🚀 go-pugleaf

     With regards to the recent exchange of ideas concerning math,trig, and
physics, I would like to share a few ideas.
     The first is that a similar line of notes was posted here some months
ago. At that time I posted a spreadsheet program I had used rather
successfully for the first time this school year. (it is in Physhare's files
as TRIGEX TEXT)
     Finding that math skills among my students are in certain, if uneven,
decline, I had to give an exceptional amount of extra help this year. I can't
even consider "higher math"- if I did so, my physics enrollment would drop
drastically.  For instance, I don't even try vector problems other than those
at right angles anymore- the degree of difficulty in getting those across is
almost too much for many of them.  So I wrote TRIGEX TEXT and gave all
students access to it on our network. It allowed them to tutor themselves in
adding vectors, which we all found to be most helpful. I INSISTED on mastery,
something which gives them trouble- it's rarely demanded of them. Most
students succeeded, and even the weakest learned it reasonably well.
     Furthermore, anyone wishing to teach addition of component vectors (I
call it R&R- resolution and resultant) who has a spreadsheet available should
use it for a simple program like the following. This program resolves the A
and B vectors into their X- and Y- components, adds them together with the
Pythag theorem, and gives the angle. (The angle is a bit of a problem, since
some spreadsheets have problems with angles beyond 90 degrees.  I have an
algorithm for IBM Lotus and Works; if anyone is interested just contact me
for further details.)

          A vector            B vector
          A*cos(theta)   +    B*cos(theta)   =    X-comp
          B*sin(theta)   +    B*sin(theta)   =    Y-comp

          R = SQRT(A^2 + B^2);  Alpha = arctan(X/Y)

         NOTE: A^2 is A the the second power.

     Another point concerning teaching of vectors concerns addition by the
graphical method.  I agree with Dave Popp that all methods should be used, or
at least presented to students.  However, the graphical method is not
necessarily an "easy" alternative.  I always insist that students solve at
least a few problems graphically, and the situation is similar to
mathematics.  That is, some students get it easily and others seem almost
unable to draw accurately.  This assumes that you specify a certain
tolerance, such as within a degree, mm, etc.

     All of this is OK in the "traditional" classroom such as mine and, I
presume, most of yours.  But the traditional classroom is, to a large extent,
the problem itself.  After all these years I finally asked my students why
they couldn't use the trig they had learned in trig class in physics.
Subsequent discussions with them and the trig teacher's similar inquiries led
to the discovery that they simply had no idea that what they had learned in
trig class- even fundamentals such as sin, cos, and tan- and what they had
learned in physics class were related.  Change texts (physics to math),
change classes, and suddenly you are in another compartment (math) which has
nothing to do with the compartment you just left (physics).
     Is this the students' fault?  School for the serious student consists of
accumulating points in order to get the highest grade.  Understanding is
secondary to this goal, but of course they realize that they should
understand this stuff because they will somehow need it someday.  That often
fails to motivate them to work toward the understanding.  I hope you catch my
drift- they are just responding to our expectations.
     Our entire paradigm is the problem.  Although mired in the traditional
system, I advocate approaches like Sizer's Essential School, an outcomes-
based approach to exploring coordinated, interrelated, essential concepts
throughout the curriculum.  The basic problem is not whether we should teach
higher math, whatever that is, or even what method we should use to teach
vectors.  What we should do is ask what outcomes we desire, then use all
resources in a coordinated fashion in order to accomplish it.  This requires
a method fundamentally different than the traditional, stand-up delivery
method most of us use all too often.
     I have tweaked my traditional program as much as I can, but it is like
trying to make a jet out of a prop airplane.  That prop job flew fine in its
day, but it is of little use in the jet age.  Meaningful change will occur
when and if the system I work within begins to support a model such as the
essential schools concept.  Unfortunately I don't foresee that happening soon
in spite of the attempts of a few of us at my school
(Brentwood HS near Pittsburgh).  Well, keep the faith and have a good summer.
<<Joe>>