________________________
Joel Rauber
Department of Physics - SDSU
Joel.Rauber@sdstate.edu
605-688-4293
|
| I'm not entirely convinced of the relative value of the
| "mental exercise" in doing algebraic manipulations. Since
| they can be done by actual robots it seems that a human doing
| them is also acting like a robot, brainlessly performing
| trained manipulations without thinking.
|
The fact that many students have difficulty with this means that it
isn't a "brainless" activity for them.
Problems can be structured so that it isn't brainless. As a good
example, one can make the angle of launch be the unknown in a garden
variety projectile problem. In order to solve this, one has to be
comfortable with algebra as it typically requires rewriting your
equation as a quadratic in COS(theta). This is well within the
capabilities of an Algebra II student who knows how to solve a quadratic
and has had trigonometry to the extent of knowing that COS^2 + SIN^2 =
1.
But if assigned this problem many if not most physics students give-up.
If they just plug the equation in Mathematica they will get an answer,
but will never have confronted their lack of understanding of algebra.
IMO, courses shouldn't be so compartmentalized so that you do your math
course and forget it when doing the physics course and vice versa. The
courses and the skills developed in them should reinforce each other.
|
| But when the robot human makes algebra mistakes and gets the
| wrong answer it gets upset and gives up. There are more ways
| to foul up algebra than physics, I think. (Then again, maybe
| two infinities are equal...)
|
I think this argues for making the human robot do some of this. One
important skill in developing problem solving skills is to not give up
when first frustrated.
| I'm quite glad my 1st grade daughter isn't being
| > taught arithmetic on a calculator for example. I don't
| think it would
| > be a disaster if students weren't allowed to touch
| calculators until
| > high school; and even then only under adult supervision. :-)
|
| Oh, it'd be a disaster, I think.
There is an existence proof that it wouldn't be a disaster. All of K-12
education prior to the original HP-35's provide the existence proof.
|I'm sure we both want
| *meaning* to be conveyed to students.
Absolutely!!
|I'm not sure manual
| labor does this just by virtue of being 'manual.'
|
Clearly, despite what I've said, calculators and computers are important
tools and students have to learn how to use them somewhere along the
line so somewhere along the line they must be introduced into the
curriculum. So, IMO, the question is more when and in what amounts does
one start to use the mechano/electro "robots". I have no definitive
answer, we probably disagree as to the when and as to the what amounts.
(But you asked in your original post for some contrary opinion.)
As an example, I think it is worthwhile for students when they first
learn about matrix inverses to have to compute a bunch of inverses by
hand. Certainly 2x2's and 3x3's if not a few 4x4's, and maybe try to
find the inverse of some non-invertible matrices.
If all you have ever done is hit the matrix inverse button , you will
miss some real understanding regarding linear algebra and matrix
mathematics and understanding regarding elementary row and column
operations.
| I've noticed the math
| > department at our university is beginning to be less enamored with
| > graphing calculators and using them for teaching the algebra and
| > calculus classes. I think this is happening for a reason.
|
| What is this reason?
|
I can only speculate, but I think it is because they felt that less
understanding of the material was happening.
_________________
On a related note, I don't like the way that a lot of newer calculators
work, for example, some of the newer "algebraic" calculators take trig
functions by the sequence hit sin -> open paren-> number-> close paren.
presumably in order to make the entry look like how you see it in the
textbook.
For older calculators the sequence is enter number -> hit the sin
button. This is preferable, IMO, as it reinforces the idea that sin is
a function (or mapping) or an operator that acts on an input to give an
output. (Not to mention that it involves less key strokes).
I don't go so far as to prefer RPN logic calculators, which would be a
logical extension of this comment. So I suppose I like the "operator"
method for unary operators but not binary operators.
:-)