The Math I Learned After I Thought Had Already Learned Math
blog.mrmeyer.com
blog.mrmeyer.com
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-...
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.
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.
[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.
“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.)
(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...
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:
{ (x,y) : y=sin(x) is true }
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 :)