Further my post dt. Dec 28, 2013 8:28 AM (http://mathforum.org/kb/thread.jspa?messageID=9352100?), some amplifications and clarifications: > > Kirby Urner (KU) posted Dec 27, 2013 12:20 PM > (http://mathforum.org/kb/message.jspa?messageID=935179 > 0) - GSC's remarks interspersed: > > > > On Tue, Dec 24, 2013 at 3:30 PM, Robert Hansen > > <firstname.lastname@example.org> wrote: > > > > > > You seem to always shoulder students with the > > > > burden of suffering a math > > > > deficit, whereas it may be the school's > > > > curriculum > > > > and/or teachers and/or > > > > the whole culture of a high school that > > > > starts > > > >closing these doors. > > > > > > That?s not true. Most of my years here were > spent > > > studying curriculums, in > > > detail. I blame the schools and curriculums for > > >not > > > nurturing the students > > > who are good in math. The only students that > > > actually continue on after > > > school and use and apply this gift. The little > > >bit > > > of math that the other > > > students pickup is soon forgotten. > > > > > > > I don't think what we call "math" in K-12 should > > at > > all try to style itself > > as some miniaturized version of what a college > > math > > major might study. > > It's its own animal. > > > > Numeric literacy includes using measurements and > > currency, telling time, > > arithmetic, appreciation for data visualization > > techniques and conventions, > > familiarity with rounding / rounding error / > > significant digits, an > > understanding of statistical arguments and > > concepts > > such as mean, mode and > > median, standard deviation (bell curve), interest, > > probabilities, the wave > > and particle nature of energy... > > > > Then I'd add some broad brush stroke overview > > perspective as a navigation > > aid, tell some history. > > > Yes indeed (and more)! Not just 'history', of course. > (Your remarks in another post about providing 'STEAM' > to STEM are significant, in this context). > I've previously been using the terms *integrative* terms of 'dyad' and 'triad' (after Charles Sanders Peirce and then Warfield) to denote important 'collections' (*integrations*) of notional elements.
I believe that with concepts like 'STEM' and 'STEAM', we may well need to expand on the earlier terms for 'collectives-of-ideas' like 'dyads' and 'triads' (perhaps we may call them 'quads' and 'quints'?)
I have not done any investigations into such 'quads' and 'quints' - and I really don't have much idea about how this is to be done. (I have found 'fairly *effective*' ways to investigate 'dyads' and 'triads' quite satisfactorily. Yes, what I call 'prose + structural graphics' [p+sg] has been the practical tool for such investigation. I do suspect p+sg would also help us understand 'quads' and 'quints' better).
In regard to the notions of 'STEM' (/'STEAM'), I am fairly sure that the 'M' does have significant "contributory" effect to help us understand and apply the other 'elements' of the integrative 'multiplex'.
However, notwithstanding the profound insights of the above-noted thought leaders, the last words on the 'utility of math to ourselves (and to help us understand our place in the world') have not yet been written or thought. I claim that investigations of our real world (including math as a 'thought-system', using the transitive relationship "CONTRIBUTES TO", based on Warfield's insights into complex systems, could help significantly. The world is eagerly waiting for the researcher who can perform this enormous task!] > > > > > This history and context part is a part normally > > left > > out I realize, but is > > what you'd find in a Time-Life trade book such as > > 'Mathematics', > > influential on my younger self: > > > > http://wikieducator.org/File:Math_cover.jpg > > > I've not encountered 'Wikieducator' earlier - thanks > for this reference. Shall try and investigate. > > > > I think many of those who survive the intensely > > dull > > presentation of grade > > schools are saved by trade books (not textbooks) > > such > > as the above, and > > today by stuff in the Internet, such as > > Mandelbrots > > and Mandelbulbs 'n > > stuff. > > > Yes, indeed. The "intensely dull" presentations at > grade school focus almost entirely on the "HOWs?" - > to the near-total exclusion of the "WHYs?". The > 'trade books' (including I would guess, Scientific > American, Discover, Quanta, and the like give us > *some* connection to the "WHYs?". *Integrating* the > "HOWs?" with the "WHYs?" is really (IMHO) the great > task of education (and the 'system' is failing, > alas!) > Should the above be called 'pop-science' mags, perhaps?
