. Kumon (tutoring) basically folow this method- they have students do many many math problems that are all similar until they are getting a certain perentage right. They don't spend any time explaining why it is important or trying to make it fun and I have heard they are very successful.
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For basic arithmetic, I cannot think of a better way to learn than just plain rote drill. All four arithmetic operators are just lookups, with the possible exception that you can do addition by counting on your fingers.
Then for trig and suchlike, you have to understand what's going on to set up the equation correctly, but after that it's rote memorisation and a small amount of intuition again. The amount of intuition increases as you go on to calculus; but I don't think there's anything taught in your basic undergrad math that doesn't benefit from quite a bit of rote memorisation.
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quote:Originally posted by TheGrimace: this conversation greatly saddens me. Using calculus to solve algebra/trig problems on the SATs/ACTs back in highschool made me happy though
I'm a senior in high school and because I used calculus on my SAT I realized i answered a question wrong.
The SAT doesn't agree that (1)/(infinity)=(0)
Edit: I was really thinking that the limit of 1/infinity=zero or would it be that the limit of 1/x is 0 as x approaches infinity. I'm not all that sure how it would be stated.
quote:Originally posted by fugu13: What I usually do is figure out ten percent (shift the decimal point), then add in about half of that more.
I just usually tip 20% but if i was tipping 15% I would do it this way as well.
quote:Originally posted by scholarette: Well, passing the test was the measure they used for determining understanding. I don't see a good way to test understanding other then that in this kind of survey. Kumon (tutoring) basically folow this method- they have students do many many math problems that are all similar until they are getting a certain perentage right. They don't spend any time explaining why it is important or trying to make it fun and I have heard they are very successful.
Ugh, I cannot think of a more effective way to get someone to hate math even more. And I don't really think that this would be a particularly effective method in the grand scheme anyway. Sure the person understands how to get that style of problem right, but without being able to extend that knowledge and apply it to other things, it's very nearly worthless.
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According to a recent SciAm article, students who learn concepts abstractly (like use x and y) score significantly better on tests using applications of those concepts then students who were taught using concrete examples (slices of pizza, distances, etc). The students taught with concrete examples were not able to apply the information to another situation. They did however, learn the concrete example faster, which is why teachers are convinced that concrete is the better way.
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quote:Originally posted by Brett Moan: The SAT doesn't agree that (1)/(infinity)=(0)
Edit: I was really thinking that the limit of 1/infinity=zero or would it be that the limit of 1/x is 0 as x approaches infinity. I'm not all that sure how it would be stated.
Right, that's how you would say it. One divided by infinity doesn't equal zero, but the limit of 1/x as x approaches infinity does.
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I believed there was unbiased truth in mathematics but eventually I realized there was no truth, only observation. If we had 8 fingers, would you count to ...6,7.8,11,12,13,14,15,16,17,18,20.....and would it still work? I was discouraged when my grade would suffer for not showing work or for using ratio and proportion in lieu of a formula. When more than the solution mattered, I was discouraged.
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quote:Originally posted by scholarette: According to a recent SciAm article, students who learn concepts abstractly (like use x and y) score significantly better on tests using applications of those concepts then students who were taught using concrete examples (slices of pizza, distances, etc). The students taught with concrete examples were not able to apply the information to another situation. They did however, learn the concrete example faster, which is why teachers are convinced that concrete is the better way.
Thanks for posting that because it's basically what I was going to mention just way more official - that knowing the concept (as opposed to why it is useful) is a BIG one. To use myself as an example, someone who was good at math in high school and ended up majoring (with teaching certification grades 7-12) in it in college... I didn't know why a variable was called a variable (note: super simple case here) until a couple years after first using them; then I realized that is it because it varies. It was a 'duh' moment, but it made some things click. Obviously, there are more complex concepts to be found in the field of mathematics, but there you go. If you understand the concept, you can often do what Sean Monahan did in his story on the first page of this thread. That's really cool, by the way.
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I'm getting back in to math after not taking any math in college since I'd taken Calculus in high school. I'm taking a Calculus II class at the community college near me and I'm loving it. I love the logic and problem solving that goes in to it. Yay for math!
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I haven't posted in months (perhaps close to a year), but you've pulled me out of my shell.
Tman, what do you think my number 1 reason is for doing math? Why in the world have I chosen to suffer through grad school (2 years to go!) instead of making lots of money, say, being an actuary? Why am I going to spend my life solving problems that will almost certainly have no practical applications, especially if we confine the effects of their solutions to my life time?
It's the beauty in it. Yes, I think mathematics is beautiful.
