Karl L. Wuensch <[EMAIL PROTECTED]> wrote:
> I think that Bob Hayden is on to something essential here ("I noted that
> Karl presented all the understandings he sought verbally on the list. Why
> not do the same in class?"). I think of the "definitional formulae" just as
> a convenient shorthand for the verbal definition of a construct. But it may
> be the case that most of our students assume that something that looks like
> a formula is just for use with mindless computations. They may have learned
> this in their first 12 years of schooling, where formulas may indeed be
> presented as nothing more than mindless recipes for getting some quantity
> not really well understood. How can we break our students of that bad
> habit? I do frequently verbalize the 'formula' after writing it on the
> board -- for variance, saying something like "look at this, we just take the
> sum of the squared deviations of scores from their mean, which measures how
> much the scores differ from one another, and then divide that sum by N, to
> get a measure of how much scores differ from one another, on average." The
> shorthand is really convenient, I don't know how I would get along without
> it.
> I have always thought that success in stats courses was much more a
> function of a student's verbal aptitude and ability to think analytically,
> rather than mathematical aptitude. Has anybody actually tested this
> hypothesis?
I think the problem here is that, at least in the US K-12 system,
"mathematical aptitude" really means "computational proficiency." In
"back to basics" math (and the overwhelming majority of K-12 math
curricula *are* "back to basics"; it's only in a few elite schools with
high-performing students that any of those educational innovations that
right-wingers claim are destroying the minds of our students have actually
been used, but I digress; see Alfie Kohn's writings), mathematical
notation is taught as a series of Taylorized (as in Frederick Winslow) job
instructions rather than as a language for precisely describing certain
types of relationships. IOW, it's taught as a bunch of hoops to be jumped
through without really understanding what you're doing; all that matters
is going through the prescribed motions. A student who is "good at math"
is one who can jump through those hoops quickly. The object of most K-12
math is to turn out human calculators (something that actually made
economic sense in the Old Days when a machine to do calculations cost a
lot more than an employee doing calculations longhand).
As John Allen Paulos pointed out in _Innumeracy_, the majority of American
K-12 students know *how* to add, subtract, multiply, and divide, but a
much smaller number of them know *when* to add, subtract, multiply, and
divide. They do just fine when given a spelled-out list of calculations
to perform, but they panic when confronted with word problems.
Over in comp.infosystems.www.authoring.html, there's a long-running and
acrimonious debate between those who see HTML markup as a *description* of
the structure of a document and those who see it as a bunch of commands
interspersed with text. The notion that formal notation can be a
description or explanation rather than a sequence of tasks to perform is
alien to a lot of people. It may very well be that some people simply
don't have the built-in cognitive ability to grasp the former, but I
suspect that a much larger number of people *could* develop that ability
if only they were taught how to think that way (and it is indeed a way of
thinking rather than a set of "skills"). Right now I suspect that most
who developed that ability picked it up through intellectual exploration
outside the formal system of schooling. I'm pretty sure I did (I was a
math major but am not involved in teaching).
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