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The Math I Learned After I Thought Had Already Learned Math

118 points by snake117 11y ago

trengrj 11y ago
I think that maths at school it still too focused on creating engineers (the non software kind) and much of the material is aimed at producing people who can do calculus. I'd prefer for there to be more focus on proofs, logic, geometry, number theory, graph theory, and cryptography etc. I think these concepts would give people a more rounded understanding of mathematics and prevent their experience of mathematics being one of pain.
- antimora 11y ago
  I would add statistics to this list. Science and pretty much all the numbers quoted out there are based on statistics.
  - jacobolus 11y ago
    It’s impossible to reasonably understand statistics without studying calculus. Everyone I’ve talked to who took a statistics course without first studying calculus had an overly formulaic, theoretically wishy-washy, and often wrong understanding of the meaning of basic statistical distributions and techniques.
    - grayclhn 11y ago
      Um... I feel like there is likely to be a lot of selection bias in your experience. Now, I would notice that, having taken calculus before statistics myself, but...
    - thaumasiotes 11y ago
      The introductory economics courses at my school didn't require calculus. This would have been less annoying if every single topic hadn't been explicitly introduced as finding the slope of, or area under, particular curves.
      - TeMPOraL 11y ago
        Those who don't learn calculus have to reinvent it, poorly. C.f. so-called "AUC" [0], which is apparently an important metric in medicine. [0] - https://en.wikipedia.org/wiki/Area_under_the_curve_%28pharmacokinetics%29 https://en.wikipedia.org/wiki/Area_under_the_curve_%28pharma...
        stdbrouw 11y ago
        What exactly bugs you about AUC? What about ROC curves?
    - stdbrouw 11y ago
      Not necessary at all. You can teach most of the interesting parts of statistics with just the binomial distribution and numerical techniques like bootstrapping. Nothing about that is wishy-washy, it's just a modified curriculum that sidesteps some interesting things that require calculus but in return allows you to focus on some other really interesting things that don't require it. There is, for example, nothing rigorous about being able to prove and use the Central Limit Theorem – the whole point of the thing is that it's an approximation and not always a very good one at that. And this is the origin of many statistical techniques and tests: computational shortcuts. The fact that these shortcuts require a good understanding of math doesn't suddenly make them any more central to a conceptual understanding of statistics than they are. Two good examples of good stats intros that are light on the math: Allen Downey's Think Stats http://greenteapress.com/thinkstats2/ http://greenteapress.com/thinkstats2/ or the more traditional/frequentist http://onlinestatbook.com/ http://onlinestatbook.com/.
- andrepd 11y ago
  On the other hand, perhaps you are biased in favour of the math relevant to computer science, such as number theory, graph theory, and cryptography?
  - trengrj 11y ago
    Haha, I could agree with that. I think though the problem with calculus is that it is so cumulative, so doing badly in one year leads to the next etc.
    - deleted 11y ago
      [deleted]
    - hugh4 11y ago
      That's the nature of mathematical knowledge though. If you can't understand basic calculus then there's a helluva lot that's downstream from there that you won't understand either.
  - kaitai 11y ago
    It might be true... but ask an American 17-year-old with collegiate aspirations what he or she will take in college. "Calculus!" What else? Blank stare. What other math exists? This number theory you speak of... as far as an average high schooler is concerned, you're making stuff up.
- hugh4 11y ago
  I don't know if there's too much calculus, but there's definitely too much of the wrong type of calculus. Congratulations, you have just learned to integrate. Let us now spend the next three years running through several thousand different examples of analytic integration of one-dimensional continuous functions. More dimensions? Path integrals? Numerical integration methods that you'll almost certainly have to apply in the real world because most functions don't have analytic integrals? Differential equations? Never heard of them. Now let's all integrate (x^2 - sin(x))/coth(x).
  - Gibbon1 11y ago
    I think there is too much focus on the mechanics and not enough on the big picture behind what you're trying to do. Mostly it's a lot of. Step one recognize the problem. Step two apply the solutionator+ and turn zee crank. Step three receive kibble. +Solutionator: Bunch of mechanical steps that the student understands in the exact same way a trained monkey understands an organ grinder.
    - gh02t 11y ago
      It's kind of funny, most people who major specifically in math or applied math will take something equivalent to a course in Real Analysis. RA is basically starting over from scratch and reteaching you the exact same material as calculus, but in a much more formal and rigorous manner. Mostly it focuses on eliminating the not-so-rigorous concept of an infintesimal quantity and formalizing the notion of limits and convergence (albeit, there is a way to rigorously treat infintesimals called nonstandard analysis, but I digress). It involves a lot more open-ended thought than the endless memorization of earlier calculus classes. It's often dreaded by undergraduates as being very proof-heavy, but it was one of my favorites. It always felt like learning the "essence" of calculus, at least for me (who sucks at/dreads rote memorization).
      - JackFr 11y ago
