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.
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.
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...
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.
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.
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.
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.
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).
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.
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).
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.
Was plane geometry your favorite math course in high school? Did you like proving theorems? Are you sick of memorizing integrals? If so, real analysis could be your cup of tea. In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is Pure Mathematics, and it is sure to appeal to the budding pure mathematician.
It depends I think. My experience has been that many people simply don't do well in "real" (i.e., formal and proof-based) math classes. Not to mention, a deep understanding of calculus isn't necessary for doing a lot of engineering, which provides most of the students in a lot of university calculus courses. On top of that, even if you start with an analysis-like approach of teaching calculus, you ultimately will have to spend some time plugging-and-chugging to get used to working with calculus on a practical level.
On the other hand are people like me, who really don't do well with learn by memorization type classes, but does well in classes that are theory first, with a thick layer of application over that.
I concur. What's worse was when they finally did get to an example of how you would use something, it was always so contrived as to be pointless.
For example, you've learned about dealing with rates of change. Now lets talk about the rate at which we pour water into a bucket.
For the love of god! Stop making problems about pouring things into other things! This has never come up in my life! And as an example, it doesn't help me relate to anything!
I remember the semester I was taking Calc II, I was also playing a lot of the game Terminus. (This was in 2000.) The game prided itself on having realistic space flight physics, so you had to spend the same amount of time accelerating and decelerating when traveling between locations. I remember realizing I could actually use the calculus I was learning to figure out how long / how far it would take to decelerate, and so plan my thruster burns accordingly.
As a kid I was very gifted at swallowing mechanics, and misled myself into thinking I was good at math. Like an ability to parse musical score very fast, but unable to play any music. Getting the concept is far more important, but far harder to do in class with quantifiable results (unless we measure the average grin on students face, maybe).
I teach these courses. I can tell you that everything in them is asked for by a client department.
Not saying that change wouldn't be good, but these things are there because we are asked to put them there. In fact, we are asked to put many more things there that don't fit in the course.
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.
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.
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?
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.
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.
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.
Having long been fascinated with computer graphics this bit was easy for me to get. I imagined a computer raster-scanning the plane and plotting a point when the condition in the equation was true. I just had to scale it up in my head from screen resolutions to infinite resolution and infinite extent and... oh boy, this is already starting to make my head hurt...
But the gist of it I got. And I admit, I'm unusual.
It's another way of thinking of the graph. Basically, the points on the graph are the points that satisfy y = f(x), so in that sense "the graph is a picture of all the points that make the function true".
From there he generalizes the idea to drawing the set of points that makes a statement true, without necessarily having a formal function defined.
He didn't seem to have realized fully what a 'graph' was. He probably went with the 'standard procedure' for drawing a graph (pick a x, and draw a point f(x) places above the x-axis, repeat).
He later realized that functions are a type of relation, (which is made explicit with the notation "y = f(x)") and that the graph is the set of points that satisfy that relation. Of course if you realize that you can also generalize to g(x) = f(y), f(x,y) = 0 etc. Which he seems to have done.
It also seems he hadn't completely realized that a graph is merely a collection of points, not a nice line-like object. This last is more explicit in non-continuous functions, but you don't usually see those in high school.
> It also seems he hadn't completely realized that a graph is merely a collection of points, not a nice line-like object. This last is more explicit in non-continuous functions, but you don't usually see those in high school.
Well, I find whole thing rather controversial, so to say. Depends on what do one mean by "collection" and by "point", eg. if we take point literally: (x, y) \in RxR, then we have to assume that "collection" is uncountable (which seems counterintuitive, esp. from the school student's pov). OTOH, one could treat "point" as representation of that point, eg. pencil mark, so that point is no longer single element, but rather an uncountable subset containing the point of interest, along with its neighbourhood => our "collection" becomes countable, so we could think about it as bunch of indexed points (much more intuitive, isn't it?), shifting our attention towards mapping's representation from the mapping itself.
tl;dr Treating graph as a collection of points is no more correct/better, compared to treating it as a continous line. Different problems require different approaches and different "angles" of view. Barely an insight from the author - the most of the mathematics is about abstractions and pattern matching.
