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Re: [Phys-l] Should equation solving be done with calculators and robots or by hand?

Currently I see many students who can manipulate equations, but they can't
write equations. The latter skill is the one that can't be done by a robot,
while the solving can be done by a robot. Part of the problem with
conventional teaching is that students are often given a handy dandy
equation that solves the common problems. One can promote understanding by
not giving such equations, and by requiring students to us reasoning to
arrive at the solution. So they only have just a few simple equations, and
then they have to construct everything they need. At that point an equation
solver can be used.

While calculators take a lot of the drudge work out of math and science,
they can be overused. However, the evidence that calculators have made
students less adept at math is tenuous. When I was young, people in stores
would perform laborious hand calculations, with little understanding of what
they were doing. I remember the lady who did long multiplication to figure
the price of selling 10 23cent items. My mother told her immediately it was
$2.30 but the lady persisted, and in the end looked puzzled and asked my
mother how she knew the answer. It is now known that the ability to do
arithmetic is in a different part of the brain from the part that does
algebra, so good arithmetic ability is weakly coupled with algebra ability.

So math should be teaching equation creation just as much as solving. Also,
math needs to pay attention to some great visualization tools such as double
number lines. The other problem is that math is usually taught as a set of
algorithms. But there is strong evidence that teaching algorithms blocks
the ability to transfer. Here is the reference and abstract to a paper on
this subject.

Cognitive Development, 6, 449-468 (1991)
Learning and Transfer: Instructional
Conditions and Conceptual Change
Michelle Perry
University of Michigan
It is widely assumed that instruction plays a role in learning and in
transfer. The
present studies examine how type of instruction (containing principle-based
vs.
procedure-based information) influences learning and transfer in a
mathematical
concept. In the first study, both types of instruction led a comparable
number of
children to learn, but principle-based instruction led significantly more
children to
transfer their new knowledge. In the second study, the types of instruction
were
combined (i.e., children received both principle and procedure information).
The
results were virtually identical to the results obtained from the
procedure-only
instructions. This indicates that principle-based instruction may be crucial
for
transfer to occur and, when children also are exposed to procedures, few
will
transfer. It is hypothesized that children may ignore the conceptually rich
information
inherent in the principle when procedures are also provided.

John M. Clement
Houston, TX




I think my earlier posts may have given the impression that I want to go
too
far in reducing algebra manipulations in physics problem solving. My goal
is to make better use of robots to take care of 'chore' sorts of tasks
that
steal attention and time that I think could be better devoted to
expressing
physical reasoning in words and interpreting the *meaning* of answers.

Piles of algebra just aren't what I'm looking for in solutions to physics
problems. That seems to be what a lot of students crank out, though. I
want language instead.

Joel Rauber wrote in part:


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 like your matrix inverse example but I think the main benefit to doing a
matrix inverse by hand is just to get a feel for how much labor you save
by
having it done for you by a machine. I don't see how doing the elementary
row operations teaches anything very deep. (Maybe I'm missing something?)

How about the quadratic formula? Does using the quadratic formula by hand
in a projectile problem teach the physics better than using an equation
solver? I don't see any advantage to doing it by hand. Using an equation
solver saves time and reduces errors.

Would anyone argue that today's students should learn how to interpolate
values between entries in tables of trig functions like I was taught when
I
was a kid? Calculators have made that totally obsolete. Why shouldn't
more
powerful calculators make these other chores obsolete as well?



| 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.

I'd like to know more about this if possible. I can imagine other
reasons,
especially of the 'technoglitch' sort. One can burn up a lot of time
trying
to track down typing or keystroke errors when one or two students'
machines
don't seem to be doing what all the others are. Making everyone wait
while
trying to figure it out can be a big headache for all involved.

Learning all the ins and outs of a piece of technology takes quite a long
while. I've seen things go wrong that I've never been able to figure out.
_________________


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 thought this change had something to do with maintaining the correct
hierarchy of operations but maybe I'm wrong about that.

Proper use of parentheses is a huge problem for students, whether they use
robots or not. I think this needs a lot more attention in their algebra
courses.


_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l


_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l