Joe Niederberger (JN) posted Jan 31, 2014 9:59 PM (http://mathforum.org/kb/message.jspa?messageID=9376536) - GSC's remarks interspersed: > > Neighbor says: > >If a beginner starts to learn math, the first > >concepts he comes upon are very likely points and > >lines. How should we explain to him these? (of > >course, not only giving the definition, but also > >explaining to him the why of this by means of > >rational arguments) That is to say, telling him In > >what way he should approach these sorts of problems. > > Now *that* is a very good question. > ... > I think one thing lacking today is a bit of frank > discussion about the big, big leap... > Indeed, that's a very good question. And why is it not generally understood in our math educational system how learners may be helped to make that big, big leap?
It is a characteristic of our educational systems in general that they fail to help much to help learners in them make those 'big, big leaps' that need to be made in all disciplines, not just math. The underlying problem is, I believe, that 'teaching' is considered to be a 'thing-in-itself' so to speak, not a member of the silver dyad of 'learning+teaching'. A few teachers do manage to help their students make those 'big, big leaps' - but those are the exceptions.
The 'ed system' as a whole lacks needed awareness.
Perhaps, instead of advocating that the "schools of education should be blown up", we might suggest that they should learn how to ensure that their trained teachers would come out of them adequately equipped to enable students make those 'big, big leaps'?
Another slogan I've seen from one of our participants here ("Children must be PUSHED to learn math!" [RH]) is possibly even more dangerous than "blowing up the schools of education". > > (I think children > learn counting, and other concepts such as fluid > quantities, long before they contemplate zero > dimensional points and 1 dimensional lines and such.) > > The "number line" comes to mind, as a possible > introduction -- something you more or less explicitly > rejected in this thread. At first, concepts like > "infinitely thin" don't need to be mentioned at all. > The first step is to illustrate numbers as lying on > the line, and numbers also get associated with > lengths that way. As the grades progress through > negative numbers and fractions, the number line is a > steady referent. By the time real numbers are > explored, the notion of "completion" (as in closed > under the limiting operation) could be broached in a > very naive and informal way, along with many other > Alice in Wonderland notions such as decimal > expansions "that go on forever without repeating" > etc. Some set theory along the way would be useful. > > I think one thing lacking today is a bit of frank > discussion about the big, big leap from say, > rationals, to a proper recognition of what is > required to understand what the real number system > *is* (which is more or less a set of rules for > playing with equations and abstract operations on > completely abstracted quantities, because one can no > longer perform the actual operations in reality.) > Given two fractions, I can "really" multiply them > (write down the fractions that represents their > product.) Given two real numbers, in general, I > cannot. > (One might object that if the fractions are big > enough I'm also stymied!) > > Anyway, such a discussion has a lot to say about what > "mathematical thinking" is, and has a lot to do with > algebra and logic. > I recall a thread here where a course on 'mathematical thinking' was discussed (Devlin's?). While the course as outlined certainly had some merits, it may not have quite succeeded, I believe, in inculcating mathematical thinking in learners. Why not? Any 'math thinking' developed by the participants in that course would have come mainly from within their own minds, only marginally aided by their learning in that course.
I don't believe the discussion here adequately identified that and pointed a way to ensure that 'math thinking' would be inculcated by way of such a course.
GSC > > Without algebra and logic, thinking about infinitely > thin lines and zero dimensional points really does > lead into a labyrinth. Hell, it leads into a > labyrinth even with those tools. > > Cheers, > Joe N