> Please reply to the list since this is a major issue at our school and in
> Maryland in general. I would particularly be interested in how
> intermediate algebra is described and why?
>
>
> On Wed, 4 Feb 1998, Nancy Sattler wrote:
>
> > hi everyone!
> > I am again trying to gather information. I am very interested in how you,
> > as mathematics educators define developmental mathematics at the college
> > level. What level is it? arithmetic? elementary algebra? intermediate
> > algebra? other?
> >
> > On the flip side, how would you define a high level of mathematics at the
> > high school level? two years of math, three years of math, four years of
> > math, other?
> >
> > at the college level? intermediate algebra? college algebra? precalculus?
> > calculus? other?
> >
You've got a sticky issue here, because the answer cannot cite only _content_,
but must get into issues of what we here at NTID call _intent_. The difference
between high-school and college-level algebra may not be so much the particular
symbolic operations that students are encouraged to master as a more mature
appreciation of what is really going on behind all of the symbols. Some idea
of what this is _for_, where it fits in one's understanding of human culture, a
beginning of a sense of being in _control_ of at least part of the agenda, all
these are really part of what a college student should be acquiring as part of
the experience of study of mathematics. Students who are still in somewhat
shaky control of _process_ may yet possess mature understandings that guide
them through problem-solving situations; these understandings are to be valued
as gold!
"Developmental mathematics" is a term that, to my thinking, should be reserved
for curriculum intended to develop conceptual understandings that were supposed
to have been inculcated by earlier schooling, typically at an early age, but
which were missed (for one reason or another) by the student. An excellent
case in point is the use of fractions. Those who see fractions only as
peculiar-looking symbols that need to be handled by certain rules have missed
an important developmental stage. Curriculum devoted to bringing students to
see the _meaning_ of a fraction, to foster the sense of _proportionality_ that
is behind the idea of "reducing" a fraction, that helps students penetrate to
the underlying ideas of magnitude and order that lie behind fractional
notation, is certainly developmental curriculum. But this can also be rich,
provcative, challenging mathematical instruction! There are many deep, subtle
aspects to fractions that have resonated down through the history of
mathematics (the concept of the existence of irrational numbers is just one
such).
A dilemma we face (which I am sure is echoed in many departments responsible
for providing introductory-college-level mathematics to students) is the
student who has (more or less) absorbed standard manipulative processes in
algebra (e.g. solution of simple linear equations) but who has been
_conceptually_ starved and does not really have the basis for moving on to
higher-level work. Such students have "seen it before" and don't need another
explanation of process. But they "hate fractions". Er. You can't send them
back to an arithmetic class, so what do you do? One weapon to wield in this
situation is the electronic calculator.
Far from "weakening" student skills, appropriate instructional use of
calculators can open up new windows, whole new worlds, of understanding.
Calculators can empower students. Learning to make judgements about proper
methods and tools to use to tackle presented problems, and being given the
_responsibility_ for making appropriate judgements of this sort, and following
through with clear carefully-documented work, is definitely appropriate content
at the introductory college level, and cuts across all levels of developmental
understanding.
I said a paragraph back that you can't send students back to an arithmetic
class, but that is not strictly true. I like to tell my students that the very
first course I took in grad school (as a math major at a well-known university)
was a course in arithmetic! It's true -- but of course the _intent_ was far
different than it was when I was first challenged to pick up the rudiments of
arithmetic fifteen years earlier. What is true is that you can't send students
back to a course that _looks like_, or _is called_, or _amounts to_ the sort of
arithmetic course that they should have succeeded in some time in their past.
You can certainly _embed_ attention to developmental understandings in courses
that are ostensibly (and honestly) focused on acquisition of new levels of
skill in abstract symbolic calculation, and "bring the students along" as you
move into the new territory.
The use of calculators is important here too. Today semi-symbolic work can be
accomplished with graphing calculators. Tomorrow (no not in the next
millenium, but _tomorrow_) true symbolic manipulation will be available for
pennies, as universal as electronic computation of numerical results is today.
In the real world workers run backhoes, and you can't get a job wielding a
shovel any longer. Very few of us develop the muscle and endurance to do this
kind of physical labor any more, anyway. So what? The same is true with
computation -- we've got to teach our students to run those symbolic backhoes
if they are to get jobs, and not just get them to build up their muscles with
pointless exercise. There's still a lot to learn about soil conditions, and
buried cables, and all that, fortunately...
Of course I am speaking from my soapbox in the last paragraph. But this is the
direction I see necessary for us to head in, and the sort of curriculum I fully
expect to see in place in the schools in 20 or 30 years if I am lucky enough to
still be around then. How are we going to get there? A lot of people ae going
to have to go on asking about and discussing (and disagreeing on) what
mathematics ought to be at the high school and college levels. So I am
delighted to see this discussion going!
RWW Taylor
National Technical Institute for the Deaf
Rochester Institute of Technology
Rochester NY 14623
>>>> The plural of mongoose begins with p. <<<<
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