Robert Hansen posted Jul 6, 2013 9:50 PM (GSC's remarks interspersed): > > On Jul 5, 2013, at 10:12 PM, "Clyde Greeno @ MALEI" > <email@example.com> wrote: > > > "Pure" mathematics entails descriptions of KINDS > of things that are being attended, together with > whatever logically substantial concepts and > conclusions are derived from those descriptions. > > This is because you think that analogy (common sense) > is the seat of reasoned thought. It isn't. It will > get you enough arithmetic to be functional though. > > > On Jul 5, 2013, at 10:12 PM, "Clyde Greeno @ MALEI" > <firstname.lastname@example.org> wrote: > > > Outrageous formalism has done far more to inhibit > mankind's mathematical progress than to facilitate > it. > > > How could a bunch of pure mathematicians with no > people skills inhibit anyone's anything? > The 'people skills' are involved with <<others>> 'learning' from the intensive (/profound/deep) studies done by the mathematicians into the variety of matters they research into. Check out, for instance, my response to Clyde Greeno (dt. Jul 7, 2013 9:44 AM at http://mathforum.org/kb/message.jspa?messageID=9159557), where I've discussed some relevant issues, in particular how Gottlob Frege would have totally failed in getting his discoveries known to the world if it had not been for Russell and others who had managed to understand what he'd done.
Clyde Greeno is not (I believe) *entirely wrong* in stating that 'pure formalism' did in some ways inhibit the progress of 'development of the understanding of the foundations of math (though it was intended to promote just that!) I have in the above-linked post recalled a couple of anecdotes indicating how Gottlob Frege's fundamental studies lay neglected for many years (till Russell and others 'rescued' those studies) - MAINLY BECAUSE OF the 'language' in which they were written!!
For instance, one of Frege's most fundamental papers on differential equations lay completely unknown even to mathematicians for decades BECAUSE no one - including very competent mathematicians! - could understand the technical 'language' that Frege had used!!! (This sorry situation continued till some good soul translated it into 'ordinary German' - at which point EVERYONE recognised the profound nature of what Frege had accomplished!)
[Greeno is, however - I believe - entirely wrong in claiming that what they (the 'pure formalists') had done was "Humbug!" The post linked above discusses some aspects of this belief of mine. The 'formalist programme is NOT "Humbug!" at all (though it sizably failed most of its aims). I have discussed this at another message, which may appear here in due course]. > > Why is it that some reformers, like yourself, make > these bold claims of how easy and common mathematics > is, yet never make it past arithmetic with their > students? > (I don't know much about the efforts of Clyde Greeno and other 'reformers' to reform math), but your claim is incorrect, evidence the following anecdote:)
A college freshman (who had more or less failed or just managed to pass) ALL his math right through his school career did, in fact, manage very successfully to do all his college math - *mainly* by constructing models showing how his own characteristics were hindering his learning of math - and by enabling him to construct a realistic action plan on how he could accomplish his goal of developing at least adequate competence in his college math.
Yes, I do wish the reformers (including Clyde Greeno) would think in terms of checking this process out. If they'd take the small trouble to do so, they might well find practical ways of countering many of your objections - mainly by finding ways to strengthen their theories and developments. > >People like us would "get it" if what you > said made any sense. > Here is what you would need to "get" as a pre-requisite, if you desire to understand the process I am discussing:
You would need, first, to find out - in some detail; through your own explorations of ANY issues of interest to you - just how the relationships "CONTRIBUTES TO" and "HINDERS" behave in systems, and how to use those relationships in real-life situations. Some (a very small amount of) learning and a fair bit of 'unlearning' is demanded.
(Here are some aspects of both the 'learning' and the 'unlearning' processes involved:
(i. You would need to understand the importance of your own ideas in any problem situation: that, in fact, they are important enough to WRITE THEM DOWN for a beginning.
(ii. You would need to understand that the underlying meanings of these relationship are very different indeed from the relationship "PRECEDES" on which you have gotten yourself hung up. Also, that 'Interpretive Structural Modeling' (ISM), INCLUDES the PERT Charts on which you have gotten yourself hung up - and that, in fact:
(ISMs in general bear a *somewhat similar* relationship to PERT Charts as does a good novel to the letters of the alphabet of which it is composed. [Words or phrases enclosed in ** contain some enhanced meaning from the standard dictionary meaning]. By and large, you can come to understand such meaning if you construct and adequately understand some models representating your own *mental moels*. In order to understand ISM's, you will need to understand something about *mental models* as well as about 'systems thinking': here are a a couple of links to some information on these crucial pre-requisites: - -- "Mental Models: a gentle guide.." - Mental models: a gentle guide for outsiders) - -- "Mental model musings" - http://www.systems-thinking.org/ - -- "Systems thinking" http://www.thinking.net/Systems_Thinking/systems_thinking.html) > > More importantly, if what you > said was true then we wouldn't even have to "get it". > It appears to me that if the thousands of reformers > like yourself deserve any credit it is for proving, > these last several decades, that mathematics is > anything but common or easy. > Even more importantly, if you wish to learn anything about, for instance, "how children learn" (in somewhat greater depth than you know now), you'd have to understand in some depth an old saw that Shakespeare had given to Hamlet to speak: "There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy" (including all your math and logic and science).
- -- I do not class myself as a 'reformer' of math.
- -- I do NOT claim that math is *common* or *easy* (assuming I am 'getting' the real meaning you intend by those words).
- --I DO claim that an 'adequate understanding' of math is essential for all of us, in every field of life.
- -- I DO claim that most of the educational system (the traditionalists and well as the 'reformers') have gotten much of it wrong to daate.
I DO claim that there is no reason whatsoever for the great majority of students who leave school to fear or loathe math. Further, that they DO need to achieve some level of math competence to help them with their daily lives AFTER they leave school - and (if they've gained sufficient competence to read a newspaper or a simple book and 'get' its meaning) it is ENTIRELY for all of them to gain the needed competence in basic math.
The 'educational system' may not be not far wrong in the level of math it has suggested for school leaving students (though I would like to test this out somewhat better than is being done in the conventional way).
However, the educational system has evidently gone VERY FAR wrong in the way that math is *taught*: the fact that most students exiting school DO fear and loathe math should be evidence enough.
Even President Obama is apparently included in that group! [i.e., membership of that 'math hating group' when he had passed out from school]. I find it IMPOSSIBLE to believe that someone who can achieve the soaring levels of eloquence with words that President Obama has is incapable of 'getting' basic math!! The ONLY possible answer is that the educational systems (when President Obama passed out from school) were 'teaching' math incompetently. From all I've been reading at this thread and elsewhere, there has not been much change.