On 16/08/2014 2:44 PM, Wayne Bishop wrote: > And in this situation, teaching students to (even just mentally) > rewrite Gary's original (for x not 0, of course) as: > > y = x^2/(x-1) = x/(1 - 1/x) and to think about what happens for > large values of x is an important part of the "toolbox" of > knowledgeable students. "Conjecturing" from a computer screen goes > nowhere. Actually proving the asymptotic behavior at some later stage > is another level of sophistication that is helpful for getting ready > for calculus but otherwise don't do it. Students can forget how easy > the idea is if they get buried in the formalities. > > Wayne > > At 09:47 PM 8/15/2014, Robert Hansen wrote: > >> On Aug 15, 2014, at 11:05 PM, Wayne Bishop >> <email@example.com> wrote: >> >> > Of course you were, so was I. My problem is that this is in the >> context of software. Any student that needs to understand that stuff >> ought to be able to sketch a graph of your original without plotting >> any points as with the one you had in mind. For wild stuff taken >> from lab data, or some such, the situation could be very different >> (and computers are better than graphing calculators that no one uses) >> but for teaching the concepts? Absolutely not. >> > >> > Wayne >> >> This seems to be the issue in this discussion. What sort of awareness >> are we talking about. Gary did say *reinforce* and I think, >> pedagogically, that is all that a visualization can do. For example, >> if I show a student what an asymptote *looks like* they are not going >> to gain any mathematical significance from that. The teacher has to >> guide them through the significance using dialog, examples and >> counter examples. Visualizations can only supplant that process, not >> reduce it nor fix it. >> >> Bob Hansen > > Anyone of those who believe the purpose of math 8 is preparation for math 9 and the purpose of math 9 is to prepare for math 10 etc will find my postings somewhat off the wall. I'm more of the opinion that a sound math education provides the student with the tools to investigate. In days of yore, could we conceive of Greek geometricians who would eschew stick, sand, straight edge & twine because they realized that mathematical truths do not require any physical 'evidence'. Is there a student of math today who disdains slide rules,calculators, computer software for the same reason. As Herodotus or his cousin may have played with stick n sand, so too Dwayne today might similarly play with his calculator and note that repeatedly hitting the sqroot button causes an interesting outcome - one that he might share with Betty Lou across the isle. They just might try to figure out why. Their teacher might be of help. At some stage students will note that a circle consists of an infinity of points & anticipate that the circle with radius 0 will just disappear - - but if they use graphing software to plot (x-1)^2+(y-2)^2=0 they may be surprised to see that the infinity of points instantaneously become a singleton.
****I believe calculators & math software will be most useful when we do not know the outcome in advance.***
Math can be fun;)
I once noted that folding carefully a strip of paper into a 'knot' provided a pentagon. Maybe we have someone in the military industrial complex who whiled his/her time away on printer tape.