Because of the extraordinary importance of the issue of *creativity* that is *intricately involved* in the act of 'learning' generally and in the 'learning of mathematics' in particular, I am in this post seeking to refer to all the responses that have appeared in the thread - but it is of course possible that I have omitted important ideas.
[This concept of *intricate involvement* is something that only recently came to mind. I shall try and make it clearer in what follows, but full clarity will probably take somewhat longer to appear - even in my own mind! It's a bit different from *inextricable involvement* (though there surely is a bit of each in the other)].
Anyway, there is clearly considerable *creativity* in 'learning something new' (in math, or in any other discipline); there is also great creativity in ideating and then developing those new ideas (in math, or in any other discipline).
Unfortunately, quirks in the scripts handling the maintenance of threads at Math-teach have led to the thread being broken up into various parts. Therefore, I may not in this post be able to capture all or even most of the ideas in the responses that have appeared.
In his generally sound introductory remarks to the excerpt he's provided to the book "The Philosophy of Creativity", Scott Barry Kaufman makes clear his appreciation of the importance of the issue for us as humans: > > "... Creativity drives progress in every human > endeavor, from the arts to the sciences, business, and > technology. We celebrate and honor people for their > creativity, identifying eminent individuals, as well as > entire cultures and societies, in terms of their > creative achievements. Creativity is the vehicle of > self-expression and part of what makes us who we are. > ... > Briefly, creativity may be said to define us as human beings (to a considerable extent).
As I'd observed earlier, Wayne Bishop (WB) certainly had it right when he agreed with Joe N.'s suggestion to the effect that creativity is (probably) impossible to 'teach' - but in then leading us to recall John Saxon's well-coined saying that "Creativity springs unsolicited from a well-prepared mind", WB also leads us to recall that "Saxon's Method for Math Learning" emphasises and enforces "memorization of algorithms".
I claim that the 'well-preparedness' of the mind in question DOES NOT SPRING only from the "memorization of algorithms": in fact, I'd go so far as to claim that this rule of "memorization of algorithms" probably leads more to a 'thoroughly bored mind' rather than to a 'well-prepared mind'.
This utterly absurd rule is surely the best escape route for lazy math teachers to enforce the 'NEXT RULE', namely, "learn by heart this and this and this... OR ELSE!"
Further, it should be entirely obvious that children who pick up the 'secret' of *effective math learning* at the start of their learning careers will automatically memorise all the math algorithms that they ever need to.
NO SUCH RULE OF MEMORISATION OF ALGORITHMS IS EVER REQUIRED (as a rule)!!! (Assuming of course, that math teaching is done effectively from the start).
I claim that (in any discipline) BY-ROTE MEMORISATION AS A PRIMARY AND FUNDAMENTAL RULE IS WRONG!!
No doubt 'some memorisation of algorithms' is appropriate in math - but I claim that it is the individual learner (who certainly should come to understand the capabilities of his/her own mind) - it is that learner who is the best judge of what is appropriate for him/her.
It's the teacher/guide's job to find out just how much is necessary for each of his/her own wards - and to guide them appropriately. No doubt, in the current teaching system, this is extremely difficult to achieve - but *effective* learning and teaching are both very difficult to achieve in the current system. (And yet, some teachers have managed to become good, even 'great' teachers: imagine what could happen within a truly effective system, which is responsive to the needs of each learner in it!)
I do not have any empirical evidence in support of the above claims - but I do know that I was highly competent in learning all the applied (engineering) math I ever required while studying engineering; and I was quite competent while studying pure math at graduate school level.
Thus I may safely claim that I did manage to learn math to that level quite effectively - which does involve a fair bit of *creativity*. (But, I observe, I rarely ever memorised an algorithm).
In this I'm NOT claiming that I ever approached the *creativity* at math in the sense that the great mathematicians I had named in an earlier post in this thread were *creative*. This particular disclaimer is put in here especially to counter Wayne Bishop who unfortunately DOES seem to have the tendency of dishonestly seeking to suggest such implications in statements that I have made. On one occasion, WB infamously [and utterly dishonestly] even suggested that I was ascribing to myself similar powers in *creative mathematics* that Grigoriy Perelman clearly possesses! (Ideas to that effect). Of course, I refuted that infamous suggestion immediately - but what is the impression that the casual reader would be left with who sees WB's dishonest suggestion and not my refutation??
I further claim that this kind of dishonesty in arguments put forth is very deliberate - IT'S NOT ACCIDENTAL! This claim may be confirmed via the 'One Page Management System' (OPMS).
In regard to WB's admiration (if that is a 'sufficient' description) of John Saxon's prowess at the 'teaching of math':
I had noted in my earlier post responding to WB that I have not really seen any Saxon books and thus I do not know if they contain any 'magic mantra' to ensure that the 'NEXT RULE' does not come into operation in the process of math teaching. I do not know whether Saxon himself enforced that 'NEXT RULE' and I'm not claiming anything along these lines.
I AM claiming that there is probably some rather serious 'confusion' operating within the 'Saxon system of math teaching'). The OPMS could probably help sort out these confusions.
However, I suggest another, further, proposition:
There is considerable *real creativity* involved in the 'learning of math' (at any level) - from the most elementary to the most advanced. There is *real creativity* (of probably a different order) involved in the *creation* of 'new math'.
There is both 'knowledge' and 'art' (in Robert Hansen's sense) *intricately involved* in the *creation* of 'new math'. In fact, there is, I claim, both 'knowledge' and 'art' intricately involved in the *creation* of 'new knowledge' in ANY discipline!
There are, for instance, VERY advanced levels of both 'knowledge' and 'art' in the creation of one of George Hart's 'geometrical sculptures'.
Anyone who claims that he is able to separate out this 'knowledge' and 'art' is just plain and simple lying.
At the current stage of development of the 'cognitive sciences' we do not know enough about differences between the different kinds of *creativity* involved in acquiring the learning (or knowledge) of a discipline like math and the development of new ideas in that discipline. We know very little about these differences in any discipline. We also know rather little about the connections between the different kinds of creativity involved in learning (acquiring existing knowledge) and in developing (*creating*) new knowledge.
Anyone who claims he is able to separate out different kinds of *creativity* (or of 'knowledge' and 'art') in different disciplines, like, say, 'pure math' - and the kind of application of mathematical ideas that George Hart creates in his 'geometrical sculptures' is also just plain lying.
Anyone who makes statements like "Even though knowledge is there on the surface, the creative mathematician is studying the art behind it" simply does not know what he is talking about. Probably because he has rarely learned anything and has never created anything - except those fabulous 30-foot PERT Charts.
I observe that Robert Hansen does not at all understand the concepts underlying *creativity in math*, nor does he understand at all the idea of what a *creative mathematician* really does. This becomes evident with almost every message he has posted at this multi-part thread. As Joe Niederberger has suggested: Mr Hansen is really a very funny guy on this issue.
Most of Robert Hansen's claims about the 'knowledge' and the 'art' of math are just so much nonsense: this can be demonstrated through an application of the 'One Page Management System' (OPMS).
I claim there was considerable *creativity* ('art', if you will) involved in discovering the concept of the 'One Page Management System' (OPMS):
Namely, the recognition that, if the construction of a single model from 'simple' elements conferred some benefits - then the construction of a 'MODEL OF MODELS' should surely confer *manifold* benefits.
I also claim that there is a fair amount of *creativity* involved in creating effective models to serve as guides for Action Planning in any complex situation.