I note that it is extremely difficult - perhaps even impossible - to get answers to our "WHYs?" in practically all our existing societal systems. If our educational (and other) systems learn 'how' *effectively* to respond to the question "WHY?", I believe they will also learn how to respond to the question "HOW?" somewhat better than they are doing today. > > However, I observe that even the trade magazines often > fail quite seriously. For an example > of a recent failure check out a recent article in SciAm > "A New Tool to Help Mathematicians Pack" - > http://www.scientificamerican.com/article.cfm?id=a-new > -tool-to-help-mathematics-pack&posted=1#comments, > (originally in Quanta, > https://www.simonsfoundation.org/quanta/20131220-nudgi > ng-spheres-closer-little-by-little/). > > I note that investigation of the 'systems' under > consideration via the relationship "CONTRIBUTES TO" > (and its 'negative' "HINDERS") could help > significantly to clarify the "HOWs?" along with the > "WHYs?". > > Some 'graphical representations' of these system > relationships as models (as, for instance, in > Warfield's Interpretive Structural Modeling) is > generally needed in order to clarify 'relational > matters' adequately for understanding all round. The > 'pure prose' to which we are generally confined in > our conventional discussions (including here at > Math-teach) often leads to a significant lack of > clarity all round. > How then to convert 'STEM' to 'STEAM'? Is that possible at all? Well, Douglas Hofstadter in 'Godel-Escher-Bach: ..." demonstrated that, yes - to a great extent - this is indeed possible.
As noted in my previous message (down the line a bit), the conventional educational system had largely failed to take up the challenge Hofstadter had thrown.
Alas, even at the 'Center for Research on Concepts and Cognition' [http://www.cogsci.indiana.edu/, where Hofstadter is Director] at Indiana University (Hofstadter's home ground), it appears that may not be quite on track towards the '*integration* of knowledge', as they do not appear to be doing much towards it (except rather superficially). This is a great pity (even, perhaps, a tragedy), as Hofstadter was one person who did appear to be capable of doing such an *integration*. (Note for Professor Bishop [in case he happens to be listening]: I'm certain I'm not capable of doing this kind of *integration*: someone like Hofstadter may be capable of it).
The desired *integration* would, I claim, be significantly enabled by application of the late John N. Warfield's insights into complex systems (and how we may cope with them) - in particular, his modeling tool of Interpretive Structural Modeling (ISM), which provides what I describe as the 'initiation of the integration process'. However, in order effectively to apply ISM and the other profound 'systems insights' that Warfield left us with, we really do need to understand the significant difference between various 'system relationships'.
In particular we need to understand the difference in complex systems between the transitive relationships "CONTRIBUTES TO" and "PRECEDES", which has been 'explored to death and beyond' by the 'management sciences via their PERT and Gantt Charts ('Project Management Systems' etc, etc).
Specifically, we do need to understand the simple 'system fact' that the relationship "PRECEDES" may well be useful in complex systems that we already understand fairly thoroughly, BUT:
The "PRECEDENCE" relationship is in fact MUCH worse than useless in systems that we have not adequately explored in regard to their fundamental 'structures' - it may well lead us to the mistaken impression that we 'understand' the system under consideration when in fact we do not understand the system at all! This is where the 'management "sciences"' have utterly failed us: they have - for near about 100 years! - been assiduously (but fruitlessly) exploring complex organisational and societal systems using the "PRECEDENCE" relationship - implicitly making the false claim that we (humans) have understood everything (or are en route to profound understanding). [The "PRECEDENCE" of Events and Milestones in a system would probably become more clear when the system as a whole has been adequately explored with "CONTRIBUTES TO"].
In fact, in regard to the complex organisational and societal systems that we have imposed on the natural systems that have evolved over eons, we have understood very little indeed: we have arrived at the mistaken notion that we (humans) are the "MASTERS OF THE UNIVERSE", a notion that is still widespread amongst (many of) us.
Of course, most of our religions and our human egos had already contributed significantly to such erroneous notions long before the 'management "sciences"' ever came to be. In fact, the 'management "sciences"' only built on that notion that we are the "MASTERS OF THE UNIVERSE". Those sciences did not at all help us understand how very dangerous this notion is.
