I recently read the book "Peak: Secrets from the New Science of Expertise" by Karl Anders Ericsson and Robert Pool. Ericsson is a respected researcher in the field of expertise formation, this is why I take his words as valuable. In a paragraph of the book it reads:

Some activities, such as playing music pop in music groups, solving crossword puzzles, and folk dancing, have no standard training approaches. Whatever methods there are seem slapdash and produce unpredictable results. Other activities, like classical music performance, mathematics and ballet, are blessed with highly developed, broadly accepted training methods. If one follows these methods carefully and diligently, one will almost sure become an expert.

I´ve written him an email, but, in the mean while, do you have any idea what "highly developed, broadly accepted methods" is he talking about? A simple reference to a book, article or website to start with might suffice.

The purpose of my question is that I feel I'm learning maths without any "highly developed" approach, just by following my intuition. If such approach existed I would really like to know about it, to improve my learning, and perhaps teaching, in the future.

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    $\begingroup$ Page 224 last full paragraph (google books) has more remarks in this direction. I have uploaded an image in which I underline in red the parts that I find questionable (and in purple the re-reference to crossword puzzles, which were in your quote-pull). The "most intriguing" curriculum reaches everyone, leads to "more than twice as much progress," but haven't been peer-reviewed/published? As the author writes of such claims, "it is hard to judge them objectively" (!). $\endgroup$ – Benjamin Dickman May 26 '16 at 19:14
  • $\begingroup$ Moreover, though you ask here about the training methods in mathematics, it is worth pointing out that there are ways to "train" in folk dancing... and crossword puzzles: for the latter, note the re-mention above alongside Scrabble, for which I am personally aware of methods that are not slapdash... $\endgroup$ – Benjamin Dickman May 26 '16 at 19:15
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    $\begingroup$ @BenjaminDickman Thanks for pointing it, but "Jump Math" is for elementary school maths (fifth-grade students in the experiment). I hope this is not what he means by "becoming an expert". $\endgroup$ – Armando May 26 '16 at 19:24
  • $\begingroup$ You are learning math by reading books and web sites (highly developed training methods). Or you are learning math by figuring it out for yourself, no books, instructors, or web sites? $\endgroup$ – Gerald Edgar May 26 '16 at 22:00
  • $\begingroup$ Oh god, Jump Math. I must say that their PR is excellent. (I've had that pulled up as a suggested training method at our college, and I also had to point out that it was K-6; ends before algebra). $\endgroup$ – Daniel R. Collins May 27 '16 at 4:41

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