Now, first and foremost, when I say "mathematics", I do NOT mean arithmetic, trig identities, solving equations, and to a large extent, I exclude calculus as well. These ideas are certainly very important in (almost?) all branches of mathematics, but they are not "mathematics" as I use the term.
To me, mathematics is the art of problem solving. It's the art of finding the most elegant reasons why a given fact should be true. It's the art of finding the perfect argument.
I think it's clear, then, that I view mathematics more as an art, less of a science.
It's more about creativity than applying formulas, more driven by the search for beauty than the search for applications.
As Lockhart says, "Music can lead armies into battle, but that’s not why people write symphonies. Michelangelo decorated a ceiling, but I’m sure he had loftier things on his mind."
In the same way, yes, mathematics often has important practical applications, but that is not it's purpose.
So, I say, give math a shot. When/if you go to college, take a beginning undergraduate math major course. Maybe you'll get a hint of the beauty of mathematics. Maybe once you see the beauty, you'll be hooked.
And as far as the idea that all math has been solved, I invite you to take a look here for a list of 171 new math papers that are coming out TOMORROW. I also suggest you check out this for a list of 324 math papers that have come out in the last week.
Math is still a vibrant field, full of innumerable problems. And more often than not, the solution to one problem merely opens the door for more questions.
Edit: I just wanted to add that if what I wrote intrigues you, I highly recommend you read a decent portion of Lockhart's Lament, especially the part where he describes the nature of mathematics.
Also, just be be clear, while I agree wholeheartedly with everything Lockhart says about the nature of math, I do NOT agree with much that he says about mathematics eduction.
posted
Although what you say is true, it is als unfortunately true that most people are never going to grasp the beauty of mathematics. It takes a certain amount of brainpower to enjoy or be good at symbol manipulation; most people don't have it.
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Sigh. Once more: Having a brain wired to do X can be restated as "my brain has the power of X". Therefore, a lack of such wiring is a lack of brainpower. See the whole argument on the previous page of this thread.
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If you lack the ability - the power - to do math, you just lack it. Where the lack comes from is not relevant.
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But you didn't say they lacked the ability to do math, you said they lacked amount of brainpower. The difference is not necessarily one of amount, but of quality.
After all, just because, given sets A and B of mental ability, A - B = C, where C is non-empty, does not mean |A| > |B|
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Sure, if you say something other than what I said, and something other than what you said, something else is true. Unsurprisingly.
quote: It takes a certain amount of brainpower to enjoy or be good at symbol manipulation; most people don't have it.
Note the use of the word amount. You're asserting that there is a quantity of brainpower required to be good at symbol manipulation, and that people lacking that quantity cannot do it. This neglects the case of people who have plenty of brainpower, but such that it isn't very useful for that sort of symbol manipulation.
Shall we break it down into symbols again? For your statement to be true requires the following statement to be true (assuming we accept the proposal of different sorts of intelligence):
"good at symbol manipulation" member of A and "good at symbol manipulation" not member of B implies |A| > |B|
And that's just not true. Or do you assert this is not necessary for your statement I quoted above to be accurate, and if so, could you explain why?
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Well, perhaps you shouldn't use set notation to discuss something that is not discrete. In arbitrary units, let 1 be the threshold of symbol-manipulation-ability (SMA) above which people are able to enjoy simple forms of math. Plainly, 1 is greater than 0.5. Also notice that SMA is usually correlated with other forms of intelligence. although I understand that there are particular forms of brain disabilities where this is not true.
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To me, mathematics is the art of problem solving. It's the art of finding the most elegant reasons why a given fact should be true. It's the art of finding the perfect argument.
To me this is only half the story. There is also an exploratory side to it, where new entities are defined and their structure investigated. I have been told of a quote by Grothendieck in which he dismisses some highly-decorated mathematician as a problem-solver (I forget who).
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KoM: there's an obvious bijection between types of intelligence with a variety of discrete levels and sets, so I feel just fine using set notation.
Perhaps this should be phrased a different way: considering two people, where one is a good painter/bad mathematician, and one is a good mathematician/bad painter, do you feel the good mathematician is necessarily more intelligent?
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Painting is a trainable skill to a larger extent than mathematics is; the mathematician can likely become a competent draftsman with practice. But assuming for the sake of argument that both have made good-faith efforts at learning the other's skill, and failed, then no, these two are about equally intelligent. (Holding their other skills equal, obviously, and ignoring the weighting factor that math has rather more applications than painting.) A splendid showing of the power of constructed examples. In the real world, as I noted, skill at mathematics is correlated with other measures of intelligence.
