Before reading this morning's TIPS Digest I had already prepared in my
mind the opening of my reply to an anticipated response from Stephen along
the following lines:
Mea (partly) culpa. I misread some of Stephen's remarks. I spend most of
Sundays away from home, and I dashed it off my message in the heat of the
moment after a quick glance through Stephens posting. I've now looked
more carefully at what Stephen wrote, and see that I extrapolated some of
it beyond the situation he was citing.
So I start by acknowledging some justice in Stephen's opening words:
>Yet if people are going to dump, it's only fair that they do so on the
basis of what I said, not on what they would have liked me to have said,
so that, in truthiness, they can indignantly refute it.<
Nevertheless, I dispute Stephen's next comments:
> And in this evidence-free environment, one person's assertions all too
> readily can be labeled "fallacies" by another.<
I believe (though I acknowledge that it is not clear-cut) that one can
distinguish between matters of opinion and fallacies. Here are what I had
in mind when I referred to fallacies in Stephen's previous posting:
1. Algorithms are "*just* a set of instructions to be followed by rote"
(my emphasis). I previously explained why this is a fallacy. (See also
below.)
2. Calculators are "a fast and *reliable* way of getting the right
answers" (my emphasis). Again I previously explained why this is not the
case.
3. Being against the teaching the use of calculators from the very
beginning of mathematics *in place of* "having to teach obsolete
algorithms" is equivalent to being in favour of "counting on our fingers"
in preference to "making marks on papyrus". Well, how one describes that
view is more debatable, but I'd call that kind of argument, which glosses
over the actual points at issue, a fallacy. But if Stephen prefers to call
it a difference of opinion, we'll agree to differ on that one.
As I said before, in my previous response I extrapolated from what Stephen
wrote, and partially responded to the extrapolation. This time I'll stick
more precisely to what he does say:
>What I argued is that the particular algorithms we were taught for long
multiplication and division have had their day. As we regretfully retired
the vinyl record for the DVD, and the slide rule for the calculator, so
must we retire these two ancient algorithms. I still play my vinyl records
on occasion but the generation now being born will find them as quaint and
useless as we find the gramophone. Tellingly, no one answered my question
concerning how many times they carry out long division and multiplication
by hand. But I know the answer: rarely or never.<
I think we can do without the analogies, which (as before) obscure rather
than illuminate the issue.
>What I argued is that the particular algorithms we were taught for long
multiplication and division have had their day.<
In his first posting Stephen wrote in favour of the "cluster" method of
multiplication that "it may help students to understand how numbers work".
So we're all agreed that students should understand how numbers work. Im
certainly not against (and nor was M. J. McDermott) the use of the cluster
method as part of this process of understanding. Now what is an algorithm?
It is nothing more than a systematic procedure for solving a standard
problem. In other words, once some degree of understanding is achieved, an
algorithm provides a set procedure that enables the method to be applied
in the most concise way. This means that the student then no longer has to
keep returning to first principles but can use the algorithm as a tool
when dealing with the next, and later, stages in learning maths. And the
standard algorithms for multiplication and division do this job better
than the cluster method, because they are more concise and more quickly
executed.
That doesnt mean that students should spend a lot of time applying the
multiplication and division algorithms to larger and larger numbers. What
is important is the *procedure*, in the course of which some understanding
of what is involved in multiplying and dividing numbers should be
developed. Beyond that I agree with Stephen: for relatively large numbers
calculators are there for use, and there is no *extra* value in getting
kids to do more complicated multiplication and division questions without
a calculator.
>Tellingly, no one answered my question concerning how many times they
carry out long division and multiplication by hand. But I know the answer:
rarely or never.<
If Stephen checks back hell find I *did* respond to his question, as
follows:
"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*."
Stephen writes:
>It might be different if these algorithms had some value in illuminating
the structure of our number system. But they don't. We can teach students
to see how these methods depend on this organization, but that's not the
same thing. If the reason for teaching a particular algorithm is that it
helps understand how numbers work, I'm confident that there are better
methods to do this than these two. Their main benefit was efficiency, not
illumination, and that efficiency has had its day and been replaced by
more efficient technology. Similarly, if understanding is our aim, there
are undoubtedly better ways to make students understand about remainders
(Jim Clark's concern) than by teaching them the algorithm for long
division.<
> It might be different if these algorithms had some value in illuminating the
> structure of our number system. But they don't... If the reason for teaching
> a particular algorithm is that it helps understand how numbers work, I'm
> confident that there are better methods to do this than these two.<
I think this illustrates why I said that Stephen's view of algorithms is
fallacious. Of course the algorithm as such does not illuminate the
structure of our number system, that's not its purpose and nor is its
purpose to help "understand how numbers work". Any half-way decent
teaching will provide that illumination while leading up to the algorithm
"illuminating the structure of our number system" should come in the
process of *deriving* the algorithm. The algorithm has the role of
*systematizing the procedure* for solving the problem so that students
have a set procedure they can rely on without having *thereafter* to go
back to first principles every time. If we're going to argue purely on the
issue of multiplication and division, I'd say (and Stephen evidently
agrees) that students need to "understand how numbers work". And as part
of the process of understanding how *mathematics* works, having given them
this understanding in relation to multiplication and division the
procedure should be systematized this is the very essence of
mathematical thinking.