        That was my experience exactly. In the first class the professor told us that everything we learned in calculus was true, but not quite for the reason they taught us. We then re-proved everything withpoint set topology. But it felt like real math.
- aratno 11y ago
  For me, this has been about proving that mathematics is about more than just numbers. The world is made up of quantities, relationships, nodes, probabilities, and much more. If students are expected to use math to make better decisions and understand the mechanisms of the world, a breadth of math needs to be taught earlier on.
  - brianclements 11y ago
    I think this was completely lost on all my math teachers growing up. It wasn't until I started teaching myself programming and exploring the patterns of the universe and reading philosophy and science and even too from being a musician and teacher did I start to FEEL and KNOW math and experience it's concepts directly. Not just memorize steps. I almost feel like math and philosophy should be combined in school into a hybrid course that is very experimental (verbal hypotheses, mathematic proofs) and should focus simply on investigating the world of ideas around you. Get them into a mode of thought where your creative and skeptical insights can get you into mental conundrum, and your mathematical skills can help you get out of it. How wonderful would it be to get students to "rediscover" fundamental insights about the world? THEN you crack the book open and teach them who discovered it first 2000 years ago.
ginko 11y ago
Is "secondary math" university mathematics? How can anyone think he's "learned" that?
- titanomachy 11y ago
  No, it's high school math. The kind you learn before university. secondary school == high school in many places.
xpinguin 11y ago
I really didn't get the article. I mean, I've got that we have some mapping f: R -> R, or equivalently f: (x, y) \in RxR, so we could represent it by drawing on 2d plane, but no further... Author claimed, that he had been missing something essential regarding that stuff, but I wonder what exactly does he talk about?
- ColinWright 11y ago
  People think of "drawing a graph of a function" as drawing a line, so if y=sin(x) they end up with a wriggly line going infinitely far left and right. What they don't think of is that the line is actually the collection of points (x0,y0) such that the equation y0=sin(x0) is true. Kids in high school don't think of the graph of a function as being a subset of points of the plane, being: { (x,y) : y=sin(x) is true } Realising that opens the doors to equivalences between different ways of thinking. We can think of a permutation of objects as both the act of permuting them, and as the result of applying that permutation to the default initial position. We can think of a vector (4,6,9) as a location in space, and as the movement to get from (x,y,z) to (x+4,y+6,z+9). We can think of "3" as a location on the number line, or as the action of adding 3 to something, or as the action of multiplying 3 my something, and so on. We can think of the graph you draw as a line, or as a subset of the plane, and we shift effortlessly between them, deliberately blurring the distinction, and from that blurring can come power. Does that help?
  - kbenson 11y ago
    In my first calculus class, they introduced integrals to us through Riemann sums, and ever increasing numbers of regions for those sums. It makes thinking in this manner somewhat easier to come by, as it was touched on in the introduction.
  - xpinguin 11y ago
    Thanks! Seems like I'd been getting it right since the beginning, so I've missed the point.
  - sago 11y ago
    This hurts when they deal with equations like x^2 + y^2 = k and can't even begin to understand how you could graph something like that. It took me a while to grok it, so I second his experience of this not being well stressed (30 years ago, admittedly). I did further maths A-level (and got an A), but didn't grok it until uni.
toothbrush 11y ago
Cool, i also had an "aha" moment while reading: “It was only after grad school that I learned (from Lockhart’s book Mathematician’s Lament) to consider natural numbers as stones that can be arranged in various patterns that illustrate the different properties of a number. For example, evens are piles of stones that can be arranged into two equal rows, and square numbers have just the right number of stones to make a square! It’s really fun thinking about various operations in this way, and there are some beautiful proofs based on this technique. For example, why the sum of the odd numbers 1 + 3 + 5… Is always a square.” I should hang out with maths teachers more often!
- roymurdock 11y ago
  This is a really helpful way to think about natural numbers. I like to remind myself that we can gain efficiency through abstraction, but true, intuitive understanding comes through concretization/deconstruction. Interestingly enough, I use round stones to represent ideas when I meditate. As the ideas come to me, I pick them up, examine them, and weigh them. If the idea is pressing, I delve into it and think it through. If not, I put the stone down and wait until my mind picks up the next. Mental stones: some of modern life's most useful tools :)
- j2kun 11y ago
  I think the vast majority of math teachers don't go on to learn new things and perspectives on math after college.
- thaumasiotes 11y ago
  > and square numbers have just the right number of stones to make a square! This one bothers me, because I don't think it's right. You can define the triangular numbers easily: 1, 1+2, 1+2+3, 1+2+3+4, ... and it's easy to arrange that number of dots into a triangle. The difference between two consecutive triangular numbers is always increasing by one. The difference between consecutive square numbers always increases by two. The squares are 1, 1+3, 1+3+5, 1+3+5+7, ... and it's also easy to arrange square numbers of dots into squares. So the difference between pentagonal numbers always increases by three. The first few pentagonal numbers are 1, 5, 12, 22, .... But pentagons don't tile the plane. How do you arrange a pentagonal number of dots into a pentagon? (Wikipedia has a proposed solution on display at https://en.wikipedia.org/wiki/Pentagonal_number https://en.wikipedia.org/wiki/Pentagonal_number , but the pentagons it constructs have no obvious internal structure. The square you construct from a square number of dots is symmetric wrt rotation.)