I claim that 'systems science', via the insights originally provided by Warfield (after accepting the fundamental laws of thermodynamics), could possibly (and I claim, 'could probably') provide a way to overcome many such dangerous notions to which we're prone. > > > > > >> "Math dependent degree" includes a lot of them, > > >> including just about > > >> anything healthcare related. Maybe 3% want to > be > > >> math majors, whereas 40% > > >> seek a math-dependent major. > > > > > > Arithmetic dependent, probably, but not math > > > dependent. And certainly not > > > pure math dependent. > > > > > > > Arithmetic for sure, but also using measurements > > and > > currency, telling > > time, appreciation for data visualization... is > > SQL > > arithmetic? I'd say > > not. Microsoft Access did a lot to expose cube > > farmers to the power of SQL > > in terms of pay scale. Get the DBA certification > > from Oracle or whatever > > and watch the income bump up. > > > > When I started my training as a high school math > > teacher (Jersey City, > > 1980s), just about every math teacher I met was > > training to go in the other > > direction: towards private sector IT. We lost a > > huge army of math > > teachers to careers writing Visual Basic. It's > > not > > too late (in theory). > > Many of those VBers would like to jump back into > > teaching math, but we > > won't let 'em (barriers to entry are high). > > > > Here's the paradox: computer science is a fancy > > extra most run-o-the-mill > > schools can't afford, but the high property tax > > areas > > can, so the only kids > > who know about XML / HTTP / CSS in any detail are > > privileged, get the good > > summer jobs and internships, join the nonprofits. > But you've doubtless observed that even 'computer sciences' and the many practical developments from the computer sciences that have transformed our world (e.g., mobiles, notebooks, cloud computing, etc) have not really changed how we think about ourselves and our place on planet earth. [The process of *thinking* is more fundamental than all the computer sciences and all the management 'sciences']. > > > People talking about the > > achievement gap never mention our solution / > proposal > > in Oregon: let math > > teachers extend their curriculum such that all > this > > vocational / applicable > > stuff stops being so elective / dispensable. Let > > them learn programming > > *for math credit* for a change, rather than making > > 'em burn out on > > calculus. I was sorry the politicians failed the > > IQ > > test and let it > > languish, but not surprised. Idiocracy is > > widespread. > > > Indeed, the 'idiocracy' is very widespread indeed. But the idiocracy of our politicians (and their tame bureaucrats) must, in the final analysis, be an outcome of the 'idiocracy' of the people at large (i.e. all of us). I claim that some investigation about how things in the real world may "CONTRIBUTE TO" each other could help us to a great extent overcome this widespread 'idiocracy' that rules us today. > > > > > >> Some students have just never had the benefit > of > > >> an optimized learning > > >> environment. > Can you identify ANY students who've had an "optimized learning environment"? I claim there are none. Some have 'better' learning environments, and some have 'worse'. None of them is really 'optimized'. Is 'optimization' possible at all? I don't know. > > > >> They have every potential to > > >> catch > > >> up and surpass, and yes, > > >> that potential is as yet unfulfilled. > > >> Notunusual. > > > Indeed, practically every child that is born does have this 'potential'. However, little of that potential will ever be properly realised if the ruling philosophy is "WE (HUMANS) ARE THE MASTERS OF THE UNIVERSE!" as prevails now. > > > > I am not saying that is not true. But this is > > > not the course I would point > > > that ?some? to. This course is not designed for > > > the student that found > > > their calling, even if they just found their > > > calling. Can this course help > > > a late student find their calling? Maybe, but I > > > would design it quite > > > differently. Similar topics but very different > > > treatment. This course lacks > > > a crucial element. It doesn?t prove to the > > > student > > > that they are really > > > good at this, and that is what a calling is. > > > > > > > You should, design something quite differently and > > put your name on it. > > You evaluate the work of others but do you ever > > compete? I'm out there > > with my alternatives at least, however incomplete > > or whatever. But then I > > ramble through STEM, don't confine myself to the M > > part. For a long time > > I've marketed as ~M (not-mathematics -- just > > something quite like it). > > > Yes, indeed: as noted earlier, 'STEM' needs to be > expanded to 'STEAM'. > > > > > > > >> That