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Re: [RUME] vocabulary
<x-html><!x-stuff-for-pete base="" src="" id="0" charset=""><html>
<font size=3>Hi,<br><br>
I know Ed is not going to agree with me, but that's fine -- Ed and I have
had a lot of fun in the past disagreeing (and occassionaly
agreeing:-)<br><br>
I think it might facilitate the discussion if we make a distinction
between "theory in a hard sense" as in physics, and
"theory in a soft sense" as in math education (and the human
sciences in general). Of course there may be many intermediate degrees of
softness between these two extremes, such as the theory of Darwinian
Evolution, on the details of which world experts still battle
ferociously.<br><br>
Also, when we try to capture what we mean by theory, do we mean this in a
descriptive sense, i.e. let's look around for the best examples we can
find in the current math ed research literature; or do we mean this in a
prescriptive sense, i.e. what a theory *should* look like, perhaps in 50
years' time, though we may not have proper examples at present.<br><br>
From a fair (though far from perfect) acquaintance with math ed
literature, appearing in major research journals such as Educational
Studies in Mathematics or the Journal of Research in Mathematics
Eucation, as well as from personal experience, I would claim that at
present we can only talk about theories in the soft sense. There may be
exceptions I am unaware of, but in general I believe the best research in
math ed falls far short from meeting Alan's criterias, especially those
three:</font><blockquote type=cite class=cite cite>
<dl><font face="Palatino" size=5>
<dd>o<x-tab> </x-tab>Predictive
power</font><font size=3><br><br>
</font><font face="Palatino" size=5>
<dd>o<x-tab> </x-tab>Falsifiability</font><font size=3><br><br>
</font><font face="Palatino" size=5>
<dd>o<x-tab> </x-tab>Replicability</font></blockquote>
</dl><font size=3><br><br>
<br>
Physicists have such a high level of agreement about the use of the
fundamental terms of their theories, that they can actually formulate
their theories in mathematical terms and prove mathematical theorems
about them -- thereby lies much of the predictive power.<br><br>
"Soft" theories are still useful, especially as heuristic tools
for making sense of a tremendously rich and complex world and huge amount
of data; but we'd better be modest (i.e. realistic) about the kind of
work we expect them to do for us.<br><br>
Uri<br><br>
<br>
At 15:11 11/07/2001 -0700, Alan Schoenfeld wrote:<br>
<blockquote type=cite class=cite cite>Hi,<br><br>
(I can't help myself, though I really should be doing other
things...)<br><br>
At 4:59 PM -0400 7/11/01, John Donovan wrote:<br>
...<br>
<blockquote type=cite class=cite cite>With regard to your comments about
falsifiability, I do not think the<br>
theories I mentioned, APOS and the theory of reification, are<br>
falsifiable. Yet they seem to satisfy the criteria that Alan<br>
suggested - "I claim that certain objects are related in certain
ways."<br>
I would like to hear the comments of others on
this.</blockquote><br><br>
I think falsifiability is essential. It's one of a series of
criteria I put forth in the _Notices_ article I mentioned earlier:
</font>
<dl><font face="Palatino" size=5>
<dd>A major question, then, is the following: how much faith should one
have in any particular result? What constitutes solid reason, what
constitutes "proof beyond a reasonable
doubt?"</font><font size=3><br><br>
</font><font face="Palatino" size=5>
<dd>The following list puts forth a set of criteria that can be used for
evaluating models and theories (and more generally, any empirical or
theoretical work) in mathematics education:</font><font size=3><br><br>
</font><font face="Palatino" size=5>
<dd>o<x-tab> </x-tab>Descriptive
power</font><font size=3><br><br>
</font><font face="Palatino" size=5>
<dd>o<x-tab> </x-tab>Explanatory
power</font><font size=3><br><br>
</font><font face="Palatino" size=5>
<dd>o<x-tab> </x-tab>Scope</font><font size=3><br><br>
</font><font face="Palatino" size=5>
<dd>o<x-tab> </x-tab>Predictive
power</font><font size=3><br><br>
</font><font face="Palatino" size=5>
<dd>o<x-tab> </x-tab>Rigor and
specificity</font><font size=3><br><br>
</font><font face="Palatino" size=5>
<dd>o<x-tab> </x-tab>Falsifiability</font><font size=3></font><font face="Palatino" size=5>
<dd>o<x-tab> </x-tab>Replicability</font><font size=3><br><br>
</font><font face="Palatino" size=5>
<dd>o<x-tab> </x-tab>Multiple
sources of evidence
("triangulation")</font><font size=3><br><br>
<br><br>
</dl>I'd argue that if APOS theory (or the theory of reification) is to
be considered a theory, then it must be falsifiable. There's a
strong interaction with "rigor and specifiability" here: If the
theory says<br><br>
<x-tab> </x-tab>"processes
become objects by encapsulation"<br><br>
then trained researchers should be able to examine (say) a video
interview in which one researcher claims that certain processes became
objects. They should be able to identify the same processes, the
same objects, and the means by which the transformation occurred - and
there should be good agreement. If the mechanism is supposed to be
universal - e.g., "with roughly so much repetition or practice, then
processes become objects by encapsulation", then you have a
falsifiable statement. <br><br>
Note that it's perfectly reasonable to insist on falsifiable claims about
mental objects. The simplest example deals with things like
short-term memory, and the claim (cf. George Miller's famous paper
"the magic number 7+/-2") that we can't keep more than 9
"chunks" in STM. Simple tests like "try to multiply
379 by 658, with your eyes closed" provide the test of
falsifiability.<br><br>
Cheers,<br>
Alan<br>
-- <br>
##################################################<br>
Alan H. Schoenfeld<br>
Elizabeth and Edward Conner Professor of Education<br>
Education, EMST, Tolman Hall # 1670<br>
University of California<br>
Berkeley, CA 94720-1670<br><br>
Phone: 510-642-0968<br>
Fax: 510-642-3769<br>
email: alans@socrates.berkeley.edu<br><br>
Home page (papers, etc.):
<a href="http://www-gse.berkeley.edu/Faculty/aschoenfeld"; eudora="autourl">http://www-gse.berkeley.edu/Faculty/aschoenfeld</a><br><br>
UCB page:
<a href="http://www-gse.berkeley.edu/Faculty/gsefaculty.ss.html#schoenfeld"; eudora="autourl">http://www-gse.berkeley.edu/Faculty/gsefaculty.ss.html#schoenfeld</a><br><br>
MARS website:
<a href="http://www.educ.msu.edu/mars"; eudora="autourl">http://www.educ.msu.edu/mars</a></font></blockquote></html>
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