There are so many fallacies in "contrarian" Stephen Blacks response on
TERC that I scarcely know where to begin. (Sorry Stephen!):
> Back when we were in school, tiny and cheap calculators were science
fiction. We _needed_ those algorithms. We don't any longer, now that we
have those really neat calculators. Tell the truth, how long has it been
since you actually carried out a long division with pencil and paper?<
And:
> It might be useful to continue to teach the algorithms if they gave
insight into the nature of mathematics, but they don't. They're just a
set of instructions to be followed by rote.<
What Stephen writes is quite possibly justified as a criticism much of the
*teaching* of "traditional" methods, but not of the methods themselves. It
is nonsense to say that the "algorithm" approach is "just a set of
instructions to be followed by rote." Any half-way decent teacher will go
through the procedure with simple examples, showing at each stage the
mathematical sense of the procedure, then moving on to more difficult
examples, until the process becomes automatic. Ideally there doesn't need
to be *any* learning by rote, the algorithm is picked up by practice. And
for those who dont fully grasp the rationale behind the procedures, the
algorithm is there for use so that the next stage can be tackled.
Now maybe what Ive described doesnt happen as often as it should. But
thats the fault of the teaching, not the method.
> Tell the truth, how long has it been since you actually carried out a long
> division with pencil and paper?<
What relevance has that question to the teaching of mathematics? Id say
none. We should be teaching kids to understand what they are doing (and,
yes, I repeat, if taught properly the algorithm approach should start with
*understanding* the procedures). Pushing buttons on a calculator certainly
doesnt do that. Equally important from the point of view of teaching
mathematics is that more advanced topics require at least a reasonable
understanding of elementary topics, and their more or less automatic use.
It is simply not possible to acquire a reasonable mastery of algebra
unless there is some facility in using numbers *without calculators*.
To reiterate: There is absolutely no reason why using an algorithm should
be "learning by rote". I taught elementary mathematics in Further
Education Colleges in the UK (roughly equivalent to Community Colleges). I
*never* presented students with an algorithm to learn by rote. The
algorithm is the last stage of the introductory process, which should
start with getting the students to understand what they are doing. Sorry
to keep going on about this, but I get so tired of people banging on about
so-called "traditional" methods being "learning by rote" when they are
talking about bad teaching, not bad methodology.
Stephen writes:
>The cluster method is better [than the lattice method], because it may
help students understand how numbers work.<
I repeat, it is a fallacy to say (by implication) that the algorithm
method does not teach how numbers work. If properly taught, it does. Now
from the point of view of facility with numbers (including how they work)
what Chris and Tim say they often do (something akin to the cluster
method) of course makes sense (and occasionally I do it when totting up a
bill or whatever). But there is no substitute for a standard procedure
which students get to know automatically and on which they can build for
the next stage in mathematics. Being able to juggle numbers around in
their heads is fine (and I suspect only a minority accomplishment), but it
is no basis on which to build the hierarchical systemisation necessary for
a reasonable mastery of mathematics.
>If I ran the schools, I'd first ensure that students had really good
facility with their calculators, so they have a fast and reliable way of
getting the right answers. Then, with the time saved from not having to
teach obsolete algorithms, I'd spend more time on teaching the
underlying concepts of math and on more advanced topics.<
First, using calculators is not a "reliable" way of "getting the right
answers". I cant tell you how many times I had students getting wrong
answers (occasionally absurd answers) because they have done something
wrong, and they dont have sufficient *understanding* of the problem they
are doing to realise they have got it wrong. Second, no one should be
"teaching obsolete algorithms", they should be teaching how to do the
procedures on the basis of understanding, out of which the algorithms
naturally arise. A decent teacher will be using the "underlying concepts
of math" in the process of teaching the procedure, they are *not* separate
things. As for then moving on to "more advanced topics", Stephen has
obviously never been in the situation where he is trying to teach such
topics when many of the students dont have a solid grasp of earlier
topics, and are unable to apply the necessary procedures automatically.
And, yes, the automatic procedures involve algorithms. And, incidentally,
the use of algorithms does not preclude the occasional reminder of the
rationale behind the algorithm. When teaching procedures involving
algebraic manipulation of fractions I expected students to know the
algorithm for adding fractions. But I still occasionally emphasized the
mathematical rationale behind the algorithm in the course of going through
examples.
