Throughout elementary, middle and high school mathematics is quite merely about memorizing concepts and formulas, understanding the theorems (without their proofs) and applying acquired knowledge in examinations. It is not perfect, I concede, but it is almost as good as is needed. I am a graduate student who is devoted to pure mathematics, and I have about 4 years of teaching experience with a great variety of students. I hope that this is gives me enough credit to discuss a matter of this sort.

It is (to some extent) true that mathematics is “art”; it is about creating patterns and connecting them with each other, etc. This is what most professional pure mathematicians like to think and say to the public. Now, there is the harsh truth that a great portion of pure mathematics undergraduates drop out or, after graduating, either run away to some applied area of mathematics or decide to teach mathematics in middle or high schools; “real” mathematics is very difficult even for people who were very good/excellent in high school math. Moreover, many of the students who safely land in graduate school find it difficult to get accepted in the graduate programs they wanted to get in because of their incompetency, and so they end up doing something they did not really want to do. I am not trying to suggest that advanced pure mathematics is a nightmare, but my point is that if the education in school was made so that students would learn to “appreciate” the beauty of mathematics and get motivated to pursue pure mathematics, most of them will fail later on.

One asks: why, then, are most professional pure mathematicians discontent with the current style of math education? In my opinion, the answer is: most of them are geniuses and have no idea what mathematics looks like to inferior beings, and the rest simply forgot what kind hard work is required for learning and digesting advanced concepts and are unaware of how it might be much more difficult for less capable people.

My conclusion is that mathematics education is in a fine state in today’s world. It is not “fundamentally wrong” and “delusive”; it is just realistic. If you are not extremely comfortable dealing with numbers, elementary functions and geometric figures, you do not stand a chance in advanced mathematics.

To make this a question, allow me to pose following:

Do you disagree with this? What important points am I missing in my defense of the current school education system?

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    $\begingroup$ Primarily opinion based $\endgroup$ – Gerald Edgar Aug 24 '16 at 18:20
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    $\begingroup$ @J. Doe: That's part of the problem. This isn't a discussion forum. I think the question is valid, though. $\endgroup$ – Jon Bannon Aug 24 '16 at 18:27
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    $\begingroup$ I would separate out the issues of (1) many students can't justify or explain their procedures, and (2) students can't discover or invent new procedures. I'd argue that the former is important for all, while the latter can indeed by reserved for a minority of aficionados. $\endgroup$ – Daniel R. Collins Aug 24 '16 at 19:57
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    $\begingroup$ I think I have to disagree with many of the premises here. If nothing else, the activity-label "mathematics" refers to wildly different things depending on context: as practiced by professional mathematicians, say, versus ultra-practical mathematics as (should be) practiced by people trying to evaluate investments by thinking about compound interest. E.g., encouraging people to understand how to manage money is not at all the same as encouraging them to try to become professional mathematicians... $\endgroup$ – paul garrett Aug 24 '16 at 21:53
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    $\begingroup$ "why, then, are most professional pure mathematicians discontent with the current style of math education?" What gives you that idea? I'm not aware of any surveys of "professional pure mathematicians" that would warrant the assertion that most of them are discontent in the way that you claim that they are. $\endgroup$ – John Coleman Aug 25 '16 at 15:53

I think this question is, probably accidentally, responding to a strawman argument. It presumes that the criticism of math education coming from pure mathematicians is some combination of "students aren't learning sufficiently advanced math", "students aren't sufficiently prepared to become mathematicians", and "students don't appreciate the beauty of math".

Some people certainly make these claims, but they're not the main ones being made by mathematicians who criticize math education.

The main criticism of math education is that that students end up learning rote procedures without any understanding or ability to use them outside of the classroom. Students don't need to learn more advanced math for these problems to become evident: consider the many students who won't notice if a calculator spits out an answer which is off by orders of magnitude, or who have great difficulty solving word problems even when they have no difficulty with the corresponding equations.

That is, the complaint isn't "why do we spend so much time getting our students comfortable with numbers when we want them to do advanced mathematics?", it's "why aren't our students comfortable with numbers?"

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    $\begingroup$ Why should they be able to use them outside the classroom? To begin with, the trivial content of the math school curriculum is such that it is not possible to effectively/meaningfully use the learned concepts outside the classroom (if you can suggest just one non-trivial counterexample, that would be great). Also, it is not difficult for people who were good students to refresh their math skills and learn the required material in case they ever needed math in their professional careers (which is what I assume the intended meaning of "using math outside the classroom") $\endgroup$ – J.Doe Aug 24 '16 at 19:57
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    $\begingroup$ @J.Doe, it seems that you think it ok that students learn something that is neither usefu nor fun to them. $\endgroup$ – Carsten S Aug 24 '16 at 21:05
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    $\begingroup$ @J.Doe 1) What purpose do you see mathematics education having at all if it is neither to be useful outside of school nor to prepare students to be mathematicians? 2) Elementary and high school curricula do or can include many basic but practical topics, including arithmetic, algebra, and basic probability. Also, "outside the classroom" includes other life activities, like managing money, choosing among medical procedures, and evaluating political candidates. 3) Comfort with numbers, at the level I mean, is useful for virtually everyone in a modern society. $\endgroup$ – Henry Towsner Aug 24 '16 at 21:32
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    $\begingroup$ Yes, indeed, "strawman argument" is exactly on the money. Innumeracy, akin to illiteracy but in some regards far worse, is a huge problem in social policy and so on: huge populations of people who cannot even correctly understand their own best interests? (Thus, allowing them to be deceived and manipulated by demigogues...) Elementary probability, compound interest, etc., if widely understood, could profoundly change public policy... $\endgroup$ – paul garrett Aug 24 '16 at 22:03
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    $\begingroup$ I've often compared all pre-university education to physical strength training. The question of how useful certain knowledge is "in the real world" is widely asked about math, but never about Shakespeare. I've used math a lot at work, but never Shakespeare. Still, we somehow have to justify math education. Again, to me it's like lifting weights to build brain muscles. The actions are repeated and targeted to certain mental "muscles" to make them stronger. Then those strong mental "muscles" can be used by the student for whatever they want, which is unlikely to be like the original exercises. $\endgroup$ – Todd Wilcox Aug 25 '16 at 3:26

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