Re: contribution of mathematics in understanding physics

[Lynn Aldrich <laldrich@miseri.edu>]

Rick,
> Are there certain clientele who really can't handle and don't need the
> mathematical treatments that many courses are wont to present?
 
My comments will be directed towards those who teach future elementary
teachers
(K-6 in Pennsylvania).  
 
> Unfortunately I didn't copy the reference, but the following is from an AIP
> or APS survey of physics trained people working outside of academia (maybe
> someone else has the reference).  It asked about the most important skills
> needed for their work.

These skills I interpreted to refer to physics majors, few of whom are
teaching elementary school, which means these skills may not be(and I would
argue are not) necessarily the skills needed by these teachers.

> The one group for which I can certainly justify the need for strong
> conceptual understanding would be future teachers, but it is this group,
> specifically elementary-ed majors, where we find the most resistance to the
> more mathematical treatments.  It is certainly true at my school (and I
> suspect many others) that the level of mathematical sophistication needed to
> be certified in elementary-ed, and therefore to teach science in elementary
> schools, often falls below basic algebra.  That is, a non-zero number of
> successful elementary ed students COULD NOT pass a typical algebra level,
> problem solving physics course.   I question whether that is the level of
> expertise we want teaching our children--at any level?
> Rick Tarara

I agree there is strong resistance to mathematical treatments among this
group and quite often major problems with mathematics and algebra.
However, I have found that a hands-on, inquiry-oriented approach allows me
to "sneak" in the math, even algebra, in a way that aids the conceptual
understanding.  A typical algebra-based physics course would lost most of
these students at my college, and I suspect at many others.

I think we ought to design our course goals based on these students' needs
as future teachers, not what WE think they ought to know.  There are now 2
sets of proposed national standards in the United States to guide us, in
addition to standards of individual states.

Benchmarks for Science Literacy standards were approved by AAAS as part of
Project 2061 in 1993. The National Science Education Standards were
approved by the National Research Council(NRC) and published by the
National Academy Press in 1995.

For example, from the NRC:

Physical Science K-4 Content Standard B

As a result of the activities in grades K-4, all students should develop an
understanding of::
1) Properties of objects and materials
2) Position and motion of objects
3) Light, heat, electricity, and magnetism

Fundamental concepts and principles that underlie this standard include:
1) Objects have many observable properties, including size, weight, shape,
color, temperature, and the ability to react with other substances.  Those
properties can be measured using tools, such as rulers, balances, and
thermometers.
    Objects are made of one or more materials, such as paper, wood, and
metal.  Objects can be described by the properties of the materials from
which they are made, and those properties can be used to separate or sort a
group of objects or materials.
    Materials can exist in different states-solid, liquid, and gas.  Some
common materials, such as water, can be changed from one state to another
by heating or cooling.
2) The position of an object can be described by locating it relative to
another object or the background.  
    An object's motion can be described by tracing and measuring its
position over time.
    The position and motion of objects can be changed by pushing or
pulling.  The size of the change is related to the strength of the push or
pull.
    Sound is produced by vibrating objects.  The pitch of the sound can be
varied by changing the rate of vibration.
3) Light travels in a straight line until it strikes an object.  Light can
be reflected by a mirror, refracted by a lens, or absorbed by the object.
     Heat can be produced in many ways, such as burning, rubbing, or mixing
one substance with another.  Heat can move from one object to another by
conduction.
    Electricity in circuits can produce light, heat, sound, and magnetic
effects.
    Electrical circuits require a complete loop through which an electrical
current can pass.
    Magnets attract and repel each other and certain kinds of other materials.


The discussion of the role of mathematics in understanding physics for
future elementary teachers should be based on what is needed to understand
the physics inherent in the above standards for students at the grade
levels of K-4.  I would argue that the mathematics used in an algebra based
physics course at the college level is NOT the mathematics needed to be
able to be a teacher in K-4 with the above student goals.

The next grade level of standards from NRC is 5-8, which poses problems in
determining what the needs of elementary teachers are (since these grades
are now considered to be middle school) in Pennsylvania since we have only
elementary (K-6) and secondary (7-12) certification.  More mathematics may
well be needed at the 5-6 and particularly the 7-8 grade levels.

Lynn Aldrich
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