Re: [Phys-l] "Looking up" results on a graph

["John Clement" <clement@hal-pc.org>]

This is a difficult thing, and students often do not know which variable is
independent and which is dependent.  I always tell students that this is a
good rule to follow if you do not have a good reason for putting specific
variables on a particular axis.

But of course when analyzing specific data, there could be other reasons.
So conventionally t is treated as independent in most graphs.  You usually
plot weight vs. t for looking at babies growth, money vs. t for the stock
market, and position vs. t when looking at motion.  The latter graph must be
done that way if you define velocity as the slope or first derivative, and
acceleration as the second derivative.  But often motion is measured by
noting the time at specific markers, so the time is actually the dependent
variable.  Modeling gets around this by having students measure the position
at regular time intervals, so the independent variable is time.  Math
teachers will tell you that time is ALWAYS the independent variable, but
what about the period of a pendulum?

The conventions in the order of independent, dependent variables are
actually inconsistent.  This confuses the heck out of students.  So in an
ordered pair the independent is first the same as in the columns of a data
table.  But in the command "plot x vs. t", the independent variable is
second and in a sense it is second on a graph because the dependent variable
appears to the left of the independent variable.  Nobody ever bothers to
point out these things to students, and to tell them that it is a
convention.  Table columns out of order are not anything to get bent out of
shape, but ordered pairs are usually rigidly ordered.

Basically students have to learn to identify the independent variables and
this is difficult for even seniors in HS.  So Modeling requires this
identification in the lab reports, and does not give them a cookbook recipe
beforehand.  But the labs are arranged at the beginning to have the
dependent variable the one you would "look up" on the Y axis.  Later on the
convention may be broken.

So students should first be able to identify the independent variable(s) and
then practice putting it on the horizontal axis.  But eventually they need
to encounter situations where you violate this rule because of the method of
analysis.  Once they know how to identify relevant variables, they can learn
when to violate the conventional rule.  The big problem is that students
often have difficulty with understanding variables, and this is never really
addressed in most science books.  It is assumed that they know what they are
because they have had math.  Thinking Science by Shayer&Adey starts out with
understanding variables, and then works towards more formal understandings.
It is quite clear from what I have observed that many HS seniors still do
not understand variables, so for them no rules make sense.

It really does matter which way it is plotted, because people are used to
seeing certain presentations, and reverse presentations often confuse them.
It also matters when you have definitions such as V=dx/dt.  Plotting t vs x
will give you the inverse slope.  So the method of analysis is probably
ultimately the important rule.

John M. Clement
Houston, TX 


> > > Students (especially at high school or lower college 
> levels) often have
> > > a hard time deciding which variable to plot along the 
> bottom of a graph.
> > > A very simple memory aid suddenly occurred to me today -- 
> "looking up"
> > > the answer on the graph.
>
> > There is a fundamental reason why it is hard for students 
> > (or anybody else) to decide which variable to plot in which
> > direction.  The reason is:  In most cases, it doesn't matter!  
>
> True.  However, I think it's useful for students to learn the 
> convention
> of plotting the independent variable on the horizontal axis and the
> dependent on the vertical axis.  The science doesn't care how we
> represent it, but following the convention makes it easier to
> communicate our findings with each other.
>
> Part of the reason I don't personally need a mnemonic for 
> which axis is
> which is because pretty much every plot I've ever worked with over
> several decades depicts y as a function of x and not the other way
> around.  I feel that one of my responsibilities is to turn 
> out students
> who not only understand the physics, but who can make themselves
> understood easily, and part of that involves being able to understand
> and follow the conventions.
>
> Conventions may be broken, but it is generally unwise to do so
> capriciously or out of ignorance.