rsy96 11y ago
The revelation he has had looks trivial to me. Graphs as equations, equivalence of expressions or matrix as linear operators are all something I learned and understood from the very beginning. I wonder if it has something to do with the textbooks. As a Chinese I often found American textbooks on mathematics so softcore. They have so many analogies, so many "real world" examples that masquerade the true mathematical meaning of the concept. Many of Chinese people argue that these are the reason that Americans are more creative, but I cannot help but wonder maybe the lack of rigor underlies some of problems with American math education. Or maybe I am just the exception. Maybe other Chinese struggle with math just the same.
- arsenide 11y ago
  American graduate mathematics student here at a US university, speaking from my perspective. This is perhaps not useful with respect to high school or first/second year undergraduate mathematics, but I choose to include this for sake of another perspective many may not be aware of. Whenever anyone says something is "trivial" -- I understand that to mean simply "it is obvious to me." I try to NEVER explain things in such ways -- to say things are trivial because it is very clear to me that is NOT trivial to many other people. And there has to be some respect given to this idea. I abhor when professors use this sort of language -- instead of stating it is trivial, explain using an extra sentence or two why it is so. If it takes more than that, it is likely not trivial. The best way for me to learn new concepts (in my graduate classes, for example) is as follows. And it is the method all of my professors currently use whether they know it or not. First, they begin with the definition of something. For example (one of my classes last week expressed this idea in particular): in metric spaces they say "A function is upper semi-continuous at x_0 if for all x approaching x_0 the limit supremum of f(x) is smaller or equal to f(x_0)." But what does this mean? You can use the "epigraph" as a way to explain this idea in a different way, explaining what it means to be a u.s.c. and/or a l.s.c. function in a more understandable way. From here, then, after understanding this, I was able to go through the strict mathematical notation of these concepts and follow it more clearly. Instead of starting from the mathematically "rigorous" notation, I am able to jump to a more familiar idea and proceed to link it back to the notation and rigor described in the definitions and associated theorems. This is a much more "efficient" way, in my opinion, than staring at the notation and trying to decipher it for hours. Having examples makes it easier to understand the context of the symbols and rigor involved: by having a specific example of a general form. Is there a lack of rigor in American math education? In general I would say yes -- especially in high school/undergraduate mathematics. But how do people most easily learn? Through examples tying back to definitions. This is more of the American way of teaching in my experience. Furthermore, you state "American textbooks on mathematics" are "softcore." I think this is true for a lot of the lower-level material, but if you pick up Rudin's Real and Complex Analysis I do not think you will feel the same way.
SteveBash 11y ago
I somehow understood the same thing(a graph is a collection of points that makes a function true..) a year ago through Gilbert Strang Big picture of derivatives: https://www.youtube.com/watch?v=T_I-CUOc_bk https://www.youtube.com/watch?v=T_I-CUOc_bk
TerryADavis 11y ago
What comes after Nonlinear Differential Equations, psy-op nigger?
jwmerrill 11y ago
It looks like the blog is down at the moment. Here's google's cached version of the post in the meantime: http://webcache.googleusercontent.com/search?q=cache:7SWRaHWxfiEJ:blog.mrmeyer.com/2015/the-math-i-learned-after-i-thought-had-already-learned-math/+&cd=1&hl=en&ct=clnk&gl=us http://webcache.googleusercontent.com/search?q=cache:7SWRaHW...
jwmerrill 11y ago
Two good ways to represent plane curves are parametric form: (x(t), y(t)) or implicit form: f(x, y) = 0 Parametric form is naturally associated with one point of view of what a plane curve is: the set of points traced out as a parameter is swept over its domain. Implicit form is naturally associated with a different point of view: the set of points that satisfy a certain relation. Functions of the form y = f(x) can be easily re-expressed in parametric form: (t, f(t)) or implicit form y - f(x) = 0 so both the parametric viewpoint and the implicit viewpoint are equally valid and useful ways of understanding the graph of a function. You could rephrase the author's insight as saying that he had always understood graphs of functions parametrically, but later learned to also understand them implicitly. Depending on the application, it may be more convenient to have a parametric representation of a curve, or an implicit representation of a curve. For example, it's easy to find a point on a parametric curve, but hard to test if a point is on a parametric curve; on the contrary, it is hard to find a point on an implicit curve, and easy to test if a point is on an implicit curve. If your curve is the graph of a function, it is easy to convert back and forth between these forms, but in general, converting from one form to the other may be quite hard. For me, the relationship between implicit and parametric representations is a piece of math that I didn't really learn until long after I thought I had already learned math.

