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Topic:  Calculus without calculus 
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Subject:  RE: Calculus without calculus 
Author:  LFS 
Date:  May 10 2011 
#1 I was very clear that I was using technology to do calculus without calculus
for specific purposes, e.g. find extreme values, find the area under a curve,
find the length of a curve, etc. in order to do real and interesting problems
from life.
#2 Any textbook that has the student use a graphics calculator to solve problems
by tracing out the extreme value (maximum) is doing this, i.e. using calculus
without knowing calculus and almost every textbook in the US does this. This
makes a great deal of interesting problems previously only accessible to
calculus students now available to every student. I am only extending this to
area under a curve and arc length (We do not have access to graphing calculators
where I live so we use freeware like geogebra.)
== So I would ask, are you using graphing calculators or software to graph
functions and find their extreme values? If yes, then you are doing calculus
without calculus. Even if you are just using technology to graph functions, you
are doing calculus without calculus. (In my day, we graphed functions by doing
all the calculus on them. There was no other way. Nowadays, we type the function
into a program and hit "graph". Calculus without calculus and thank
goodness!)
== My point was only that we should extend this process.
#3 Here is a wonderful arc length problem. How far does a projectile travel
(given h0, v0 and angle)? Currently textbooks only ask about its maximum height
and the horizontal distance traveled. Using Length[], you can find the actual
distance traveled and discuss things like average speed, etc. Why should we wait
for Calculus 2 and a very, very difficult solution that only a very, very few
will understand? This is fun! And useful and understandable.
#4 Proving Pythagoras' theorem is interesting. Knowing that its converse also
works is useful. However, knowing how to use the theorem is vital.
Proving the quadratic formula by completing the square is interesting.
Finding roots and the vertex of a quadratic function using completing the square
is a mathematical exercise  fun if you like to do that stuff. Factoring ditto.
However, knowing how to use the quadratic formula completely and understanding
that quadratics are symmetrical, understanding what roots are and not being
afraid of irrational roots and not trying to graph complex roots is vital.
#5 Your BTW is NOT a proof of the formula for the area of a circle in any way,
shape or form. It is an "illustration". It will only "become" a proof (and a
very difficult one to actually write down) if you let the number of pie pieces
> infinity. Hmm. Think that is called calculus.
My real point is that it seems to me that mathematics is taught the way we
mathematicians think is interesting (I hold a doctorate in theoretical
mathematics). This serves only to propagate mathematicians, i.e. it fulfills the
needs of the very, very tiny group of students who will become mathematicians
and leaves the rest bewildered (and angry).
However, in my opinion the real tragedy is that the math we teach is almost
completely inapplicable to the STE of STEM. For example, sin²x+cos²x=1
(a) Have a student actually use this formula with x=45° to see whether he will
get 1 (technology  does he understand that sin²x=(sin(x))^2?). My college kids
certainly do not.
(b) Have a student use the formula to find the parametric equations for a
circle(engineering/physics  the formula is "bad" because it uses x as the
argument and "backwards" for applying to parametric equations since x "goes
with" cosine), i.e. the parametric form of the unit circle is x=cos(t),
y=sin(t).
So we are teaching mathematics for those students who will become
mathematicians. Doesn't seem to be a really good plan.
We are trying to teach children that need to drive a car safely and properly how
the engine works. Yes, you should be able to fill up the windshield water fluid.
Yes, should should be able to check your tire pressure. But you don't need to
know how the motor works. And for sure, it is more important to know to check
your mirrors before changing lanes (i.e. check your units match before
solving).
Rigor is important. Learning how to look at a problem is important. Logical
thinking is important. Checking units is important. Checking that the answer
makes sense is important. Thinking about how to use technology to check your
solution is important.
Knowing how to complete the square is not.
 
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