The fallacy behind Stephens remarks quoted above is that unless there is
some automation of the procedures in elementary mathematics
("algorithms"), students are not in a position to build systematically on
their knowledge to enable them to tackle more advanced mathematics.
>Sometimes we cling to old technology longer than we should because it's
familiar and it served us so well.<
No one is suggesting we return to, e.g., the use of log tables (though as
preparation for University mathematics more advanced high school students
should certainly be introduced to the basic concept of logarithms), only
that calculators should be used discriminately. Clearly the ubiquitous use
of calculators at an early stage (in place of pen and paper calculations)
does not facilitate understanding of the mathematical procedures.
Moreover, it leads to students lacking the facility with numbers that is
necessary for some elementary algebra (e.g., solving quadratic equations).
This is not, as Stephens tone suggests, a question of being progressive
or retrogressive, but of trying to use the best methods for teaching
mathematics without bothering about whether or not one is being "advanced"
in ones thinking.
A qualification to the above: When I tutored in LA in 1977-1979 I taught
the occasional pre-University school student and was surprised to find
what I regarded as a certain rigidity in the approach to teaching
mathematics. Students seemed to be expected to go through a pre-ordained
procedure, e.g., in geometry, specific "steps" were expected in a proof,
whereas I was happy for any procedure as long as it was mathematically
logical and each step was briefly justified and methodically presented.
But that was over 25 years ago, and I have no idea how typical this was at
the time.
Allen Esterson
Former lecturer, Science Department
Southwark College, London
http://www.esterson.org/
------------------------------------------------
Date: Sun, 28 Jan 2007 02:11:00 -0500
Author: [EMAIL PROTECTED]
Subject: Re: Why I pull my hair out grading statistics assignments
> Michael Scoles said:
>
> http://www.youtube.com/watch?v=Tr1qee-bTZI
>
> And the responses have been predictable. Time for a contrary opinion.
>
> True, what Ms McDermott complains about does have some merit. The math
> textbooks she describes, particularly the one which spends its time on
> geography (!), seem rather uninspired and irrelevant in content.
>
> But is she right to claim that it's a crime against education to fail to
> teach the standard algorithms for multiplication and division we all
> learned and loved when we were in school? And is Ms McDermott right that
> not teaching them leads to calamity in calculus and are TIPSters right
> that it leads to disaster in statistics? Maybe not.
>
> Back when we were in school, tiny and cheap calculators were science
> fiction. We _needed_ those algorithms. We don't any longer, now that we
> have those really neat calculators. Tell the truth, how long has it been
> since you actually carried out a long division with pencil and paper?
>
> It might be useful to continue to teach the algorithms if they gave
> insight into the nature of mathematics, but they don't. They're just a
> set of instructions to be followed by rote. So it seems unlikely that
> the cause of student difficulties in college math can be blamed on not
> knowing how to do traditional long multiplication and division.
>
> The authors of these textbooks seem to recognize this, although why they
> think the lattice method is an improvement is beyond me. The cluster
> method is better, because it may help students understand how numbers
> work. And as Chris says, it's useful when you have to do it in your head.
> So not everything in those books is outrageously misguided. But they
> don't go far enough.
>
> If I ran the schools, I'd first ensure that students had really good
> facility with their calculators, so they have a fast and reliable way of
> getting the right answers. Then, with the time saved from not having to
> teach obsolete algorithms, I'd spend more time on teaching the
> underlying concepts of math and on more advanced topics.
>
> Sometimes we cling to old technology longer than we should because it's
> familiar and it served us so well. I'd guess there was the same debate
> when we first decided to give up counting on our fingers in favour of
> making marks on papyrus. But I think it turned out to be a pretty good
> decision in the end.
>
> Stephen
>
> -----------------------------------------------------------------
> Stephen L. Black, Ph.D.
> Department of Psychology
> Bishop's University e-mail: [EMAIL PROTECTED]
> 2600 College St.
> Sherbrooke QC J1M 0C8
> Canada
>
> Dept web page at http://www.ubishops.ca/ccc/div/soc/psy
> TIPS discussion list for psychology teachers at
> http://faculty.frostburg.edu/psyc/southerly/tips/index.htm
> -----------------------------------------------------------------------
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