118 points by snake117 11y ago

I think that maths at school it still too focused on creating engineers (the non software kind) and much of the material is aimed at producing people who can do calculus. I'd prefer for there to be more focus on proofs, logic, geometry, number theory, graph theory, and cryptography etc. I think these concepts would give people a more rounded understanding of mathematics and prevent their experience of mathematics being one of pain.

antimora 11y ago
I would add statistics to this list. Science and pretty much all the numbers quoted out there are based on statistics.
- jacobolus 11y ago
  It’s impossible to reasonably understand statistics without studying calculus. Everyone I’ve talked to who took a statistics course without first studying calculus had an overly formulaic, theoretically wishy-washy, and often wrong understanding of the meaning of basic statistical distributions and techniques.
  - grayclhn 11y ago
    Um... I feel like there is likely to be a lot of selection bias in your experience. Now, I would notice that, having taken calculus before statistics myself, but...
  - thaumasiotes 11y ago
    The introductory economics courses at my school didn't require calculus. This would have been less annoying if every single topic hadn't been explicitly introduced as finding the slope of, or area under, particular curves.
    - TeMPOraL 11y ago
      Those who don't learn calculus have to reinvent it, poorly. C.f. so-called "AUC" [0], which is apparently an important metric in medicine. [0] - https://en.wikipedia.org/wiki/Area_under_the_curve_%28pharmacokinetics%29 https://en.wikipedia.org/wiki/Area_under_the_curve_%28pharma...
      - stdbrouw 11y ago
        What exactly bugs you about AUC? What about ROC curves?
  - stdbrouw 11y ago
    Not necessary at all. You can teach most of the interesting parts of statistics with just the binomial distribution and numerical techniques like bootstrapping. Nothing about that is wishy-washy, it's just a modified curriculum that sidesteps some interesting things that require calculus but in return allows you to focus on some other really interesting things that don't require it. There is, for example, nothing rigorous about being able to prove and use the Central Limit Theorem – the whole point of the thing is that it's an approximation and not always a very good one at that. And this is the origin of many statistical techniques and tests: computational shortcuts. The fact that these shortcuts require a good understanding of math doesn't suddenly make them any more central to a conceptual understanding of statistics than they are. Two good examples of good stats intros that are light on the math: Allen Downey's Think Stats http://greenteapress.com/thinkstats2/ http://greenteapress.com/thinkstats2/ or the more traditional/frequentist http://onlinestatbook.com/ http://onlinestatbook.com/.
andrepd 11y ago
On the other hand, perhaps you are biased in favour of the math relevant to computer science, such as number theory, graph theory, and cryptography?
- trengrj 11y ago
  Haha, I could agree with that. I think though the problem with calculus is that it is so cumulative, so doing badly in one year leads to the next etc.
  - deleted 11y ago
    [deleted]
  - hugh4 11y ago
    That's the nature of mathematical knowledge though. If you can't understand basic calculus then there's a helluva lot that's downstream from there that you won't understand either.
- kaitai 11y ago
  It might be true... but ask an American 17-year-old with collegiate aspirations what he or she will take in college. "Calculus!" What else? Blank stare. What other math exists? This number theory you speak of... as far as an average high schooler is concerned, you're making stuff up.
hugh4 11y ago
