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For research about cognitive information retention, I'm trying to find a scientific reference where they explain the difference between the capability to apply real mathematical skills (entailing symbols, patterns, the works basically) and the capability to follow simple recipes as in normal arithmetic or calculus.

So, I'm not talking about arithmetics or calculus, but more like anything that proves which kind of skills or insights you need in order to be good at "real" mathematics.

It seems that I keep bumping into articles where people just build on assumptions and I really need to find it soon. Any ideas?

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    $\begingroup$ I think you are asking about a systematic treatment of the difference between formulaic pattern matching which sometimes passes as "calculus" or "arithmetic" or for that matter "algebra I or II" in highschool (US) and real math. The trouble is, calculus, arithmetic and algebra are all fantastically deep topics for which real proficiency does indicate mathematical prowess. I was just reading: toomandre.com/my-articles/engeduc/ARUSSIAN.PDF so the idea of a fake calculus is fresh on my mind. I'm sure someone here can help you, but, your question needs some clarification... $\endgroup$ – James S. Cook Jun 23 '17 at 0:59
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    $\begingroup$ One problem is that "real mathematics" is a very broad field. You might need pattern recognition or structured thinking for algebra, where on the other hand you need graphical, 3D thinking for analysis. For combinatorics, it helps to be very creative, etc. However, these skills are not limited to math, they can also be useful in other fields. Thus, it might be really hard to define what is a "mathematical skill". Furthermore, "real mathematics" is also not well-defined. There is a huge difference, for example, between solving exercises (even on PhD level) and doing independent research. $\endgroup$ – Dirk Jun 23 '17 at 8:53
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    $\begingroup$ @JamesS.Cook That is an awesome paper. While I think it portrays the situation as being a bit bleaker than it is (I was never at liberal arts college though but rather a 1st tier research university) I remember the incredible shock when I taught probability for business students. $\endgroup$ – DRF Jun 23 '17 at 9:13
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    $\begingroup$ @JamesS.Cook Thanks for the great paper. Although it is over 20 years old and written about the US, it is still true today that non-mathematicians who have to take basic courses often see them only as an obstacle and not as a chance to learn (speaking from a German university here). $\endgroup$ – Dirk Jun 23 '17 at 9:37
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    $\begingroup$ @DRF thanks. I should give credit to the MESE user who posted it in another comment, I forget who sadly. You're right, the situation is not as bad at research schools. At least in my experience, the problem is much worse at SLACs. Moreover, I actually don't think the US mathematicians don't care about the sad state of business calculus and the like, it's just that we know better than to fall on our sword for the sake of students who are just here to seek a degree. It is sad. $\endgroup$ – James S. Cook Jun 23 '17 at 14:51
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I think the OP asks a difficult question that will not have a succinct answer. Permit me to point to one publication,

Schoenfeld, Alan H., ed. Assessing Mathematical Proficiency. Mathematical Sciences Research Institute Publications. 53. Cambridge University Press, 2007. (PDF download of book.)

which addresses the question of what constitutes mathematical proficiency in at least two chapters:

  1. James Milgram "What is Mathematical Proficiency?"

  2. Alan Schoenfeld. "What Is Mathematical Proficiency and How Can It Be Assessed?"

Let me quote one passage from Milgram's chapter (p.33):

Realistically, in describing what mathematics is, the best we can do is to discuss the most important characteristics of mathematics. I suggest that these are:
   (i) Precision (precise definitions of all terms, operations, and the properties of these operations).
   (ii) Stating well-posed problems and solving them. (Well-posed problems are problems where all the terms are precisely defined and refer to a single universe where mathematics can be done.)
It would be fair to say that virtually all of mathematics is problem solving in precisely defined environments, and professional mathematicians tend to think it strange that some trends in K–12 mathematics education isolate mathematical reasoning and problem solving as separate topics within mathematics instruction.

A very crude summary is that the OP's "capability to apply real mathematical skills" requires an appreciation and understanding of precision, and a facile skill at problem solving.

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  • $\begingroup$ Thank you so much! This is exactly what I was looking for! $\endgroup$ – Tanika Jul 17 '17 at 15:26
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I think there is a strong belief in the people who comment here that proof skills are more important than calculational skills. But I am not so clear that this is really true, is not completely how the human mind works.

I mean does understanding Dedekind cuts (or whatever they are called) really help you do arithmetic? If you learned that and never learned long division, versus someone who did the converse, who would be more powerful if you had to calculate something?

Some manipulative skill can be very helpful when doing trials or developing new mathematics. And the algorithms and methods developed over the years for arithmetic, algebra, and calculus have a lot of power in them. Represent a lot of distilled experience and effort to make things clear and simple and useful.

Ideally you should have both. Understanding of concepts and manipulative skill. And move back and forth between them. And have them inform each other.

Remember there a lot more people who need to understand math like Feynman did so they can follow physics derivations or do engineering problems than there are who need to play Bourbaki. And even for the Bourbaki soldiers, some combat with manipulation will make them better mathematicians.

This doesn't mean there is no value to concepts either. They can help with remembering methods, can help with understanding them, etc. But having a toolbag of tricks is valuable too. Don't look down on it.

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    $\begingroup$ -1: The question here is, at its core, a reference-request asking for a scientific reference, and this answer does not appear to provide one at this time. Perhaps converting some part of your answer to a comment would be more appropriate in this situation, although I do not remember what the reputation cutoff is to leave comments. --Chris $\endgroup$ – Chris Cunningham Jun 23 '17 at 17:49
  • $\begingroup$ Someone negated your minus 1. Did they do it because they like my answer, like me, or are messing with you? Create a Bayesian table to analyze! ;-) $\endgroup$ – guest Jan 12 '18 at 2:00

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