> Well, I find whole thing rather controversial, so to say. Depends on what do one mean by "collection" and by "point", eg. if we take point literally: (x, y) \in RxR, then we have to assume that "collection" is uncountable (which seems counterintuitive, esp. from the school student's pov). OTOH, one could treat "point" as representation of that point, eg. pencil mark, so that point is no longer single element, but rather an uncountable subset containing the point of interest, along with its neighbourhood => our "collection" becomes countable, so we could think about it as bunch of indexed points (much more intuitive, isn't it?), shifting our attention towards mapping's representation from the mapping itself.
The whole paragraph does not make sense to me. For example, how is a collection uncountable somehow unintuitive? Why should a point not be a single element, but a subset or a pencil mark? Are you confusing the theoretical graph with the graph actually drawn (which has finite width)?
- enumerable (countable) sets are more intuitive to reason about
- no matter, how do you comprehend the function, you'll end up drawing it by means of composing continuous chunks
> Are you confusing the theoretical graph with the graph actually drawn (which has finite width)?
Probably. From my point of view, graph is the something that is drawn/plotted, whilst relation/mapping being something that you've called "theoretical graph". So I'd question, how treating graph as a bunch of points is superior concept, giving that one could easily slip into substituting actual mapping's points by their graphical representation, for the sake of simplicity, thus possibly hiding mapping's behaviour from own mind.
PS
Nevertheless, I do now understand, that the whole thing author has meant to say, was: "Given x = f(y), it is not only some explicit line on a plane, but also a mapping f: (x, y) \in RxR (which also could be drawn, btw)".
The whole subthread is more of a dialectical excercise, with the definition of "graph" not being synchronized among participants :)
Treating a graph as a collection of points is certainly more correct and better if the condition you are trying to satisfy gives a non-continuous set of points. If you can only deal with continuous functions you will be lost in these more interesting cases.
Some people cannot see the trees for the forest. Others cannot see the forest for the trees. If you do not understand that they are both true and real and right there in front of you, you have missed something important somewhere along the way.
Actually, only tree is real. "Forest" is a mental construction - just region of space with the density of trees being high enough. Play with that "high enough" variable, and a meaning of "forest" will drift.
"Tree" is also a mental construction. It is just a bunch of particles that form a recognizable pattern in the features and scale we've evolved to discern in the world. Examine two trees in enough detail, and the vast differences may make the tree-scale similarity seem utterly insignificant. But alas, "particle" is also just a mental construction...
“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.”
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 :)
> 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 , but the pentagons it constructs have no obvious internal structure. The square you construct from a square number of dots is symmetric wrt rotation.)
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.
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.
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.
Hmm are you in Applied Math? Or by "graduate" you mean perhaps some kind of terminal (non-PhD track) Masters program? Because "staring at the notation and trying to decipher it for hours" (when lucky; sometimes it could be days, or even weeks) is precisely the bulk of the Math graduate experience IMO --especially after quals...
Also, do know that after a certain point, the whole "examples then general form" approach is not just barely applicable (e.g. the 'example' alone often requires so much setup that it ends up being harder than the formal statement itself!) but is also a serious handicap to your capacity for thinking 'syntactically' from formal statements alone. Terry Tao has an excellent post on this[1]; although it might or might not be exactly applicable to your situation...
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First year pure math: that statement in reference to these introductory courses.
I don't agree that using this is a "serious handicap" to my capacity for thinking syntactically. It is just another tool to be used when applicable to generate understanding: use an example and see how it follows the rigorous definitions. I lose nothing by doing this when it is useful.
Even in this link, Dr. Tao agrees that it is not a good idea to look at statements on a strictly formal level. In a greater sense this is what I was getting at: play with some examples or with some of the assumptions and see what happens in order to get initial or further understanding. Did I misunderstand the stated point? What was said just prior but in reference to "this[1]" is not supported by Dr. Tao in his post.
Relevant quote: "'fuzzier' or 'intuitive' thinking (such as heuristic reasoning, judicious extrapolation from examples, or analogies with other contexts such as physics) gets deprecated as 'non-rigorous'. All too often, one ends up discarding one’s initial intuition and is only able to process mathematics at a formal level, thus getting stalled at the second stage of one’s mathematical education."
