
Re: The number line as the main model for operational concepts > numerals
Posted:
Apr 16, 2012 1:51 PM


On Sun, Apr 15, 2012 at 8:52 PM, Clyde Greeno @ MALEI <greeno@malei.org> wrote: > Kirby's "... The "numeral versus number" meme was big, fashionable to share > with kiddos like me." > > Unfortunately, the "numerals" attended by the authors (and so, the teachers) > of the "New Math" were simply marks on paper ... rather than being > mathematical systems for representing numbers. For example, their "Arabic > numeral" for 345 was the written expression, "345" ... rather than the > tuplic sequence (3,4,5). >
What I was taught was these tuplic sequences map to positions on the abacus which, in base 10, count in successive powers of 10.
Lots of time writing stuff like 254 = 2 * 10*2 + 5 * 10 + 4.
Then, we were told, the base didn't have to be 10. It could be like 5 or 2 or 16. Or it could even be mixed (like datetime).
Converting between bases was something we did.
Tom Lehrer came out with his 'New Math' spoof and the adults around me laughted even louder when I said "what's so funny, he's right?"
> Because of that shortcoming, "the New Math" authors' attention to their > kinds of "numerals" did essentially nothing toward improving the > effectiveness of scholastic education in mathematics. But if they had given > serious *mathematical* attention to numerals ... within the context of > mathematical theories or representation ... they would have been able to > make arithmetic commonsensible. [Their failure to do so is the primary > cause of America's educational dilemma in mathematics.] They also would have > come much closer to computermathematics. >
I thought they did a pretty good job with this bases stuff, but then my education was atypical. I got dropped into a British school in third grade and there got a serious dose of drill and kill, which made me much more proficient than my USAbased peers. When I came back to the US system, then in the throes of New Math, I stood out because of my better numeric abilities.
My current boss thinks Americans become Eloi (H.G. Wells allusion) because they just passively read and/or watch vids without really getting off the couch much. They sound smart, but have no skills. Maria Droujkova's 'Making Math' or 'Make Math Your Own' is more his approach, which he developed after a visit to Russia. The owners of "Make:" magazine found him struggling in middle America and asked him to join the open source revolution.
His business is now based in Sepastopol, named by Russian trappers and traders pushing down from the north and encountering the Hispanicized ethnicities of the south (the line between the two was essentially the Bay Area). Talking about Scott. He joined us earlier to talk calculus, as a disciple of Jerry Uhl.
> It turns out that all of arithmetic is done by operating with and on the > representation spaces (of "numerals") ... rather with the lines/planes of > numbers. Those are the only spaces in which computers can operate. >
I don't have these concepts of "numeral" versus "number" so hardwired anymore. That kind of name>object representationalism has limited appeal, though I do think Guido van Rossum has done an excellent job giving it expression in his mechanized logic, the one I teach. This logic is all about "namespaces" as containers for names referring to objects. The same object may have many names.
When the bus system uses numerals for bus routes, where it might have used letters or names, you see another kind of mathematics, all about partially overlapping trips, hubs, a topology, schedules. Lots of mixed based computations with time, lots of GPS. Add the database, which knows about each street intersection, what's one way, what's closed to through traffic, and you have a generic set of language games in which lots of numbers are involved. Is this a different meaning for "2" (as in Route 2) than in number line arithmetic? That case could be made.
In my geometry of blobs curriculum, there's emphasis on the "two twonesses" of blobs:
axial spinnability (around)
versus
concave / concavity and radial change
Take a globe. its "spin delta" is all about poles, axes, great circles, networks of great circles, spinning left, spinning right.
The globe's concave / convexness is about divergently expanding outward from a radiant center, growing and shrinking along all axes, having an outwardly convex aspect and inwardly concave one (these are topological properties of containers, hulls)
We build up geometric vocabulary on this basis, slowly sharpening the picture with Vertexes (corners, centers, hubs), Faces (lawns, fields, facets, fenced farmlands) and Edges (roads, pathways, hyperlinks). This sharpening brings the polyhedrons into focus, and only then do we start bridging back to arithmetic, with our figurate and polyhedral number sequences. Triangular numbers, square numbers, icosahedral numbers  our first computer programs (so they've learned some typing, not just with thumbs).
> Re: Kirby's post, below: "In sum, I think the New Math picture of numeral > > number has been >> >> replaced by more sophisticated and nuanced gamebased thinking in >> higher education, ..." > > > I am not quite ready to equate "more specialized" with "more sophisticated > and nuanced ..." >
The idea that "2" does NOT need to "represent something" in the sense of pointing to some imaginary entity off camera, has been exciting to many. Symbols gain meaning from having a role in a process, not from pointing, much as pointing may be part of some games.
"2", like the "pawn" in chess, is a cog in the machinery that gets its meaning from the whole machine, what it does (e.g. a transportation system).
This isn't a matter of nouns naming but of verbs acting.
In broad brush strokes: this conceptual shift from nounbased to processbased concepts has taken place across many departments over many decades.
Philosophy was an instigator with its linguistic turn but the zeitgeist has been active in other nooks and crannies as well.
There's no turning back the clock I don't think.
>> .... and this has a trickle down effect, especially through the computer >> side of things, as that's where more people >> encounter lots of mathy concepts. Real numbers are losing ground, as >> more hypothetical seeming than they used to seem, joining the >> imaginaries. > > > "The imaginaries", being simply points along the second axis of a > (possibly discrete) 4quadrant plane, can be even less "hypothetical" than > the continuum. >
Instead of "hypothetical" I might have said "metaphysical". We inherit our idea of an infinite infinitely thin plane, a perfect continuum, from the Mediterranean cultures.
The idea that "no one uses real numbers in computations" is somewhat silly as a dogma but is enlightening to debate, and therefore to defend or subvert.
People were computing long before the real numbers came along, and now that computers do our arithmetic, we've had to trade the reals in for more realistic / practical types of number, such as IEEE floating point and extended precision. The real numbers proved impractical.
Lots of complex number machinery in extended precision as there's a way of multiplying analogous to using Fourier transform with roots of 1 as a base, then adding and converting back.
> The computercatalyzed focus on discrete representations by no means reduces > the underlying importance of doublyunbounded continua of numbers. Rather, > such continua provide the context for the evolution of computeraccessible > theories of discrete approximations. True, one may choose to neglect the > context, and focus only on the representations .... so training computers > much like training children to roteprocess numeratoroverdenominator pairs > of integers. But nonetheless, those representations are mathematical > "numerals." >
You say "discrete approximations" but I flip that around. A continuum is an approximation of something more discrete. The conceptual apparatus just failed to keep things separable (like grains of sand) but you loose precision when you insist on "smooth". That's more an aesthetic than a dogma. A style / fashion / brand.
Kirby
> Cordially, > Clyde > >   > From: "kirby urner" <kirby.urner@gmail.com> > Sent: Sunday, April 15, 2012 11:19 AM > To: <mathteach@mathforum.org> > Subject: Re: The number line as the main model for operational concepts > > complexes
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Message was edited by: kirby urner

