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Re: [Phys-l] explanatory and response variables (was calibration )



We are having another of those contrapuntal discussions, with
some messages focusing on the best way to teach something at
the elementary-school level, and other messages focusing on
the most exact and sophisticated formulation.

That's all OK by me. We just need to keep track of which is
which.

On 08/08/2007 05:19 PM, Michael Edmiston wrote:

When determining the location of a point in relation to the axis of a
coordinate system, students appear to have been taught they should construct
a line through the point and perpendicular to the axis of interest. That is
not correct. Rather, they should construct a line through the point and
parallel to the other axis.

Yes yes yes yes yessssss.

It is interesting that if you look up abscissa and ordinate in Wictionary,
it is stated correctly... "the abscissa of the point is the distance cut off
from the axis of X by a line drawn through it and parallel to the axis of
Y." It has to be viewed that way if there is any possibility you might
sometime use non-orthogonal axes.

Hmmmm, yes, that is more sophisticated than the usual formulation,
but IMHO it is not 100.0000% correct. The problem is that it
assumes the X-axis is a contour of constant Y=0. There are plenty
of practical situations where it is not.

For that matter, there are plenty of practical situations where
the whole idea of "axes" is dead on arrival. Thermodynamics is
a familiar example.

It turns out that the usual educational trajectory is non-monotonic,
in the following sense:

1) In third grade, students make graphs using _graph paper_.
The graph paper is ruled horizontally and vertically. The
vertical lines can be used as _contours of constant x_ while
the horizontal lines can be used as _contours of constant y_.

You can find the x-value of any point by seeing what contour
it sits on. _No axes are required._ All we need are contour
lines, with each line (or a reasonably dense subset of the lines)
appropriately labeled. The labels can appear almost anywhere
on the line; they do not need to be clustered along any sort
of axis.

2) In middle school, students are taught to draw axes on an otherwise-
blank piece of paper, and plot points relative to the axes. This may
be a step up in convenience, but it is a giant step backwards in
generality and reliability. It has all sorts of bugs, as various
people have pointed out.

Just to make clear the silliness here: The x-values depend on the
labels associated with the *tick marks* on the x-axis ... not on
the x-axis line itself.

3) In upper-division college courses, or grad-school courses, physics
majors learn to rely on contours again, without recourse to axes,
just like they did back in third grade!

Some pictures illustrating what I mean, and some additional discussion,
can be found at
http://www.av8n.com/physics/spacetime-momentum.htm




However, I have asked the math professors
here, and they admit they teach their math-education students the
perpendicular construction. They are considering switching, but they say
most math textbooks explain the perpendicular construction. They say it
isn't exactly wrong since they are clearly discussing a system with
orthogonal axes. It seems to me that if you are trying to teach the concept
of locating something with respect to two axes, and maybe even drawing a
grid, you ought to teach it in a way that allows non-orthogonal axes, and
that the "grid" might be parallelograms rather than rectangles.

Yes! Grids are good. Grids are the gold standard. This works for
third grade and for grad school. It "should" be good enough for
everything in between. When in doubt, draw the contours!

Note that the cells in the grid might not even be parallelograms;
they might have curvy sides.Example (from thermodynamics):
http://www.av8n.com/physics/thermo-forms.htm#fig-pardev