I teach a class that is about 80% Chinese students who have come to the US. Many have much better technical skills but I do see gaps in these concepts like graphs as the locus of points satisfying a logical claim, the equivalence of various approaches to integration (numerical/limit of Riemann sum/fundamental theorem of calculus), etc.
You're right that American textbooks on math are often very soft. We try to make things so understandable that they are impossible to understand. There's nothing to hold on to. I am occasionally guilty of this in my teaching, although it can be very effective with people who do understand the technical basics.
Americans generally struggle with mathematics, and not just advanced concepts. The problem is cultural, because it is acceptable to be poor at math, it's viewed as wholly unnecessary. Some examples are A&W's Third Pounder vs McDonald's Quarter Pounder (3 < 4 so quarter pounder is bigger) [1], how long does it take to go 80 miles if you're driving 80 miles per hour [2], and Verizon Math [3].
As for Chinese struggling with math, I can give you firsthand experience at my American University, where the foreign (Chinese and Indian) students are known for rampant cheating, in both the Undergraduate and Graduate levels.
We have a serious problem with logic, math, and science. People who are good at the three are ridiculed and alienated rather than celebrated, although it's not a hard rule, just something I've noticed. Reading for pleasure is the exception rather than something normal. We're very anti-intellectual once you get out of the big cities.
The passenger sit girl state of mind is the same I experience whenever I struggle with any "foreign" concepts. Whether it was fancy recursive typing or anything post graduate level where the words felt meaningless until 2 years down the road where I had an eureka moment.
Should we order problems by some notion of difficulty or is it fractally similar at every level ?
ps: I can't take the Verizon chat, it's too painful.
FWIW, I'm a first year student at the University of Twente, and I'm only learning about sets, logic, proofs and combinatorics in the first quarter. None of the numbers stuff.
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
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.
66 comments
[ 2.8 ms ] story [ 128 ms ] threadI'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.
[0] - https://en.wikipedia.org/wiki/Area_under_the_curve_%28pharma...
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/ or the more traditional/frequentist http://onlinestatbook.com/.
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).
+Solutionator: Bunch of mechanical steps that the student understands in the exact same way a trained monkey understands an organ grinder.
Was plane geometry your favorite math course in high school? Did you like proving theorems? Are you sick of memorizing integrals? If so, real analysis could be your cup of tea. In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is Pure Mathematics, and it is sure to appeal to the budding pure mathematician.
As an example, I remember that as a high school maths student I had real trouble believing that 0.9 repeater equals one.
Nowadays, post university, it seems trivially true.
On the other hand are people like me, who really don't do well with learn by memorization type classes, but does well in classes that are theory first, with a thick layer of application over that.
For example, you've learned about dealing with rates of change. Now lets talk about the rate at which we pour water into a bucket.
For the love of god! Stop making problems about pouring things into other things! This has never come up in my life! And as an example, it doesn't help me relate to anything!
Haven't used it since, though... It's a pity.
Not saying that change wouldn't be good, but these things are there because we are asked to put them there. In fact, we are asked to put many more things there that don't fit in the course.
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.
Author claimed, that he had been missing something essential regarding that stuff, but I wonder what exactly does he talk about?
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:
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?
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.
But the gist of it I got. And I admit, I'm unusual.
This is literally the entire point of graphing inequalities.
From there he generalizes the idea to drawing the set of points that makes a statement true, without necessarily having a formal function defined.
He later realized that functions are a type of relation, (which is made explicit with the notation "y = f(x)") and that the graph is the set of points that satisfy that relation. Of course if you realize that you can also generalize to g(x) = f(y), f(x,y) = 0 etc. Which he seems to have done.
It also seems he hadn't completely realized that a graph is merely a collection of points, not a nice line-like object. This last is more explicit in non-continuous functions, but you don't usually see those in high school.
Well, I find whole thing rather controversial, so to say. Depends on what do one mean by "collection" and by "point", eg. if we take point literally: (x, y) \in RxR, then we have to assume that "collection" is uncountable (which seems counterintuitive, esp. from the school student's pov). OTOH, one could treat "point" as representation of that point, eg. pencil mark, so that point is no longer single element, but rather an uncountable subset containing the point of interest, along with its neighbourhood => our "collection" becomes countable, so we could think about it as bunch of indexed points (much more intuitive, isn't it?), shifting our attention towards mapping's representation from the mapping itself.
tl;dr Treating graph as a collection of points is no more correct/better, compared to treating it as a continous line. Different problems require different approaches and different "angles" of view. Barely an insight from the author - the most of the mathematics is about abstractions and pattern matching.