I don't know if there's too much calculus, but there's definitely too much of the wrong type of calculus. Congratulations, you have just learned to integrate. Let us now spend the next three years running through several thousand different examples of analytic integration of one-dimensional continuous functions. More dimensions? Path integrals? Numerical integration methods that you'll almost certainly have to apply in the real world because most functions don't have analytic integrals? Differential equations? Never heard of them. Now let's all integrate (x^2 - sin(x))/coth(x).
- Gibbon1 11y ago
  I think there is too much focus on the mechanics and not enough on the big picture behind what you're trying to do. Mostly it's a lot of. Step one recognize the problem. Step two apply the solutionator+ and turn zee crank. Step three receive kibble. +Solutionator: Bunch of mechanical steps that the student understands in the exact same way a trained monkey understands an organ grinder.
  - gh02t 11y ago
    It's kind of funny, most people who major specifically in math or applied math will take something equivalent to a course in Real Analysis. RA is basically starting over from scratch and reteaching you the exact same material as calculus, but in a much more formal and rigorous manner. Mostly it focuses on eliminating the not-so-rigorous concept of an infintesimal quantity and formalizing the notion of limits and convergence (albeit, there is a way to rigorously treat infintesimals called nonstandard analysis, but I digress). It involves a lot more open-ended thought than the endless memorization of earlier calculus classes. It's often dreaded by undergraduates as being very proof-heavy, but it was one of my favorites. It always felt like learning the "essence" of calculus, at least for me (who sucks at/dreads rote memorization).
    - JackFr 11y ago
      That was my experience exactly. In the first class the professor told us that everything we learned in calculus was true, but not quite for the reason they taught us. We then re-proved everything withpoint set topology. But it felt like real math.
aratno 11y ago
For me, this has been about proving that mathematics is about more than just numbers. The world is made up of quantities, relationships, nodes, probabilities, and much more. If students are expected to use math to make better decisions and understand the mechanisms of the world, a breadth of math needs to be taught earlier on.
- brianclements 11y ago
  I think this was completely lost on all my math teachers growing up. It wasn't until I started teaching myself programming and exploring the patterns of the universe and reading philosophy and science and even too from being a musician and teacher did I start to FEEL and KNOW math and experience it's concepts directly. Not just memorize steps. I almost feel like math and philosophy should be combined in school into a hybrid course that is very experimental (verbal hypotheses, mathematic proofs) and should focus simply on investigating the world of ideas around you. Get them into a mode of thought where your creative and skeptical insights can get you into mental conundrum, and your mathematical skills can help you get out of it. How wonderful would it be to get students to "rediscover" fundamental insights about the world? THEN you crack the book open and teach them who discovered it first 2000 years ago.

ginko 11y ago

Is "secondary math" university mathematics? How can anyone think he's "learned" that?

titanomachy 11y ago
No, it's high school math. The kind you learn before university. secondary school == high school in many places.

xpinguin 11y ago

I really didn't get the article. I mean, I've got that we have some mapping f: R -> R, or equivalently f: (x, y) \in RxR, so we could represent it by drawing on 2d plane, but no further... Author claimed, that he had been missing something essential regarding that stuff, but I wonder what exactly does he talk about?

