[rsy96]

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.

[zintinio3]

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.

[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

[wfunction]

The Verizon Math was gold.

[agumonkey]

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.

[brador]

> The revelation he has had looks trivial to me.

If HN had a catchphrase this would be it.

[kaitai]

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.

[arsenide]

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.

[gone35]

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...

[1] https://terrytao.wordpress.com/career-advice/there’s-more-to...

[arsenide]

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."

[pepijndevos]

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.