The whole paragraph does not make sense to me. For example, how is a collection uncountable somehow unintuitive? Why should a point not be a single element, but a subset or a pencil mark? Are you confusing the theoretical graph with the graph actually drawn (which has finite width)?
- enumerable (countable) sets are more intuitive to reason about
- no matter, how do you comprehend the function, you'll end up drawing it by means of composing continuous chunks
> Are you confusing the theoretical graph with the graph actually drawn (which has finite width)? Probably. From my point of view, graph is the something that is drawn/plotted, whilst relation/mapping being something that you've called "theoretical graph". So I'd question, how treating graph as a bunch of points is superior concept, giving that one could easily slip into substituting actual mapping's points by their graphical representation, for the sake of simplicity, thus possibly hiding mapping's behaviour from own mind.
PS Nevertheless, I do now understand, that the whole thing author has meant to say, was: "Given x = f(y), it is not only some explicit line on a plane, but also a mapping f: (x, y) \in RxR (which also could be drawn, btw)".
The whole subthread is more of a dialectical excercise, with the definition of "graph" not being synchronized among participants :)
“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!
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 :)
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 , but the pentagons it constructs have no obvious internal structure. The square you construct from a square number of dots is symmetric wrt rotation.)
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.
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.
Hmm are you in Applied Math? Or by "graduate" you mean perhaps some kind of terminal (non-PhD track) Masters program? Because "staring at the notation and trying to decipher it for hours" (when lucky; sometimes it could be days, or even weeks) is precisely the bulk of the Math graduate experience IMO --especially after quals...
Also, do know that after a certain point, the whole "examples then general form" approach is not just barely applicable (e.g. the 'example' alone often requires so much setup that it ends up being harder than the formal statement itself!) but is also a serious handicap to your capacity for thinking 'syntactically' from formal statements alone. Terry Tao has an excellent post on this[1]; although it might or might not be exactly applicable to your situation...
[1] https://terrytao.wordpress.com/career-advice/there’s-more-to...
I don't agree that using this is a "serious handicap" to my capacity for thinking syntactically. It is just another tool to be used when applicable to generate understanding: use an example and see how it follows the rigorous definitions. I lose nothing by doing this when it is useful.
Even in this link, Dr. Tao agrees that it is not a good idea to look at statements on a strictly formal level. In a greater sense this is what I was getting at: play with some examples or with some of the assumptions and see what happens in order to get initial or further understanding. Did I misunderstand the stated point? What was said just prior but in reference to "this[1]" is not supported by Dr. Tao in his post.
Relevant quote: "'fuzzier' or 'intuitive' thinking (such as heuristic reasoning, judicious extrapolation from examples, or analogies with other contexts such as physics) gets deprecated as 'non-rigorous'. All too often, one ends up discarding one’s initial intuition and is only able to process mathematics at a formal level, thus getting stalled at the second stage of one’s mathematical education."
You're right that American textbooks on math are often very soft. We try to make things so understandable that they are impossible to understand. There's nothing to hold on to. I am occasionally guilty of this in my teaching, although it can be very effective with people who do understand the technical basics.
As for Chinese struggling with math, I can give you firsthand experience at my American University, where the foreign (Chinese and Indian) students are known for rampant cheating, in both the Undergraduate and Graduate levels.
We have a serious problem with logic, math, and science. People who are good at the three are ridiculed and alienated rather than celebrated, although it's not a hard rule, just something I've noticed. Reading for pleasure is the exception rather than something normal. We're very anti-intellectual once you get out of the big cities.
[1] http://www.nytimes.com/2014/07/27/magazine/why-do-americans-...
[2] https://www.youtube.com/watch?v=m2eyq9qTOQY
[3] https://www.youtube.com/watch?v=MShv_74FNWU
Should we order problems by some notion of difficulty or is it fractally similar at every level ?
ps: I can't take the Verizon chat, it's too painful.
If HN had a catchphrase this would be it.
(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.
http://webcache.googleusercontent.com/search?q=cache:7SWRaHW...