we probably need some statistics beyond the usual anecdotes.

The TIOBE Index for May 2016 sees $Mathematica$ on a place 50+. It is a programming community index. If we concede that Nobel Laureats as well as Fields Medallists ... and Mr Wolfram ... are using Mathematica or WL, we have a statement about the fitness of $Mathematica$ to do things such people usually do.

A language must not be everybody's darling but it should encourage original ways to get things done -- to phrase it as inaccurate as possible.

A propos Fields medallists, read the funny "Hello World!" experiences by Timothy Gowers for recreation.

Side-stepping the naming of languages and returning to the original prompt, I am putting together a education platform to gather data and answer these types of research questions, http://www.codeseal.org . The terminology being used lately to describe the type of cognitive problem solving strategies learned through training in computer programming is "computational thinking". There are centers at places like Carnegie Mellon studying the topic in general https://www.cs.cmu.edu/~15110-s13/Wing06-ct.pdf. My work is focused quite explicitly on looking at how learning to program within the context of a particular discipline affects learning, e.g. "does modeling a polymer improve one's understanding of polymer formation better than deriving the mathematical formulas?", as opposed to the bigger question of "does programming in general affect how one thinks?". You pose another interesting question which is "does programming in different languages affect the learning outcomes for students in different ways?".

i just had to google those two guys to find out that they created Python and Ruby, respectively (i'm a theoretical physicist and those are the nerds i know). you can't compare the two s'es (steve jobs and stephen wolfram) - btw, i assume you know that steve suggested the use of the name Mathematica to stephen

totally irrelevant story. i was at the first Mathematica Conference (the reception buffet served shrimp - or prawns - the size of your hand) which i recall being held in Redwood City, or maybe it was in SF) and i was walking around the meeting hall and i saw steve and stephen conversing in a small room. my instinctive reaction was 'how can those two guys breathe with their combined egos taking up all of the oxygen in the room' LOL

jobs had charisma and the ability to create a 'reality distortion field'. it was incredible just to be in his vicinity. stephen's public 'gift' is that he exudes intelligence. i've known stephen since 1986 - when Mathematica was being created at the U of I - but i didn't talk to him for many years out of trepidation that i would say something that he would think was stupid. (and i don't get intellectually intimidated easily, not even by my nobel prize winning friends).

anyway to return to the theme of this discussion, i think that WL is just now beginning to reaching beyond STEM nerds to non-STEM nerds as Big Data and Digital Humanities become increasingly important. this is part of an ongoing revolution. i expect that the search for 'cause' that characterizes science will be (is being) replaced in large measure, by the search for 'correlation'. this drives us (were?) scientists nuts but that's their problem (not mine - i don't think we can ever understand reality; we can only model it).

and i think WL's pattern matching capabilities will become increasingly important. human thought is characterized by analogical reasoning which is really a type of pattern matching (by semantic rather than syntax... or do i have that backwards)? without question, the pattern-matching capabilities in WL are unsurpassed in any other language that i know of. i realize that pattern matching in WL is relatively slow (and discouraged by some excellent WL programmers) but i follow stephen's dictum expressed 30 years ago " you shouldn't develop [or use - my view] software based on hardware limitations because hardware keeps getting better and faster and you don't want it to outgrow the capabilities of the software". besides using pattern matching in WL is just too much fun (even more than making nested anonymous functions which i love to do) not to not utilize.

here's something to think about - it has recently been found that women do not avoid going into STEM fields specifically but rather they avoid going into fields that are perceived as requiring brilliance or inherent ability rather than hard work (google gender field brilliance). yet every time that stephen gives a lecture (e.g. at SXSW), the advertisements for and discussions of his talk emphasize that he is (was?) a brilliant theoretical physicist and a child prodigy (apparently there is no age limit on having been a child prodigy being a citable credential - note: it's NOT stephen's fault that others mention it when they refer to him. they even even credit him for writing scientific books as a youth that were not published (i wish i was given credit for the scientific work that i did but didn't publish). this cult of celebrity pervades the scientific community as much as it does the entertainment community. but i wonder if this emphasis on stephen's personal brilliance/genius might actually be counter-productive in promoting WL. and his live demos of WL coding might add to this problem, causing some people to conclude after watching stephen hack WL code, that they can't master WL themselves (we know that isn't true. i've taught WL programming to literally thousands of people (men and women) - close to 10,000 at this point - and it is far easier to become competent programming in WL than in ANY other programming language (at least when it is properly taught).

WL is most definitely a programming language. the fact that you can or may be able to do other things (e.g. computations within Mathematica) in no way takes away from WL being a programming language. and the fact that it is difficult, and sometimes impossible, to transfer what you can program in WL to other programing languages is irrelevant. As a term rewriting system, WL using different programming constructs and methods of evaluation than programming languages that follow different paradigms but that doesn't make WL any less of a programing language. i personally think that the first instantiation of WL within the Mathematica application, the confusion created by (mis)identifying Mathematica with WL, the ongoing inability to be able to create WL notebooks rather than Mathematica notebooks (you should be able to create a WL notebook (having the wolf logo) within Mathematica), the statements made that WL is a multi-paradigmatic language may have been useful in getting WL used initially - remembering that in 'the old days' programming was something done by physical scientists and engineers (STEM people), not by liberal arts types or even mathematicians, and Stephen, recognizing this, stated that he was using Mathematica as a trojan horse to smuggle in a new programming language to people who had no interest in learning a programming language per se - has caused some problems in getting WL recognized as a programming language (somewhat like having Donald Trump run for the Republican presidential nomination may have brought new people into the Republican party but at the cost of fundamentally changing the definition of what the Republican identification represents, for better and for worse). In fact, i think that it is the recent influx of people interested in doing big data analysis, internet (smartphone and tablet) app development and other 'non-technical' tasks that is now allowing WL to come out from hiding beneath the skirts of STEM and stand on its own as a programming language.

Interesting question! Are there studies on the benefits of the more general edge cases: (1) Learning to program, regardless of language, and perhaps not even with maths as a primary focus, and (2) using software to "look at" maths, plot things, solve stuff, but more in the "graphing calculator" sense than in the "programming" sense. Both of these aspects may give you ideas about how to proceed here, and there must be some data on that already.

And as you said, there also needs to be a way to put these benefits in perspective and compare them with other activities, because school time is a limited resource.

(By the way, "traditional maths" can teach a very different perspective and skillset from computational thinking, so I really wouldn't throw either one overboard so quickly. You know... it's fine to enjoy the book and the movie.)