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Re: AP students

I think Joe's analysis here is bang on. And I don't see a great
disagreement with what I said. The lesson we should take from this is
that we ought to try to teach science the way it is done, which
implies that guided discovery and constructivist ideas should be our
guidelines. If we had the time. The problem is that we have to
condense up to 300 years of discovery learning into about a year or
two, or less, so we simply cannot do it all by discovery. Deductive
methods are one way to speed this process up. Of course working
scientists use deductive methods as well. Once a theoretical
explanation has been proposed, deductive methods are often used to
make the predications that will be used to test the theory.

But I have a few other caveats. New science is usually not well
understood. Sometimes it takes decades or longer before the new
concepts are properly interpreted. The lack of a fundamental theory
of superconductivity didn't keep condensed matter physicists from
learning a lot about the phenomenon in the fifty years between its
discovery and the explanatory theory. But the scientists working with
those concepts press on in spite of their lack of understanding. They
have developed methods of coping with the problems they deal with
that I have called algorithmic, although I presume Jack would want to
use a different term. It seems to me that one of the ways we learn at
any level is by what I have called algorithms. They let us make
progress while the ideas take root and mature. And when the insight
of understanding comes, we are ready to take advantage of it.

Of course, if we could just do something about the way physics is
taught (all in one year), and give the students the 6 or 7 years of
physics instruction in school, we could do a lot more discovery stuff
and have to rely a lot less on the pure deductive methods.

Hugh

Joe Bellino wrote:

I think there is a deeper issue here as well, having to do with the
nature of science and what we do as scientists/learners and what we do
as teachers of science. It seems to me that science as learned by
scientists is knowledge learned by induction, whereas science as taught is
frequently presented as deductive...look at most textbooks. I think of
algorithms as the result of deductive thinking...hence we see algorithms
in the texts. Induction on the other hand does not resort to algorithms,
but instead seeks to find generalities from experience...there are of
course algorithms involved...in physics, they are frequently mathematical
in nature, that is rules about manipulating the results of quantitative
experience, as opposed to rules about the experiences themselves.

If science as taught is seen as primarily knowledge by deduction from
principles, then the claim that algorithms can precede understanding makes
sense since the algorithms are the deduced logical statements, or the
deducted problem solving rules. You might make the argument, as some
have, that this approach might work well for some students, just as
balancing a checkbook does not require a degree or even a course in
accounting. But even if science is knowledge by deduction, that assumes
that the language of the teacher matches the language of the student, and
thus the deductive argument of the teacher will pass to the student
without misinterpretation. The evidence is clear, I think, that such
passage does not always happen.

So even if you assume science as taught by us is deductive, it becomes an
inductive process for our students just as it is an inductive process of
us as scientists. The parallel, I think, is that when scientists learn
science they deal inductively with the natural world, and when our
students learn science they learn inductively in the classroom experiences
we create. So the process is always inductive.

In this sense, algorithms are always suspect, and algorithm users are
always at risk.

joe


On Fri, 27 Apr 2001,
Hugh Haskell wrote:

Jack Uretsky wrote:

> I totally don't understand Hugh's logic. Yes, concepts do not
>come easily. How in the world does this fact justify teaching students
>a bunch of algorithms?

I didn't say to just teach algorithms. What I said was don't be too
disappointed if that's how they learn, in spite of our best efforts
to the contrary. It may not be all that bad, since understanding is
much more difficult, it will usually come later, if, indeed, it ever
comes. If we try to force understanding on students before we let
them learn to solve problems, the result could be that they learn
> neither.
>
> These comments, of course, do not apply to the Feynmans of the world,
> only to us lesser mortals.
>
> >Algorithms teach students to substitute numbers
> >into formulas - nothing more.

I disagree with your definition. Plug-and-chug is at the lowest skill
level. I would put algorithmic learning a step or two higher--where
they can follow a prescribed or learned route to a problem solution
that may involve some algebraic manipulation, several process steps,
and more than one stage of computation. The algorithm may be complex
and require lots of practice in order to get good at it, but in the
end, it enables the student to solve a particular class of problems
without actually understanding the process. I can think of several
advanced concepts that I first learned algorithmically and only later
came to understand what was behind them.

In fact, much of what we do every day is done algorithmically. We
have routines that we follow, methods that we invoke to solve
problems. When the methods or routines give the wrong results, we
modify them as necessary (a learning process). We don't stop to
completely analyze every situation we encounter from first
principles. We frequently try one or more learned
routines--algorithms--to see if they will work, and if they don't
then we start trying other things that may not appear to fit, or
start a more careful analysis. Some people are better at this than
others.

In many fields, flying, for instance, training for emergencies is
highly valued because it enables the practitioners to quickly choose
the correct algorithm to deal with a particular emergency. Granted,
physics isn't necessarily like flying (neither is driving, my point
there was simply to teach one skill at a time, rather than try to
have the student learn all of them at the same time), and at the
frontier of physics, we are forced to rely less on algorithms and
more on careful analysis (which itself often involves analytic
algorithms that are different for different areas of the subject),
but when first learning the subject, reliance on algorithms can
enable a student to make progress and develop confidence, even if the
deep understanding isn't there yet.

I suspect that our disagreement is over the use of the word
"algorithm." I doubt that you will deny that you follow certain
routines when attacking a problem, and then modify the routines as
necessary as you encounter obstacles. The deeper the understanding,
the easier it is to deal with the obstacles. This is what I have
always though of as algorithmic behavior, and, call it what you will,
it is, I believe, widely practiced by nearly everybody, and many
animals as well. Where we differ from the animals is that we can
often modify our algorithms as necessary, while the animals usually
cannot, their algorithms being pretty much hard-wired in.

Hugh
--

Hugh Haskell
<mailto://haskell@ncssm.edu>
<mailto://hhaskell@mindspring.com>

(919) 467-7610

Let's face it. People use a Mac because they want to, Windows because they
have to..
******************************************************


Joseph J. Bellina, Jr. 219-284-4662
Associate Professor of Physics
Saint Mary's College
Notre Dame, IN 46556

--

Hugh Haskell
<mailto://haskell@ncssm.edu>
<mailto://hhaskell@mindspring.com>

(919) 467-7610

Let's face it. People use a Mac because they want to, Windows because they
have to..
******************************************************