Hi, Sorry for the late reply.
I don't know if this is helpful, but something people have told me before is that if you can get it started independently (i.e. without administration support/money but with permission), and you subsequently demonstrate success, then you have a more powerful argument for progress than merely an argument, no matter how coherent-- ESPECIALLY if money is involved. So I would ask: do you have a computer lab available? Can you maybe convince some below- and above-average students to take on a new after- school activity once a week, where they experiment with math in Sage? (My personal experience was that SOME below-average students would perk up if you gave them attention, and might even engage; the above- average students were usually up for something new & exciting. Maybe you can combine them in teams.) For the program itself, you could start by teaching them rudimentary programming, using as much as possible "real-world" problems to motivate each step. If you don't have access to a computer lab, then--as much as I hate graphing calculators, they do at least have programming facilities. So if nothing else you could get students working with that. Last remark: having read the AP exam twice, I understand why they require graphing calculators. They want to ask certain kinds of questions on the AP exam, and those are generally beyond the scope of what one can do by hand. QED. I don't like it much, but parts of the AP exam are calculator-free if I recall, and I allow graphing calculators on my exams anyway, so I guess I can't complain too much. If this sounds like I'm contradicting something I wrote earlier, well... it's possible. :-) But I like the idea. I've remarked to people in Math Ed at my institution that I think computer programming should be required in high school, and they look at me funny. :-) regards john perry On Jan 1, 2:26 pm, michel paul <[email protected]> wrote: > After reflecting on the responses I received last year (and happy new one!), > here is a condensed version of points I'd like to express to my > administration: > > - Given the ubiquitous nature of freely available and powerful > computational technology in our culture, what should high school students > learn? > - I believe they should learn math in a way that simultaneously empowers > them to make the most effective use of this technology. > - Fluency in graphing calculator use is not sufficient for contemporary > computational literacy. In fact, it is neither necessary nor sufficient. > The only possible reason for insistence on their continued use would be > founded in an interest in promoting the product. The AP and SAT exams > promote the use of these products. QED. > - A problem in promoting the use of such products is the limited > understanding of mathematics they encourage. > - The standard of mathematical and computational literacy required (not > necessarily by current state standards, but in the larger world) by today's > high school students can be addressed through the judicious study of > computational language. > - Not all programming activity leads to mathematical insight. However, a > central core of what we call programming is in fact a form of pure > mathematics, and many aspects of this way of thinking are in fact relevant > for the high school math curriculum. The sooner America gets on task on > this, the better off we'll be. We can begin to address the deficiencies in > both our secondary mathematical and technological literacy simultaneously. > > Again, please let me know if I'm off base with any of this. Is any of this > irrelevant or tangential? > > In my initial list > > >7. Instead of spending so much time teaching kids how to isolate variables > > in equations, perhaps it would be better for them to learn how to construct > suites of simple interacting functions? > > I think is clearly a mistaken expression. It should *not* be 'instead of > ..'. Rather, > > >7. *In addition to* learning how to isolate variables in equations (and > > explaining their reasoning), kids also need to learn how to construct suites > of simple interacting functions to model and test ideas. > > Again, please correct me if I'm off, but I think this is one of the central > differences between what we do in traditional high school math classes vs. > what one does using a computational language/environment - *construction*. > When using something like Sage, most of one's effort is not engaged in > 'solving equations' but in constructing computational models of ideas, and > this is important for today's math students to learn to do. Our traditional > curriculum doesn't touch that kind of stuff - or only rarely. > > I completely agree with and appreciate the importance of getting them to > isolate variables in symbolic formulas. I think that's where a lot of > problems arise in students' understanding of what algebra even is (and I > think the emphasis on calculators has promoted this misunderstanding) - they > think it's all about finding particular numeric solutions for individual > equations or for systems of, at most, 2 or 3 equations. Then, when it's > purely symbolic, their reaction is "Why are there so many letters? Why > can't you use more numbers?" But this really is where they need to focus. > The reasoning required to manipulate symbolic expressions is directly > related to the reasoning required for computational constructions. > > There seems to be lots of agreement about the importance of writing in > math. Perfect. I hope this can be a major point in persuading my > administration that integrating something like Sage - not treating it like > it's something foreign - would be extremely valuable. Again, kids could > create their own math reports in Sage, little mini-papers, that would > actually *do* stuff while explaining ideas. > > And along with writing - reading. I deeply appreciate the recommendation > that if kids learn to *read* a math text that everything else becomes > secondary. Yeah, that's great. I'm going to make a point of incorporating > that into my classes. > > As for 'concept maps' I will replace the example of the quadratic formula > with the example of standard deviation. I think that conveys the point > better. > > What I now need is a simple, direct, knock-down, and hopefully fatal > argument against the entrenched position that 'graphing calculators are > enough'. That's really the whole source of the opposition I constantly face > in the high school world the AP and SAT are considered sacred and anything > 'else' is too much. > > My position has been that, no, this is not some other layer on top of the > math, this IS math itself, this is how mathematicians do things these days. > > How accurate am I in making statements like that? I want to create as > effective and accurate an argument as I can. > > Also - has it become the norm for college math departments these days to use > some form of CAS, whether Mathematica, Maple, MatLab, or Sage? Or do only > some use these things? If it has in fact become the norm, and if we think > we're trying to prepare kids for the world they'll be entering, well, why > NOT show them these things? > > Again, thanks very much for the constructive dialog on this. > > Happy New Year. > > - Michel Paul
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