Stephen writes in defence of his view of algorithm as procedures for
learning by rote:
>Well, the Random House Dictionary defines an algorithm as "a set of rules
for solving a problem in a finite number of steps", which is pretty close
to my definition.<
Stephen previously wrote that algorithms "are just a set of instructions
to be followed by rote", with the clear implication that understanding
doesn't come into it. But algorithms are (or should be) the *end result*
of a *process of understanding*, at which point the required procedure is
set out concisely. If Stephen didn't mean to imply that using algorithms
directly indicates a method of *learning by rote*, why did he write that
they are "just" a set of instructions to be *followed by rote*?
>I don't see the point of the distinction that Allen makes between
"learning by rote" and "picked up by practice". The end result is the
same.<
"The end result is the same" only in a limited sense. As I think my
(rather lengthy) previous posting made clear, I indicated three stages.
First the teacher endeavours to get the students to understand the
underlying principles, and uses them to suggest ways of solving the
problem in hand. Then the procedure is systematized, so that students have
a set procedure that they follow. With some practice the procedure becomes
automatic, in other words, it has been "picked up by practice". If Stephen
thinks this is the same as "learning by rote", then again well have to
agree to differ.
>And for one more iteration, however you get there, the end result is that
the student now has a set of rules which are automatically applied to
produce a result.<
Of course they do. Thats how mathematics works. Once one has understood
the process, one needs a simple method of application (i.e., an algorithm)
so that one doesnt have to keep going back to first principles. At a
higher stage it is absolutely essential that the lower levels of knowledge
are automatically applied if students are going to get to grips with the
matter in hand *at that stage*.
>That hard-won illumination imparted by the teacher in the process is
likely to fade pretty fast. How many of us can now explain why they work?
Does this impair our ability to use mathematics?<
Ideally, that illumination should remain part of the understanding which
is foundational to what follows. Now I concede that is an ideal. But part
of the process of building confidence in mathematics is that students have
some understanding of earlier stages, and get to know a solid base of
mathematical procedures (algorithms) for use in later stages. At those
later stages it doesn't particularly matter if the student can't easily
recall the justifications for the earlier results. What matters is that
the student has a sense of building on foundations they understood at the
time, so they dont feel the need to revisit the basic processes on which
the foundations rest. In that way they can move on from stage to stage,
for which process it is necessary that they can *take as given* the
earlier work they have (hopefully!) mastered.
> How many of us can now explain why they work? Does this impair our ability to
> use mathematics?<
Well, it depends on the precise circumstances. My experience of teaching
students coming into UK Further Education colleges straight from school is
that because the anti-traditional ethos had taken over the Colleges of
Education from the 1970s, teachers no longer thought that methodological
procedures were necessary as long as the students came out with the
right result, what did it matter how they presented, or justified, their
answers? The result was that students had little idea how to present a
mathematical argument, and most of them had an insecure grasp of basic
principles. Because they lacked this understanding, it most certainly
*did* impair their ability to use mathematics. Which is why lecturers at
even some "elite" universities in the UK regularly complain that students
starting physics, engineering and mathematics courses are insecure in
their knowledge of, and application of, quite basic algebraic,
trigonometrical and calculus procedures.
>If the reason for teaching a particular algorithm is that it helps
understand how numbers work, I'm confident that there are better methods
to do this than these two. Their main benefit was efficiency, not
illumination, and that efficiency has had its day and been replaced by
more efficient technology.<
Once again, an algorithm is *not* intended to "help understanding how
numbers work", it is the *end result* of a process of understanding.
>Similarly, if understanding is our aim, there are undoubtedly better ways
to make students understand about remainders (Jim Clark's concern) than by
teaching them the algorithm for long division.<
If taught well, the processes leading up to the division algorithm provide
a good understanding of remainders. Seen in these terms, the application
of the algorithm *assists* in the understanding of remainders (because it
enables them to be obtained by a mathematically logical procedure).
>Using calculators is a skill which students should have. I see no reason
why they can't be taught to perform this skill with high reliability.
Second, I don't see how knowing the old algorithms for long division and
multiplication will help them recognize when they've gone wrong. Good
estimating skills would, but that's something different.<
I wasn't suggesting that knowing the "old algorithms for long division and
multiplication" in itself will help students recognize when theyve gone
wrong. What I am saying is that unless students learn mathematics from the
beginning in a *systemised way* they tend to have only a hazy grasp of
basic principles and that hinders their ability to discern that results
they get by pushing buttons on a calculator may be manifestly in error.
But we probably don't differ on this point.