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Where is the line drawn for what skills are representative of intelligence? In place of painter, put sprinter.
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Correlated? Sure, but there are numerous extremely intelligent people in the world who aren't very good at math, and numerous mathematicians who have areas they are very deficient in. I doubt I would have any problem finding a mathematician who was very bad at painting and a painter who was very bad at math, in the real world. Or are you asserting such individuals could not exist when you call it a "constructed example"?
And I won't even ask you to give a cite for this unsubstantiated assertion
quote:Painting is a trainable skill to a larger extent than mathematics is
quote:I doubt I would have any problem finding a mathematician who was very bad at painting and a painter who was very bad at math, in the real world.
Could you find such a pair who had both made a good-faith effort at really learning the other guy's skill? And in any case, what's with the airy dismissal of the correlation? I never claimed it is a 100% correlation, nor do I need it to be. Your counterexamples demonstrate only that the correlation is not 100%.
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natural_mystic: a wonderful question, but unfortunately one we aren't very good at answering. Mostly it is a situation of, "I know it when I see it".
I should say that I don't consider everyone who makes very good paintings very intelligent; in some sense I was using that as shorthand for being very good at thinking about how to make good paintings, and applying those thoughts systematically, which I think is pretty definitely a sort of intelligence.
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Yeah, it might require asking around, and possibly having a few potential candidates put forward a good faith effort over the period of a few months or years, but I don't see any real difficulty.
And where did I dismiss the idea of there being correlation? I think you're imagining I'm saying things that I have not said. I've been discussing existence, not correlation.
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Then you've been barking up the wrong haystack. I have not claimed that there are no intelligent people unable to do math. I have claimed that, other things equal, those with math ability are more intelligent than those without, and further that (dropping the requirement of other things equal) math ability does correlate with other ability. Existence is uninteresting in this context.
quote:And where did I dismiss the idea of there being correlation?
You didn't, but you dismissed its significance.
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In your quote, I'm ignoring the rare special cases and using the correlation, which is a much more powerful predictor.
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At best that's dismissing the relevance of correlation to a question of existence. Not dismissing its significance in general.
Your statement that I was arguing against, which I can quote yet again if you've forgotten yet again, was not about correlation. I am quite happy to agree that there is a lot of correlation between many types of intelligence.
Also, you might note what I was replying to:
quote:A splendid showing of the power of constructed examples. In the real world, as I noted, skill at mathematics is correlated with other measures of intelligence.
You were implying that the correlation meant my example didn't mean much in the real world. But even an incredibly strong correlation (and I doubt most forms of intelligence correlate at anything above .8 -- I mean, I know several mathematicians who can't work outside their specific sub-fields very well, despite a lot of trying, because their brains work better with topology or analysis or whatnot) in a world of this population, with this many intelligent individuals, will leave millions of people with intelligences that don't overlap much with millions of others. That's not the realm of a 'constructed example', that's the realm of an important distinction that can't be waved away with "correlation".
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To me, mathematics is the art of problem solving. It's the art of finding the most elegant reasons why a given fact should be true. It's the art of finding the perfect argument.
To me this is only half the story. There is also an exploratory side to it, where new entities are defined and their structure investigated. I have been told of a quote by Grothendieck in which he dismisses some highly-decorated mathematician as a problem-solver (I forget who).
Fair enough, though I would also say that the exploratory side can be subsumed in both "finding the most elegant reason why a given fact should be true" and "finding the perfect argument". One will almost certainly have to engage in exploratory activities to understand why things are true. However, at the end of the day, there should be a proof, i.e. a well reasoned argument. And the more beautiful the argument, the better.
In fact, as Hardy said "The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics."
I suppose the point to everything I'm saying is that mathematics is much more about discovery and/or creation than most people think. In this way, it is, in my opinion, best thought of as an art which happens to have occasional practical applications.
Further, it is quite often the case (and admittedly quite often not the case) that there is NO scientific component to mathematical discovery, in the sense of experimental evidence. Sometimes you just play around with a handful of axioms and out pops some beautiful result.
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Well, there's also the fact that the physical world consists of mathematics. Matter is energy. Energy is fields. Particles are like knots in the fields. Fields are just numbers in space. Physical laws are mathematical formulae without any mechanical explanation or underpinnings. As best as we can tell given our current knowledge, nothing that exists is realer than math.
There was a great quote by someone who said Mathematics is the language God speaks when creating the universe, or something to that effect that sounds a lot niftier.
Anyway, math is extremely cool, but I happen to be an applied scientist, and I think it's coolest of all when it teaches us how to build neato stuff using its principles.
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