Teaching Math It was Harold Fawcett, one day when we were walking around the Mormon Square in Salt Lake City, who made a statement I have never forgot. He said, "It is not how much you cover but how much you uncover." It was B Peirce. who defined Math as: "The science of drawing necessary conclusions." Which brings up the question, What are conclusions based on? (All conclusions are based on undefined terms, defined terms, basic assumptions, and previous conclusions or laws and in Math theorems.) The check list, put out by the NCTM in1947, with a few additions is appropriate today as then. (Most people do not understand the forms of implications and valid or invalidity A teacher needs to feel that their subject is the most valuable and sells the subject to the students as needed for success. Yes, a teacher is a salesman. I could go on!
On Tue, Mar 5, 2013 at 10:42 PM, GS Chandy <email@example.com> wrote: > Lou Talman posted Mar 2, 2013 1:22 AM (delay because i'D not opened Math-teach for a few days): >> On Fri, 01 Mar 2013 08:43:47 -0700, GS Chandy >> <firstname.lastname@example.org> wrote: >> >> > I personally believe that the 'quantitative >> techniques' (and >> > tools/processes) are not really relevant or >> appropriate to apply to >> > human beings and their problem-solving or learning >> processes. >> >> Here I must disagree. Mathematics has been, is, and >> will remain, central >> to human endeavor; it is one of only two effective >> tools we have for >> manipulating ideas. >> > But I completely agree that mathematics is surely central to human endeavour! However, mathematics is more than numbers, or 'quantitative techniques'. > > I certainly should have phrased my ideas more accurately (/exactly?): I still do believe that the 'quantitative techniques' have to follow on acquiring basic 'tools for thought', and this was what I was trying to communicate. (Warfield's 'modeling approach, outlined in the attachment "What is Modeling?" - provided with my post to which you're responding - reflects this idea). I must make my belief clear that a great deal of our failures - in society; in organizations; as individuals - stem from our tendency to seek out numbers before we have adequately understood the 'qualitative aspects of the system under consideration'. THAT was the point I was trying to make in my previous 'miscommunication'! (I don't know whether I have succeeded in making it even now - see below). > > Very often, we find there could be errors in basic thinking 'below the numbers', so to speak, that may be source(s) of confusion; equally, there could easily be errors in communication even when the thinking may be correct. For instance, the inexactness in the way I phrased my (prose) response above had created this particular confusion, for which I apologise - though my 'mental model' of what I had wished to communicate was quite clear (to me, at least!) > > I also claim that it is very easy indeed to mis-communicate when we depend *entirely* on what I describe as the 'prose mode' of communication. We can quite significantly reduce such mis-communication (and misunderstanding of ideas attempted to be communicated) by using 'prose + structural graphics' (p+sg), where models constructed using 'structural graphics' help to elucidate the relationships perceived between factors in the 'system' under consideration: these perceived relationships often remain rather ambiguous or are hidden in prose mode communications. At Math-teach we are alas restricted entirely to communication in 'pure prose': this, I have claimed, is often a source of quite serious disputes. > > Some basic information about tools that can quite significantly enhance the effectiveness of our communications within and about complex systems is provided at the attachments to my message heading the thread at "Democracy: how to achieve it?" - http://mathforum.org/kb/thread.jspa?threadID=2419536 . > > (The message itself discusses some issues about democracy as an 'ideal' of sorts; and how it has rarely [if ever] been achieved within societies, in practice on the ground. However, the tools discussed are readily applied to any 'Mission' whatsoever. Several exapples of 'Missions' [successfully accomplished as well as failed ones] are provided in the PowerPoint presentation attached). > > (A Mission of particular interest to me is "To develop effective education systems in India" - though I must confess I've not gotten very far with it - work on it continues regularly, so it is not a 'failed' Mission. There have recently been some quite encouraging signs that some influential people in our educational systems are beginning to pay attention. > > (One 'contributory cause' for the lack of progress on this Mission is, probably, that people thinking in conventional ways do [for quite good reasons] find it difficult to believe that the above HUGE Mission can be successfully accomplished simply by stakeholders in it writing up, in appropriate formats, their ideas about it! [Of course, it is the subsequent modeling of those ideas that leads to clarification of those ideas] > > (I recall that Haim and Robert Hansen once used to have themselves much rather foolish enjoyment by poking fun at the very thought that people should write down their ideas in 'element form' about their Missions of interest. Of course, that initial writing down of ideas is only a start, and there is certainly a long, hard route to get to places with it. > > (One quite successful - though minor - individual Mission was worked on by a young freshman student: "To understand thoroughly all topics of my math syllabus, and THEREBY to improve, very significantly, my results in my math exams, tests and quizzes". The Mission was not 'minor' to the student himself! It is interesting that I gave that student NO math tuition at all! I only guided him in showing him how to construct and interpre t his models; on "what to do next" at various stages of development of his his ideas on his Mission). >> >> In his little monograph, "Liberal Education," Mark >> van Doren distinguishes >> three families of arts. There are the Useful Arts, >> with which we >> manipulate objects and there are the Fine Arts, with >> which we create. >> > I don't currently have ready access to Mark van Doren's "Liberal Education" (which I have read) - but I fully agree also with the portions you've quoted (though I recall I'd had some (ra ther minor) quibbles with some of van Doren's ideas when I had read the monograph. >> >>In >> between these two are the Liberal Arts, with which we >> manipulate ideas. >> According to van Doren, there are but two liberal >> arts (though he follows >> the ancients in dividing them into seven disciplines >> divided into two >> categories named the Trivium and the Quadrivium). >> They are language and >> mathematics. Of these two, he says: >> >> "'Language and mathematics are the mother tongues of >> our rational >> selves'---that is, of the human race---and no student >> should be permitted >> to be speechless in either tongue, whatever value he >> sets upon his special >> gifts, and however sure he may be at sixteen or >> eighteen that he knows the >> uses to which his mind will eventually be put. This >> would be like >> amputating his left hand because he did not seem to >> be ambidextrous. The >> languages of art and science are of twin importance. >> It is crippling to >> be illiterate in either, and the natural curriculum >> does not choose >> between them. They are two ways in which the student >> will have to express >> himself; they are two ways in which the truth gets >> known." >> >> In the paragraph just above, van Doren places the >> first sentence in >> quotation marks, so he is quoting someone. He doesn't >> give a source, and >> I've never been able to find the original. I think it >> was probably Scott >> Buchanan, but I can't support that statement with >> anything more than a gut >> feeling. >> >> - --Lou Talman >> >> Department of Mathematical & Computer Sciences >> Metropolitan State University of Denver >> <http://rowdy.msudenver.edu/~talmanl>
- -- Jim
Jim has 5 new CDs on the market. CD1:TGIF MATH (A 100+ activities to make a hectic math period on days like prior to homecoming into an rewarding learning day.) CD2: EVERYDAY DECISION MAKING VIA GEOMETRY ESSENTIALS (A Logical development of the essentials of PL. & Solid Geometry and applying it to decision making.) CD3: EVERYDAY DECISION MAKING VIA MATHEMATICAL BRIDGES FOR A BETTER FUTURE (Liberal Art "bridges" emphasizing critical thinking.) CD4: EVERYDAY DECISION MAKING FOR A BETTER CAREER (Mathematical topics needed for skills and for better decisions) CD5: BASIC HIGH SCHOOL MATH REVIEW (Review for SAT, ACT or other tests with Dec ision Making skills) For more info: http://sites.google.com/site/mathfordecisionmaking/