Hansen's reasoning (by his use of the humanly natural creative and rational processes that constitute the psychological meaning of "common sense") contradicts his own argument against common sense being "the seat of human reasoning."
My "humbug" to formalism refers not to the role of formalism as a vehicle for advancing professional knowledge in mathematics. [When writing mathematical theories, I tend to be formalistic.] Rather, the worldwide reliance on formalism as a mathematical foundation for core-curricular education in mathematics has filtered the majority of students out from mathematics-dependent studies, out from college studies, out from school, or (all too often) out from civilized life. Those formalistic underminings of humans' personal mathematical potentials have been of enormous costs to mankind and thus to the progress even of professional mathematics. "Humbug" not *within* mathematics, proper, but within the context of mass education in and about it.
I reject all claims that I am a "reformer!" I am a clinical researcher, part of whose work is disseminate clinically ascertained findings which can be readily replicated by any other fully qualified clinical researcher. Clinical research is all about advancing professional knowledge about the individual human's personal mathematical well-being ... NOT about trying to sustain or to "reform" any entrenched curricular practices.
Hansen's quarrel with (MACS) Mathematics-As-Common-Sense (to the students, themselves) is easily explained and excused by the scarcity of serious, enlightening literature about the common-sensibility of school-level mathematics [from infancy through elementary calculus, statistics, and linear algebra]. Throughout those levels, the mathematics (in psychomathematical perspective) is common-sensible to the degree that the learners can be instructionally guided to achieve its concepts and facts through their own reasoning ... without being TOLD those mathematical "points" by the instructor. Of course, such *eductive* methods of learning-guidance heavily rely the learners' concurrent, progressive development of personal emerging powers of rational, creative, analytic mathematical reasoning. To minimize needless quibbling, that is *the kind* of common-sense with which the MALEI Clinic is concerned. [Many mathematicians would call it, "guided discovery."]
Clinical eductive instruction thus discloses that school-level mathematics, itself, is quite common-sensible to whatever learners who can be so-guided to so re-create the mathematical theories underlying the core curriculum. [But over the 34 years of of the Clinic's intermittent operations, we have found NOT ONE case of a fully functional child or adult WHO CANNOT so-learn school-level mathematics.] To be sure, there is much variation in their viable modes and rates of MACS-learning.
Clinical research thus discloses that the mathematics, itself, is adequately common-sensible for functional humans (in that psychomathematical meaning of the phrase). It means that when students have trouble digesting school/college mathematics into personal common sense, it is NOT because the mathematics, itself, is non-common-sensible. It is NOT because (fully functional) students lack adequate personal mathematical aptitude. [It is not because some have the capacity and some do not.]
Instead, the learners' floundering results from whatever their lifetime experiences, to date, have contributed toward or against digesting curricular mathematics into MACS. [Some can "get" MACS from what they happen to experience in math courses; others need alternative kinds of experiences.] The household and community environments often resolve the mathematical sensibility in ways that the curricula do not. Perhaps Robert Hansen eductively guided his young son to digest his present mathematical knowledge into personal common sense ... and might proceed with MACS-education through the calculus, et al. Most likely, he will use some "prompting" and some "presenting."
Clinical eduction is an R&D *tool.* By no means does clinical research imply that eductive ("tell no math") instruction is somehow "better teaching" than is its antithesis, didactic ("tell all math") instruction. Curricular mathematics instruction is an *industry*, and as such, its *productivity* is increasingly under scrutiny. Within that industry, "better" means "more productive", especially when at lower costs. [Unfortunately, present technologies for "testing" are such mis-oriented "measures of effectiveness" that they threaten to curtail actual productivity.]
Although the relative common-sensibility of curricula is of humanitarian concern [especially for captive audiences, it is a major factor in personal educational health], popular concerns about industry-productivity force common-sensibility to the forefront. The more common-sensible the curricular experiences, the more efficient the production, the less the waste, the better the products, and the greater the market-satisfaction.
However much the curricular educational practices might or might not be "reformed", the fact remains that the degree to which the educational practices fail to make the mathematical "points" and processes common-sensible to students is the degree to which the curricula are contra-productive ... undermining their mathematical evolutions and their personal mathematical health. [Repeatedly re-teaching for re-learning results from it not being earlier made common-sensible.]
It often happens that highly formal, strictly didactic, direct instruction is sufficiently common-sensible for the intended audiences ... and often it miserably fails. It often happens that "indirect instruction" is contra-productive because it tangentially goes off into the la-la land of irrelevance ... likewise failing to make foundational mathematics common-sensible to students.
Unfortunately, relatively few American educators are equipped to make curricular mathematics common-sensible to students ... because the needed professional (MKTE) Mathematical Knowledge for Teachers' Education has yet to be surfaced and shared. The role of the MALEI Clinic is to disseminate such knowledge ... NOT to promote any specific ways of implementing it.
[Persons interested in making normally troublesome topics of school-level mathematics fully common-sensible to adults or children may wish to participate in an emerging Special Interest Group. Request details from email@example.com .]
- -------------------------------------------------- From: "GS Chandy" <firstname.lastname@example.org> Sent: Sunday, July 07, 2013 2:31 AM To: <email@example.com> Subject: Re: Is logic part of mathematics - or is mathematics part of logic?
> Robert Hansen posted Jul 6, 2013 9:50 PM (GSC's remarks interspersed): >> >> On Jul 5, 2013, at 10:12 PM, "Clyde Greeno @ MALEI" >> <firstname.lastname@example.org> 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" >> <email@example.com> 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. > > GSC