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Re: Enrichment
Posted:
Aug 22, 2012 12:12 AM
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On Sun, Aug 19, 2012 at 8:44 PM, Robert Hansen <bob@rsccore.com> 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.
http://www.flickr.com/photos/kirbyurner/497128497/
(simulator at local science museum)
There's a feeling of being pressed back, and therefore
of the plane tilting up, even before it tips up, because
of acceleration. An inner ear thing -- those fluid sensors
that affect balance.
None of this is calculus per se, but the idea of
"rate of change" from "smooth" to "sudden" is hard
wired into human experience and may be discussed.
The speed with with trends catch on.
Exponential growth, geometric growth...
now we're starting to talk about algebra a little.
Here's some middle school machinations:
>>> square_numbers = [x**2 for x in range(11)]
>>> square_numbers
[0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100]
>>> intervals = list( zip(square_numbers, square_numbers[1:]) )
>>> intervals
[(0, 1), (1, 4), (4, 9), (9, 16), (16, 25), (25, 36), (36, 49), (49, 64),
(64, 81), (81, 100)]
>>> deltas = [(b-a) for a,b in intervals] # compute successive differences
>>> deltas
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19]
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.
Kirby
> 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 <pduveen@yahoo.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.
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