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I find this to be a very interesting, very important discussion.
Robert Cohen <Robert.Cohen@po-box.esu.edu> wrote in part:
In my own physics book, I no longer include any of the "magical"
kinematic equations (like x = x_0 + v_0 t + 1/2 a t^2) as I find
they just allow students to bypass understanding. Without them,
students must solve problems based on scientific laws (like Newton's
laws) and definitions (like average velocity). At least that is my
intention. Has anyone else tried this?
I've seen it, especially at the better colleges and universities.
At the high-school level there seems to be a stampede in the opposite
direction, which is verrrry unfortunate.
On 05/16/2012 11:06 AM, Joe Wise wrote:
I have and have found it very exciting.
The equations generally taught are for specific contitions that the
students never see. They constantly used to plug instantaneous speeds
into equations requiring average speed etc.
I make students create a graph sketch of the problem and then derive
the equation that represents that graph.
They complain a lot, but understand much more. They also start
looking for salient points vs superficial elements.
IMHO this is a huge step in the right direction.
IMHO the next step is to reduce the complaining while still
promoting thoughtful understanding. In other words, it is a
marketing issue. In teaching as in toymaking or anything else,
it is important to have a technically-sound product *and* good
marketing.
Here is how I see the marketing issue. Offer the students an
explicit choice:
A) Memorize lots of factoids. Because each factoid is narrowly
specific, you rarely get an opportunity to use it. That leads
to a vicious circle: Factoids that aren't often used can't be
remembered ... and factoids that aren't remembered can't be used,
even when the opportunity arises. By the end of June, you will
have forgotten everything covered in this class, so the whole
enterprise is a huge waste of time and resources.
B) Memorize very few ideas. Because the ideas are simple,
elegant, and powerful, you will use them again and again,
throughout the course and throughout the rest of your life.
A) The alleged advantage of the factoid-based approach is that
*if* you can recognize a situation you have seen before and
*if* you can remember the right formula to use, you can solve
the problem in one step.
The disadvantage is that in real life, there are millions of
different situations, and even seemingly-similar situations
have millions of different variations, so you cannot possibly
learn all of them by rote.
B) The disadvantage of the principle-based approach is
that solving the problem generally requires multiple steps,
combining the principles and working out the consequences.
The obvious advantage is that this allows you to handle a
huge variety of different situations, including everything
from mildly-unusual situations to unprecedented emergencies.
The advantage is bigger than it looks, because this /style/ of
multi-step thinking is not limited to physics. It can be used
in all branches of science and engineering, and also in game
strategy, military strategy, business strategy, et cetera.
So, dear students, which would you prefer? Shall we learn a large
number of useless things, or a small number of useful things?
In alg based physics we are more often than not, solving for a slope,
axis, or area under the curve.
That's the high-water mark of the math part. As others have pointed
out, before we can fill the tank to that level, we have to plug the
leaks at lower levels, including not being able to do long division,
or even short division.
Obviously there is a physics part also.
On 05/15/2012 07:23 PM, Richard Tarara wrote in part:
first semester ...
Are these questions asked "cold" i.e. early in the first semester,
... or much later, after the students have had extensive training
in answering questions of this type ... or something in between?
Also: If the answer is √ ½ is it acceptable to write it as √ ½ ...
or is a decimal numerical answer expected?
[interesting quiz-questions snipped]
Students usually (but not universally) get #32 right. A reasonably
large percentage even get #36 right. Less than 25% get 33, 34, and
35 correct (over the past 10 years) and only one or two students a
year will get all 5 correct.
Let me offer a hypothesis seems to fit the *very limited* data:
It may be that they can handle non-negative integer powers, but
have trouble with negative powers and/or half-integer powers
i.e. roots.
Such a situation could easily arise if they understand powers at
the level of iterated multiplication but not at the more general
abstract level.
To say almost the same thing another way: integers are easier than
fractions, and fractions are easier than irrational numbers.
This hypothesis should be super-easy to check. Surely this is
not the whole story, but it might be part of the story.
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