>Hi everyone,
>
>In this day and age of calculators, does anyone talk about slopes as
>0.08333/1 (which is approximately 1/12 the maximum slope approved by ADA for
>a short ramp), or 4781/1 which might represent an increase in sales per
>month for some company over a particular time period? Although students
>should no doubt start with slopes of 2/3, 1/2, 4, -3/2, I do not understand
>why at any level we limit ourselves to "baby numbers." Introducing the
>notion of the slope as a tangent certainly lends itself to a discussion of
>slopes other than nice ratios.
I'm in complete agreement with Martha here, that in the
calculator age there's no reason to stick to 'nice numbers'
when doing exercises. Nice numbers (NNs) are helpful when
learning algorithms on paper, but once you understand the guts
of a process, it's time for 'off road terrain' -- no more Mr.
Nice Guy, no more hand-holding, Crocodile Dundee heads for the
hills in his Subaru Outback.
Also, I'm of the school that kids intuitively recognize that
those little calculator buttons are the keys to many secrets,
the distilled wisdom of engineers through the ages. Like,
these (mostly unary and binary) operators are the ones that
deserve 'hall of fame' acceptance on a tiny keypad and, more
impressively, microcode implementation in our microchips.
In line with this, as soon as students are 'required'
('encouraged' 'pressured' -- whatever) to bring a calculator to
class (or maybe the school issues a loaner), I think it's time
for a run through on what all those little symbols mean, right
off the bat -- not a good idea to cloak half a tool's functions
in mystery saying "you'll learn about TAN and COS a couple
years from now" -- yucky!
Even if you don't spend time doing exercises, you should have
the fog of unknowing lifted, burned off, ASAP, so then you can
legitimately carry this thing around with a "knowing" air, not
needing to act stupid if someone (e.g. a parent or peer) says
"what does this key do?" and all you can say is "duh, they
didn't teach us about that one yet."
One of my main complaints with yesteryear's textbook-based K-12
was its single-minded commitment to "plodding"[1], deliberately
keeping kids in the dark re even the gist of specific symbols
of obvious import. A lot of "disconnect" or "turn off" in math
comes from drilling for years in classrooms, and yet feeling
just as ignorant as ever about what the Riemann symbol, or
Sigma, or even TAN() have to do with. The student thinks "if
it takes 8 years of drilling and _still_ I can't follow what a
physicist is writing on the chalkboard, even just a little,
then it must be at least another 8 years to go before that
stuff even starts making sense". A propensity to nod off, to
go unconscious, may set in at that point -- a strategy to avoid
facing a sense of utter hopelessness.
But really it _could_ all start making sense a lot earlier --
not that you become an Einstein at 15, but that you at least
have the insight that both the Riemann symbol and Sigma have to
do with accumulating lots of values in a running total.
Spreadsheet programs use the Sigma as an icon to mean 'total
this column of numbers'. The Riemann symbol is 'smoothed
Sigma' -- marks the transition from 'discrete calculus' to
'differential calculus' i.e. the increments have become
'vanishingly small' and not 'jaggy' (no need to refine this ad
nauseum -- just impart the gist).
Likewise, we're seeing in this thread how 'slope' (easily
anchored in everyday experience) relates to rise/run or
vertical/horizontal (e.g. a skateboard ramp -- becomes near
vertical as you go up the side, in one of those stunt tubes --
an official winter Olympics sport). Slope stuff connects us
directly to triangles, and the right triangle in specific (one
90 degree angle, 2 acute angles). TAN(angle) = opp/adj and,
inversely, ATAN(opp/adj) = angle. The utility of the TAN key
is revealed. I want to lean a ladder against a building to
paint the 2nd floor window, what will be its angle with the
ground?...
In sum, I think the goal should be to have curriculum threads
which feature exercises, sequence, skill-building, as per
usual, but with "Doppler radar" turned on, looking over the
horizon, around the bend, keeping a "bigger picture" in view at
all times, so that even as you're chugging up the grade,
factoring polynomials, playing with trig identities or
whatever, you have frequent breaks, including movie-
documentaries, which integrate, tie together, and "review"
a lot of concepts you haven't ever really exercised with yet
(but nevertheless appreciate, have an intelligent layperson's
grasp of, are reasonably literate regarding).
Like, it's OK to watch heavy weight lifting on TV even if you
never try this at home (really makes a lot of sense, in fact,
if you're scanning the horizon for future possibilities, trying
to give yourself a rationale for 'wasting' so much of your
youth messing with homework 'workouts' involving these cryptic
little symbols staring back at you from the calculator keypad).
Kirby
Curriculum writer
Oregon Curriculum Network
http://www.inetarena.com/~pdx4d/ocn/
[1] "Plodding" links to walking, marching, (ped, pedestrian,
pedantic) -- but whatever happened to flying? Why should
everyone be "a good foot soldier" marching in rank and file,
seeing only the trees in the immediate neighborhood, when
today we have satellites with which to view whole forests
(whole deserts) at a glance?
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