On Sun, Aug 19, 2012 at 8:44 PM, Robert Hansen <firstname.lastname@example.org> wrote: > I don't think a student can do anything meaningful in calculus, statistics or any higher math if they don't get algebra. It's like asking me if a student can be taught creative writing when they don't understand grammar. It isn't an arbitrary decision that algebra precedes calculus. >
In spiraling towards an understanding of physics, you have first person experience of riding a roller coaster and know that changes in velocity have a visceral aspect.
Fast changes in velocity may include a sense of being pressed back in one's seat.
Then we switch to a simulator, or just showing one, and people see it tilting back. Inside, the people are sensing acceleration as the move more quickly down the runway.
What's they're investigating are the differences between successive square numbers.
You get this sequence: [1, 3, 5, 7, 9, 11, 13, 15, 17, 19]
They would immediately see (some would) that the next set of intervals resolve to the constant 2, i.e. (3-1) == (5-3) == (7-5) and so on.
by adding "two more than last time each time", I get a 2nd power growth rate of my cumulative total (like integration). This is slower growth than successive doublings.
This difference between linear and exponential growth is incipient in middle school and may be used to anchor the idea of successive differences having successive differences which in turn have successive differences, down to where there's no difference anymore, e.g. once we get to [2, 2, 2, 2, 2, 2, 2] we have [0,0,0...] for our end.
But how many steps did it take to get there?
What if we'd started with 5th powering?
[ see plaintext mode for correct syntax (whitespace sensitive) ]
>>> def get_values(f, domain=range(15)): """computers f(x) from a domain""" return [f(x) for x in domain]
>>> def get_deltas(values): """computes deltas between successive values""" intervals = zip(values, values[1:]) return [(b-a) for a,b in intervals]
>>> def f(x): return pow(x, 5) # 5th power function
>>> values = get_values(f) # getting f(x) where x=0...14 and f(x) is 5th power of x >>> deltas = get_deltas(values) # getting deltas between those values (akin to 1st derivative) >>> deltas = get_deltas(deltas) # akin to 2nd derivative >>> deltas [30, 180, 570, 1320, 2550, 4380, 6930, 10320, 14670, 20100, 26730, 34680, 44070] >>> deltas = get_deltas(deltas) # akin to 3rd derivative >>> deltas = get_deltas(deltas) # akin to 4th derivative, down to linear (1st power) growth >>> deltas [240, 360, 480, 600, 720, 840, 960, 1080, 1200, 1320, 1440] >>> deltas = get_deltas(deltas) # 5th derivative and we flatline (no growth) >>> deltas [120, 120, 120, 120, 120, 120, 120, 120, 120, 120] >>> deltas = get_deltas(deltas) # no change in y per whatever change in x >>> deltas [0, 0, 0, 0, 0, 0, 0, 0, 0]
So, if after a couple difference of difference reductions (differentiations) as above, you're still looking at something exponential, then the original change rate must of been really high.
The differences between the differences, and the differences between those: we can grok that. Is this algebra at all? Even calculus?
A lot of teachers would want to play such games on a spreadsheet but my attitude is why bother when you have an interactive chat window such as the one above, available free of charge.
> Can a demonstration of calculus inspire students in general. The preponderance of evidence that already exists says "No", resoundingly. Can a demonstration of calculus inspire this one particular student? Only you (his teacher) would know this better than anyone here. > > Bob Hansen > > On Aug 19, 2012, at 2:55 PM, Peter Duveen <email@example.com> wrote: > >> Well, Bob, boiled down, my topic is, 1. can calculus be taught earlier than it is generally and 2. is "enrichment," which could include such teaching, as well as all kinds of other teaching that might to some be considered off the beaten path, be used to stimulate the love of learning on the part of students who get bogged down for one reason or another in other parts of the curriculum they are required (forced under pain of law) to "learn." Something like that.