ColinWright 11y ago
People think of "drawing a graph of a function" as drawing a line, so if y=sin(x) they end up with a wriggly line going infinitely far left and right. What they don't think of is that the line is actually the collection of points (x0,y0) such that the equation y0=sin(x0) is true. Kids in high school don't think of the graph of a function as being a subset of points of the plane, being: { (x,y) : y=sin(x) is true } Realising that opens the doors to equivalences between different ways of thinking. We can think of a permutation of objects as both the act of permuting them, and as the result of applying that permutation to the default initial position. We can think of a vector (4,6,9) as a location in space, and as the movement to get from (x,y,z) to (x+4,y+6,z+9). We can think of "3" as a location on the number line, or as the action of adding 3 to something, or as the action of multiplying 3 my something, and so on. We can think of the graph you draw as a line, or as a subset of the plane, and we shift effortlessly between them, deliberately blurring the distinction, and from that blurring can come power. Does that help?
- kbenson 11y ago
  In my first calculus class, they introduced integrals to us through Riemann sums, and ever increasing numbers of regions for those sums. It makes thinking in this manner somewhat easier to come by, as it was touched on in the introduction.
- xpinguin 11y ago
  Thanks! Seems like I'd been getting it right since the beginning, so I've missed the point.
- sago 11y ago
  This hurts when they deal with equations like x^2 + y^2 = k and can't even begin to understand how you could graph something like that. It took me a while to grok it, so I second his experience of this not being well stressed (30 years ago, admittedly). I did further maths A-level (and got an A), but didn't grok it until uni.

toothbrush 11y ago

Cool, i also had an "aha" moment while reading: “It was only after grad school that I learned (from Lockhart’s book Mathematician’s Lament) to consider natural numbers as stones that can be arranged in various patterns that illustrate the different properties of a number. For example, evens are piles of stones that can be arranged into two equal rows, and square numbers have just the right number of stones to make a square! It’s really fun thinking about various operations in this way, and there are some beautiful proofs based on this technique. For example, why the sum of the odd numbers 1 + 3 + 5… Is always a square.” I should hang out with maths teachers more often!

roymurdock 11y ago
This is a really helpful way to think about natural numbers. I like to remind myself that we can gain efficiency through abstraction, but true, intuitive understanding comes through concretization/deconstruction. Interestingly enough, I use round stones to represent ideas when I meditate. As the ideas come to me, I pick them up, examine them, and weigh them. If the idea is pressing, I delve into it and think it through. If not, I put the stone down and wait until my mind picks up the next. Mental stones: some of modern life's most useful tools :)
j2kun 11y ago
I think the vast majority of math teachers don't go on to learn new things and perspectives on math after college.
thaumasiotes 11y ago
> and square numbers have just the right number of stones to make a square! This one bothers me, because I don't think it's right. You can define the triangular numbers easily: 1, 1+2, 1+2+3, 1+2+3+4, ... and it's easy to arrange that number of dots into a triangle. The difference between two consecutive triangular numbers is always increasing by one. The difference between consecutive square numbers always increases by two. The squares are 1, 1+3, 1+3+5, 1+3+5+7, ... and it's also easy to arrange square numbers of dots into squares. So the difference between pentagonal numbers always increases by three. The first few pentagonal numbers are 1, 5, 12, 22, .... But pentagons don't tile the plane. How do you arrange a pentagonal number of dots into a pentagon? (Wikipedia has a proposed solution on display at https://en.wikipedia.org/wiki/Pentagonal_number https://en.wikipedia.org/wiki/Pentagonal_number , but the pentagons it constructs have no obvious internal structure. The square you construct from a square number of dots is symmetric wrt rotation.)

rsy96 11y ago

The revelation he has had looks trivial to me. Graphs as equations, equivalence of expressions or matrix as linear operators are all something I learned and understood from the very beginning. I wonder if it has something to do with the textbooks. As a Chinese I often found American textbooks on mathematics so softcore. They have so many analogies, so many "real world" examples that masquerade the true mathematical meaning of the concept. Many of Chinese people argue that these are the reason that Americans are more creative, but I cannot help but wonder maybe the lack of rigor underlies some of problems with American math education. Or maybe I am just the exception. Maybe other Chinese struggle with math just the same.

