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Topic: Teaching Mathematics as a Science
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Subject:   RE: Teaching Mathematics as a Science
Author: tpowers
Date: Aug 10 2004
Thanks for responding, Bethany.  I guess I must have misunderstood you because
you're telling me I've been sticking you with positions you don't support.

> 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.

Maybe I didn't make myself clear.  As you point out later in your reply, I
support formal mathematical proof as a method of discovery and learning.  Also,
I can't contradict the fact that some students are more interested in
mathematics when it is presented in a fun way.  When applied correctly, I
believe that the discovery method of pedagogy can be very effective.  I just
don't believe it should be used all of the time for everybody.  For example,
among my peers, I know many students who love math but find the process of
informal discovery tiresome.  Also, the math-hating students I know feel
oppressed when they can't just memorize something and get away with it.
Clearly, turning people off mathematics is not something we would want to

> 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 article…)

I completely agree with you here, I just don't know if "discovering" complex
mathematical results is the best way to teach this (are there studies which show

> 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.

My fault -- I didn't make myself clear here.  I meant to say that formal proof
should be introduced only to those who like discovery in mathematics.

> 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.

Again, I don't know too much about studies in this area.  I was merely going on
the assumption that since rote learning had been widely accepted for a long
time, it was tolerable to students.  But I could very well be wrong here.

> 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.

My mistake for misunderstanding you.

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

I never considered mathematics as an outgrowth of the study of nature.

> 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.

My mistake again for misunderstanding.

> 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
> impossible.

I guess this idea is somewhat heretical, but I believe that learning to write
math belongs in English class.  I wouldn't do away with it.

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