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Topic: Teaching Mathematics as a Science

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Subject:   RE: Teaching Mathematics as a Science
Author: Bethany
Date: Aug 10 2004
Please see thoughts to your response below:

On Aug  7 2004, tpowers wrote:
> From the student's perspective. . .
>Bethany, you said that
> mathematics education was becoming more and more of a "discovery"
> process.  That's true, but I don't think that this is uniformly
> exciting.  Fundamentally, we must accept that there are people who
> do not enjoy mathematics especially, and for whom mathematical
> "discovery" is worthless, considering that they will never need to
> use anything above a 5th-grade math education in life.  We must also
> accept that these people are the overwhelming majority of students,
> probably over 97%.  By stressing communication and discovery
> processes, math educators are trying to make students more like
> themselves: career mathematicians who are responsible for making
> discoveries and communicating them to students.  But, really, it's
> hard to make the case that every student wants to become a math
> professor.

When did I say that every student wants to become a math professor or engaged in
a career of mathematics?  From my experience of working with students, students
learn better when involved in the discovery of their own knowledge rather than
it spoon fed to them.  By “stressing communication and the discovery process” we
strive to make mathematics more exciting and engaging for students.  The only
way that makes them “more like us” is that they become more prone to enjoy math
rather than it being a dull subject to sit around and have to absorb.  Or course
this doesn’t work for all students but it does engage the students moreso than
only having them sit and work math problems in order to learn a concept.

>I would say that at least 85% of students will never
> need to use mathematics beyond simple calculation when a calculator
> isn't handy.  And by cutting off the much-derided "learning by
> rote," we're making mathematics that much less useful.

As our society becomes more technology driven, I disagree with this statement.
The ability to understand and analyze data and numbers is imperative.  A student
needs to understand the mathematics in order to tell if what is input into the
calculator or computer will produce the right answer.  That student needs to be
able to look at the graphs on USA Today and tell with a little thought they are
completely misleading and why.  That student needs to be wide eyed at reading a
newspaper article that argues a point based on the difference between “about a
third” and a “little more than 30%”.  (Let me know if you’d like to read this

>Of couse,
> it could be argued that learning by rote "stifles creativity," but
> that's not really even the case.  The discovery method just
> substitutes activities for proofs (in my very limited experience),
> but it could hardly be said that proofs are "not creative."  The
> only thing that stifles creativity is teaching a process without any
> justification and forcing students to do it over and over.
> Creativity needs motivation, and a contextual void is simply
> unreasonable in this situation.  Also, proofs demonstrate the
> fundamental truth of mathematics, whereas activities merely portray
> it as some sort of fuzzy science.

And now you seem to be completely reversing your stance.  First you say you
don’t think students need to be pushed towards career mathematics but now you
say students need to be exposed to formal mathematical proof??  If a student
only grasps the mathematics of basic number sense (integers, decimals,
fractions, percents, ratios) an educator has no business presenting a proof
demonstrating the fundamental theorem of arithmetic.  Discovery based activities
help lead students to an implicit understanding which makes the teaching of the
more formal explicit mathematics much more readily available to them, when
called for.

>Let me quote you: "The
> prevading philosophy of teaching and learning mathematics is pushing
> us as educators in this direction, to make math more of a discovery
> process than just giving students the concepts and trying to process
> them."  What I have been saying is that, for a majority of students,
> giving and practicing the concepts is perfectly acceptable, and
> "discovery" is just another temporal roadblock to success in
> mathematics.

What exactly is a “temporal roadblock?” Do you mean that it will take students
longer to understand the mathematics presented as a discovery process than they
would if you lecture to them?  It depends on your students and how they learn.

 And, yes, I well understand that you are saying the traditional method of
presenting a math concept and having students practice that concept by working
problems is perfectly acceptable to you. Many students cannot learn mathematics
in this manner.  Even many of the students who can learn mathematics in this
manner will understand it better by experimentation/visualization.  And often,
if an activity is well designed, the learning process actually becomes quicker
and better understood.

>You als said that this new direction of education
> is bringing mathematics closer to what it really is.  But what is
> math, really?  You define it as a "science of quantity," and I don't
> think there would be anyone to disagree with you.  However, were
> even the ancient practices of rote memorization not related to the
> "science of quantity"?

I also don’t recall deriding the necessity of rote and practice in my post as
you seem to want to make it as though I did.  There is absolutely a need for
straight memorization in mathematics.  All learning must be a blend of
memorization complemented with conceptualization.  Is believe the pendulum of
educational theory has swung too far in the other direction to the exclusion of
the importance of memorization.  The truth is it needs to be a blend of both
memorization and conceptualization.  One is no good without the other.

>I suppose as a humble peon-student that
> perhaps I am founding this argument overmuch on semantics.  I assume
> that you meant mathematics had a lot to do with modeling
> experimental results, and, in fact, I would wholeheartedly agree.
> However, that does not mean the classroom should be used as a place
> of scientific experimentation.  Mathematics is NOT "natural
> philosophy," i.e., things like physics or biology.  Much as a hammer
> is not part of a finished woodwork, applied mathematics is neither a
> subset nor a superset of any of these other disciplines.  If
> mathematics be a science, it is not practiced by practicing physics
> or chemistry, but by practicing the application of math to those
> sciences, or to any other problems.

Could you explain why you believe mathematics is NOT a part of natural

>What I am basically saying is> that we should not have students build and
launch rockets just to
> introduce the idea of a parabola.  A better thing to do would be to
> tell the students that rockets do these things, and have them come
> up with the theoretical justification based on the Law of
> Gravitation.  Applied math is math, and we should be practicing math
> in math classes, not physics, chemistry, biology, or English.

Again, your argument here seems to contradict itself.  Applied math is math yet
should not be taught that way in the classroom.  And again, I never claimed that
all mathematics taught in the classroom must be linked with real world
phenomena.  What I did say though is that the move towards presenting
mathematics in such a way that allows students to experiment with cause and
effect results of mathematical concepts is good and helps engage students in the
learning and comprehension of mathematics.

Though as a closing thought here, I also must say that to exclude English as
part of mathematics hampers conceptualization.  Language is the fundamental
piece to conceptualization of any subject.  Without language thinking is

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