doesn't mean it's wrong to ill advised to > > >> steer them to courses > > >> like Devlin?s. > > > > > > My point was that Devlin?s course wasn?t actually > a > > > ?course?. The topics > > > are good for the stated purpose, but the > treatment > > > is off. You?d have to > > > read the textbook I guess. And I?ll wait for the > > > course in February to make > > > a final judgement. > > > > > > > I'm looking forward to when Robert Hansen's course > > is > > one of the > > offerings. Why not put your insights to good use? > > We're all EAGERLY awaiting the formal launch of this! > > > > > > > >> Making sharp distinctions between pure and > applied > > >> mathematics is not > > >> essential. > Broadly I agree with you - but we DO need to make the distinctions in a better way than is conventionally done. I am NOT claiming, Professor Bishop, that I know exactly how this 'distinction' is to be made. I DO believe that some *effective* research into the 'systems' using the transitive *system relationship* "INCLUDES" could help us make such distinctions. The late John N. Warfield did, at some stage (both at the University of Virginia and at George Mason University) initiate some work by both physics and maths departments at those institutions on exploring the perceived 'INCLUSION' structures relating to topics in the respective disciplines. That work has not, I believe, been continued, developed further. > > > > > > Making sharp distinctions is always essential. > > > And > > > this one isn?t > > > complicated. I like pure math, but my colleagues > > > and I didn?t find our > > > calling in solving everyday technical problems > > > from > > > pure math. I am talking > > > about engineers, software engineers, financial > > > analysts, the list goes on. > > > That is why they call it ?applied? math. Can you > > > also study pure math? Yes, > > > but it obviously isn?t a requirement for > > > everyday > > > problem solving because > > > the vast majority of people solving everyday > > > problems don?t have a lick of > > > pure math in them. To say it is a different path > > > is > > > an understatement. When > > > Lou makes comments about engineers not knowing > > > ?math? my first thought is > > > ?Who doesn?t know that?" > > > Most - not all - of these distinctions between 'pure' and 'applied' math made by Robert Hansen are only superficial (IMO). Perhaps some deeper understanding of how 'pure' math "LEADS TO" 'applied' math would help. ("LEADS TO" is a 'transitive' system relationship, which probably would yield some useful insights. However, in my view, structures based on "LEADS TO" would be *effectively* understood after some basic understanding is gained of the systems being investigated via the relationship "CONTRIBUTES TO"). > > > > Making sharp distinctions is often a complete > > waste > > of time unless you > > really love nit-picky religious wars that even > > further degenerate. > > Broadly, I agree. > > > What stands out about you is you seem quite > > conservative in having > > swallowed whole many of the very bad design > > decisions > > made by your > > ancestors. You seem to really like the status > > quo. > > > I agree. > > > > > That's unusual in my > > neck of the woods where everyone is raging against > > the machine. We almost > > forget your kind exists. > > > Such 'forgetting' could be very dangerous! > > > > > In my view, STEM is STEM and attempts to atomize > > it > > further are usually > > quite misguided. > But 'atomization' and 'reductionism' ARE the prevailing philosophies! How to overcome that? "CONTRIBUTES TO" could help, perhaps! > > > What the letters stand for hints at > > the general domain, > > but if you admire polymaths like Hofstadter you > > have > > likely absorbed their > > oft-cited aversion to academic > > compartmentalization. > > > A great many people will claim to admire a polymath like Hofstadter, mainly because of the vast reputation he gained after Martin Gardner praised 'G-E-B:...". I have met a great number of 'educators' expressing such admiration for Hofstadter - but practicing the most rigid and hide-bound teaching in their courses! They often 'shush' students up when questions are asked! (Partially, this happens because the conventional education system has no *effective* way to deal with 'inconvenient' questions. Any education based on an adequate understanding of 'systems science' would invite the student to explore those inconvenient questions further and to present his/her findings. Perhaps those findings would be significant for the education of the other participants in the educational process (including the 'educators'). > > Hofstadter is (IMHO) one who has pretty successfully > applied 'STEAM' in his writing (in particular in that > great book "Godel-Escher-Bach:..."). Alas, the > 'educational system' has not taken up the challenge > he