>No, I think the problem is that the old generation has trouble letting
go. I can see the Egyptian scribes in the shadow of the just-erected
pyramids telling their young students, "before you can make scratch marks
on papyrus (or on clay tablets), you first have to learn how to do it with
stones. How else are you going to be able to understand counting? "<
That make good rhetoric, but hardly advances the discussion of the
specific issues. Extrapolating beyond what Stephen has been writing about
here, but taking this sentiment as a generality, the result of this kind
of thinking in the UK ("away with the fuddy-duddy old methods" rather than
"how can we improve on the old methods?") has been that large numbers of
students right up to pre-University level have little idea how to think
mathematically, and their mathematical knowledge-base is weak and
insecure.
Allen Esterson
Former lecturer, Science Department
Southwark College, London
http://www.esterson.org/
------------------------------------------------------------------
Date: Sun, 28 Jan 2007 22:25:51 -0500
Author: [EMAIL PROTECTED]
Subject: Why I pull my hair out grading statistics assignments
> On 28 Jan 2007 at 5:42, Allen Esterson wrote:
>
> > There are so many fallacies in "contrarian" Stephen Black's response on
> > TERC that I scarcely know where to begin. (Sorry Stephen!):
>
> One of the consequences of taking a contrarian position is that people
> are going to dump on you for taking it. I would have been disappointed
> if no one did. Fortunately, Allen and some others have spared me that
> ignominious fate. Yet if people are going to dump, it's only fair that
> they do so on the basis of what I said, not on what they would have liked
> me to have said, so that, in truthiness, they can indignantly refute it.
> And in this evidence-free environment, one person's assertions all too
> readily can be labeled "fallacies" by another. Allen's fallacies-- er, I
> mean assertions--can be similarly challenged.
>
> I never said that I'm opposed to all rote learning in mathematics
> education. Basic facility with, for example, the multiplication and
> division tables is essential. What I argued is that the particular
> algorithms we were taught for long multiplication and division have had
> their day. As we regretfully retired the vinyl record for the DVD, and
> the slide rule for the calculator, so must we retire these two ancient
> algorithms. I still play my vinyl records on occasion but the generation
> now being born will find them as quain and useless as we find the
> gramaphone. Tellingly, no one answered my question concerning how many
> times they carry out long division and multiplication by hand. But I know
> the answer: rarely or never.
>
> It might be different if these algorithms had some value in illuminating
> the structure of our number system. But they don't. We can teach students
> to see how these methods depend on this organization, but that's not the
> same thing. If the reason for teaching a particular algorithm is that it
> helps understand how numbers work, I'm confident that there are better
> methods to do this than these two. Their main benefit was efficiency, not
> illumination, and that efficiency has had its day and been replaced by
> more efficient technology. Similarly, if understanding is our aim, there
> are undoubtedly better ways to make students understand about remainders
> (Jim Clark's concern) than by teaching them the algorithm for long
> division.
>
> > 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.
>
> Well, the Random House Dictionary defines an algorithm as "a set of rules
> for solving a problem in a finite number of steps", which is pretty close
> to my definition. I don't see the point of the distinction that Allen
> makes between "learning by rote" and "picked up by practice". The end
> result is the same. The student follows a series of steps to an answer.
> The student can be made to understand why it works, but doesn't have to
> in order to get the answer. As I said above, if the reason for teaching
> the algorithm is because it helps the student to understand, there are
> better methods available to achieve enlightenment.
>
> > It is simply not possible to acquire a reasonable mastery of algebra
> > unless there is some facility in using numbers *without calculators*.
>
> Absolutely. But that doesn't preclude abandoning an obsolete set of
> instructions which are followed automatically to an answer.
>
> > To reiterate: There is absolutely no reason why using an algorithm
> > should be "learning by rote".
>
> And for one more iteration, however you get there, the end result is that
> the student now has a set of rules which are automatically applied to
> produce a result. That hard-won illumination imparted by the teacher in
> the process is likely to fade pretty fast. How many of us can now
> explain why they work? Does this impair our ability to use mathematics?
>
> > First, using calculators is not a "reliable" way of "getting the right
> > answers". I can=B4t tell you how many times I had students getting wrong
> > answers (occasionally absurd answers) because they have done something
> > wrong, and they don=B4t have sufficient *understanding* of the problem
> > they are doing to realise they have got it wrong.
>
> Using calculators is a skill which students should have. I see no reason
> why they can't be taught to perform this skill with high reliability.
> Second, I don't see how knowing the old algorithms for long division and
> multiplication will help them recognize when they've gone wrong. Good
> estimating skills would, but that's something different.
>
> No, I think the problem is that the old generation has trouble letting
> go. I can see the Egyptian scribes in the shadow of the just-erected
> pyramids telling their young students, "before you can make scratch marks
> on papyrus (or on clay tablets), you first have to learn how to do it
> with stones. How else are you going to be able to understand counting?"
>
> Stephen
> -----------------------------------------------------------------
> Stephen L. Black, Ph.D.
> Department of Psychology
> Bishop's University e-mail: [EMAIL PROTECTED]
> 2600 College St.
> Sherbrooke QC J1M 0C8
> Canada
---
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