arsenide 11y ago
American graduate mathematics student here at a US university, speaking from my perspective. This is perhaps not useful with respect to high school or first/second year undergraduate mathematics, but I choose to include this for sake of another perspective many may not be aware of. Whenever anyone says something is "trivial" -- I understand that to mean simply "it is obvious to me." I try to NEVER explain things in such ways -- to say things are trivial because it is very clear to me that is NOT trivial to many other people. And there has to be some respect given to this idea. I abhor when professors use this sort of language -- instead of stating it is trivial, explain using an extra sentence or two why it is so. If it takes more than that, it is likely not trivial. The best way for me to learn new concepts (in my graduate classes, for example) is as follows. And it is the method all of my professors currently use whether they know it or not. First, they begin with the definition of something. For example (one of my classes last week expressed this idea in particular): in metric spaces they say "A function is upper semi-continuous at x_0 if for all x approaching x_0 the limit supremum of f(x) is smaller or equal to f(x_0)." But what does this mean? You can use the "epigraph" as a way to explain this idea in a different way, explaining what it means to be a u.s.c. and/or a l.s.c. function in a more understandable way. From here, then, after understanding this, I was able to go through the strict mathematical notation of these concepts and follow it more clearly. Instead of starting from the mathematically "rigorous" notation, I am able to jump to a more familiar idea and proceed to link it back to the notation and rigor described in the definitions and associated theorems. This is a much more "efficient" way, in my opinion, than staring at the notation and trying to decipher it for hours. Having examples makes it easier to understand the context of the symbols and rigor involved: by having a specific example of a general form. Is there a lack of rigor in American math education? In general I would say yes -- especially in high school/undergraduate mathematics. But how do people most easily learn? Through examples tying back to definitions. This is more of the American way of teaching in my experience. Furthermore, you state "American textbooks on mathematics" are "softcore." I think this is true for a lot of the lower-level material, but if you pick up Rudin's Real and Complex Analysis I do not think you will feel the same way.

SteveBash 11y ago

I somehow understood the same thing(a graph is a collection of points that makes a function true..) a year ago through Gilbert Strang Big picture of derivatives: https://www.youtube.com/watch?v=T_I-CUOc_bk https://www.youtube.com/watch?v=T_I-CUOc_bk

TerryADavis 11y ago

What comes after Nonlinear Differential Equations, psy-op nigger?

jwmerrill 11y ago

It looks like the blog is down at the moment. Here's google's cached version of the post in the meantime: http://webcache.googleusercontent.com/search?q=cache:7SWRaHWxfiEJ:blog.mrmeyer.com/2015/the-math-i-learned-after-i-thought-had-already-learned-math/+&cd=1&hl=en&ct=clnk&gl=us http://webcache.googleusercontent.com/search?q=cache:7SWRaHW...

jwmerrill 11y ago

Two good ways to represent plane curves are parametric form: (x(t), y(t)) or implicit form: f(x, y) = 0 Parametric form is naturally associated with one point of view of what a plane curve is: the set of points traced out as a parameter is swept over its domain. Implicit form is naturally associated with a different point of view: the set of points that satisfy a certain relation. Functions of the form y = f(x) can be easily re-expressed in parametric form: (t, f(t)) or implicit form y - f(x) = 0 so both the parametric viewpoint and the implicit viewpoint are equally valid and useful ways of understanding the graph of a function. You could rephrase the author's insight as saying that he had always understood graphs of functions parametrically, but later learned to also understand them implicitly. Depending on the application, it may be more convenient to have a parametric representation of a curve, or an implicit representation of a curve. For example, it's easy to find a point on a parametric curve, but hard to test if a point is on a parametric curve; on the contrary, it is hard to find a point on an implicit curve, and easy to test if a point is on an implicit curve. If your curve is the graph of a function, it is easy to convert back and forth between these forms, but in general, converting from one form to the other may be quite hard. For me, the relationship between implicit and parametric representations is a piece of math that I didn't really learn until long after I thought I had already learned math.