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Re: Ability grouping (was:Re: Affective learning in mathematics; What is math?)
Posted:
Dec 13, 1996 10:16 AM
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On Thu, 12 Dec 1996, Paul MacFarlane wrote:
{snip}
>Once we all switch to multiage classrooms, then much of the hetero/homo
>debate will be moot.
>
[snip}
>
> -Paul
>
>
> --
>
> vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
> |Paul MacFarlane - Danville,California - pmacfar@value.net |
>
> ************************************************************************
>
>
Dear Paul,
A minor point -- perhaps you should have said, " Once we _return_ to ..."
The one-room school was a multi-age classroom. It was a setting, at its
best, that provided a superb educational foundation for students.
The combination of independence, responsibility, and mutual support that
this setting provides cannot easily be duplicated in a homogeneous-age
classroom.
Of course, at its worst, it was as bad as the worst of our present
situation. The key still remains, "How does the teacher interact with each
student and what sorts of interactions among students are initiated or
encouraged by the teacher?"
Any approach that gets the minds of the students into gear, a dynamic
lecture, a powerfully instructive TV video or computer game, a
mind-stretching book, an intense but mutually-supportive discussion, a
skills stretching project, or whatever, will serve the educational purpose.
The book I have written, that is now being reviewed for publication, discusses
all of these approaches and more, within the context of using the course
content as a vehicle to teach thinking skills. I show how these skills can
be applied in many different subject areas, and, therefore, refer to them as
being _content independent._
For instance, there is a strategy we can call "substitution." We may
already be familiar with this in algebra, where we replace a literal
value in an equation with its numerical value to check the accuracy of our
solution.
But substitution has many other applications, about which I am
"sanguine." If we look this word up in the dictionary, its main meaning
is "bloody." But, from the alchemistic viewpoint of "humors" it can also
mean "healthy, sturdy or cheerful," that is, "of good humor." In reading
this sentence, we can substitute this latter concept in the place of the
less familiar word, "sanguine," and apprehend the intended meaning.
Of course, good readers, tend to substitute their own apprehension of
meaning to unfamiliar words, usually without recourse to a dictionary.
This is a teachable skill. Poor readers tend to make inappropriate
substitutions and lose the meaning and the flow of ideas. The algebraic
analogy is using an incorrect substitution procedures or having an incorrect
root. Once again, there are teaching procedures that will help readers make
better substitutions.
In cooking we can replace butter in a recipe with olive oil or margarine.
In banking we can replace checks with electronic funds transfers. In
music we can change the key of a song to fit the voice of the singer. My
point is that this list is endless. Substitution is a thinking strategy
that is _content independent._ The necessary and sufficient condition is
that the substitution be "functionally equivalent" to whatever it is
replacing.
In order to use substitution, we need to know how to determine
"functional equivalency." This is how we teach this procedure as a
thinking strategy. We need to explore both examples and non-examples of
appropriate replacements in order to develop the needed recognition skills.
Here is an interesting non-example. When we prepare a test, we try to
produce a representative sample of the course content. We then assume
that the total-correct score on the test is proportional to the amount of
"knowledge" each student possesses. However, I tested this hypothesis by
interviewing a group of students. In the interview I scored their
responses to the test questions "understands / does not understand." I
then correlated the interview scores with the "right-answer" scores and
found them unrelated to each other.
It is obvious, of course, that students can substitute memorization for
understanding. We will not be able to tell the difference from the fact
that they get answers "right." Thus the substitution of test scores (and
grade-point averages) for a statment of how much each student has
learned, is invalid. These performance ratings tell us how much our
students remember, not how much they understand.
The criticisms we keep hearing about our present educational system can be
summarized quite simply as having been produced by the outcomes of the
invalidity of this current method of measurement.
Those who doubt this claim can try this simple experiment for themselves.
It is an observation that is obvious, once it is recognized. It is an
observation that is easy to miss, in the process of trying to keep up with
the marking, reporting and awarding of placements, scholarships, degrees,
certificates, and the like.
We cannot blame ourselves for not knowing something. Once the truth has
been discovered, however, it behooves us to change our ways as soon as we
find it out.
Regards,
Jay Powell
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