had thrown. > > > > Call it the revenge > > of the natural philosophers; they're snobbish > against > > "over-specialists" > > though respectful of competence, in whatever form. > > > True 'competence' involves, I believe, *effective* > (i.e. usable) understanding of both the 'HOWs?' and > the 'WHYs?' of factors in the system. > > > > > > > >> Textbooks are very often formulaic, having you > > >> find roots of this and > > >> that polynomial, slopes of this and that line. > > >> What may not fall out of > > >> such formula-based learning is any confidence > of > > >> being able to solve > > >> "generic problems" that life my pose. > > > > > > That depends on the textbook. Unfortunately, the > > > textbooks of today are so > > > remedial in design that they never get to the > > > enjoyable part of applying > > > all these new skills and insights to a > compendium > > > of problems like our > > > textbooks did. They are so focused on the > majority > > > of students that are not > > > good at math that they never get to the pace > that > > > occurs with students who > > > are good at math. Because of the algebra > mandate, > > > the purpose of math > > > education in secondary school has changed from > > > choosing math as one of your > > > life's pursuits to just learning some math. And > > > this has had a large impact > > > on qualified students finding their muse. It was > > > easy in our day, you could > > > be good at math, like you could be good at > > > music, > > > you could be good at > > > sports. Now you can only be good at music or > > > sports, but not math. > > > > > > > > > > I wouldn't go back though. Don't take away my > > Google, my Youtube. > > > > > > > > > >> At some point in a student's career we let them > > >> know about "numerical > > >> recipes" and the fact that whole books of them > are > > >> out there for > > >> consultation. > > > > > > Let?s not paint a rosy picture where there isn?t > > > one. Numerical recipes is > > > not a best seller. And schools are not turning > out > > > as many math artists as > > > they once were that could appreciate ?Numerical > > > Recipes? or even understand > > > it. > > > > > > > Knowing what algorithms apply: that was useful to > me > > for example. I knew > > how to use the Internet to find Qhull, free > software > > for finding a maximum > > convex hull given a smattering of points in XYZ. > > Just what I needed. And > > free. I incorporated Qhull into my studies of > > Waterman Polyhedrons without > > knowing in detail how it worked. But yet I could > use > > it well enough to get > > sensible results. Quite a bit of math savvy > resides > > in such "fitting the > > pieces together" skills, is what I'm saying. > > > Yes, indeed. I do strongly believe that Bucky > Fuller's insights, specifically into what I call > 'natural engineering systems' do need clarification - > and I believe you may be appropriately qualified to > do that. Why not try? (I mean, you would need to go > a bit firther than 'Biz-Mo', I feel). > > > > > > > >> Those who use mathematical techniques to > grapple > > >> with problems are less > > >> in need of great memorization skills than great > > >> research skills. > > > > > > The evidence I see is that you can?t have one > > > without the other. There is > > > no shortcut to being smart and good at what you > do. > > > You need the reasoning > > > and you need the memorization. I have never seen > > > someone make it with > > > "research skills". Those candidates stand out > right > > > away. > > > > > > > I'm thinking a strong mathematician often has > great > > research skills, > > including the ability to find the five or six > other > > people in billions > > right now focusing on similar problems. I have > this > > friend that calls me > > to talk about phi-scaled tetrahedrons, assemblies > > thereof, variations on > > that theme. It's not a conversation he can have > with > > just anyone. People > > glaze over when you start talking polyhedrons at > 'em. > > > Yes. How to overcome this tendency? Problem still > unsolved!!! > > > > His name is David > > and he's the guy who invited me to Minnesota so we > > could go visit Magnus > > Wenninger, which we did. Magnus, age 91 (92?) is > a > > big fish in that tiny > > world of polyhedron fanatics. > > > Had not heard earlier of Magnus Wenninger - can you > provide some links to his writings/work? (I shall, > of course, check out on Google, if possible). > > > > > > > >> Where are the cookbooks? How do you match a > > >> problem to an algorithm? > > >> Those are skills too, beyond proving. > > > > > > Now you are agreeing with me on the difference > > > between applied and pure > > > math. But there are plenty of cookbooks out > there, > > > in everything. And I use > > > them all the time, but you have to be smart to > know > > > where to look and how > > > to apply them. Again, there is no shortcut to > being > > > smart and good at what > > > you do. The lack of google was never a barrier > to > > > being smart and good at > > > what you do before the internet, so how can it > be > > > an enabler of the same? > > > It isn?t. > > > > > > > Just what I was saying: "smarts" includes knowing > > where and how. > > > I'd suggest the *real smarts* must involve the > "HOWs?" and the "WHYs?" of the things - scarcely EVER > are these clarified!! > > > > > > > >> Finding the theorem and realizing it's > relevance, > > >> locating the right > > >> recipe in the library, is all part of what one > > >> needs. Good road maps. > > >> Overviews. Most K-12 textbook math is fairly > > >> bereft of such concept maps. > > > > > > No. If this were true then over the years I > would > > > have seen a steady surge > > > in qualified candidates for the technical > positions > > > we post for. I have > > > seen a study surge in the breadth of expertise > in > > > the qualified candidates > > > but no surge at all in the number of qualified > > > candidates. A sharp decline > > > in fact. > > > > > > > Why? Do you think they're teaching research > skills > > any better? The tools > > have improved, definitely, but some of the high > > school libraries I've > > visited actively discouraged using the Web. They > > call it the Free Web and > > point out that the most trustworthy and best > > information still costs money, > > so if you want good grades, look through stuff you > > have to pay for. > > > Amazing! I guess it will take another generation or > so to ensure adequate acceptance of the Internet and > the Web. The underlying problem is (I believe) that > we humans really no longer have all the time in the > world to sort 'our problems' out. The major problem > (in my view) is that we have too long believed > ourselves to be "MASTERS OF THE UNIVERSE", which is > clearly a false belief. Problem is, too few of us > realise this. > > > > You > > may think I'm kidding, but I have photographs of > the > > posters in the school > > library, spinning just such falsehoods. Poor > USAers, > > so oppressed by mis- > > and dis-info (and yet they consider themselves > "well > > off"? "Developed"?). > > > The major falsehood: "WE ARE THE MASTERS OF THE > UNIVERSE!" > > > > > Think about what you are saying. > > > > > > Has not the internet made research incredibly > more > > > easy than our day? > > > Certainly you remember the before time. Before > the > > > internet. > > > > > > > Easier yes, but are research skills being taught? > My > > teachers (an unusual > > crew, mostly expats) consciously taught them and > told > > us that's what they > > were doing. Concept maps, card catalog, note > cards, > > thesis, outline, > > footnotes -- all stuff mathematicians need to know > > about, and then some. > > > Yes, indeed. However, I'd suggest: > > Doodling; outline; footnotes; card catalogue; concept > maps - the importance of the first and the last of > these is not yet adequately understood! > The above sequence should actually be: "doodling; ... outline; ... ootnotes;... card catalogue;... bibliographies of bibliographies... concept maps", where the "..." express concepts that may intervene.
I do believe our understanding of the human mind would improve very significantly when the 'cognitive sciences' learn how effectively to use 'doodling' and 'concept maps' (at the beginning and end of the above sequence. I see very few explorations along these lines. > > > > Bibliographies of bibliographies... Princeton had > > lots of those. > > > > > > > > If road maps and research had anything to do > with > > > conquering these > > > technical fields, wouldn?t we have seen a surge > > > rather than a decline? > > > > > > > Not if no one is learning to drive. > > > Only those who are already hopelessly lost claim that > road maps aren't important! > I iterate strongly: 'road maps' are HUGELY important in all that we need to do. > > > > > Being smart and good at what you do is the key. > > > That is what needs to be > > > nurtured. Not excuses. > > > > > > > Yeah, I know. I find approximately no one is as > > smart as they could be, > > but we live in a culture that expects and rewards > > dummies. Being smart > > means being lonely at parties. > > > Absolutely! NO ONE IS AS SMART AS HE/SHE COULD BE. > > > But that's cultural and we have subcultures. I put > > a > > lot of hope in > > those. The speech and debate world, for example. > > Go Cleveland Cannibals! > > > Alas, the 'speech and debate world' is still stuck in > the 'pure prose' mode of communication! (I guess, > 'Cleveland Cannibals' WON'T go very far unless we > being to understand the "HOWs?" and the "WHYs?" of > communication). > > GSC > In the last paragraph, that should be "begin to understand", not "being to understand".